Combinatorial constructions in Smooth Ergodic Theory · 2014-11-17 · Smooth Ergodic Theory. Apart...

Combinatorial constructions in Smooth ErgodicTheory

Dissertationzur Erlangung des Doktorgrades

der Fakultät für Mathematik, Informatikund Naturwissenschaftender Universität Hamburg

vorgelegtim Fachbereich Mathematik

Philipp Kunde

aus Hamburg

Hamburg, 2014

Als Dissertation angenommen vom FachbereichMathematik der Universität Hamburg

Auf Grund der Gutachten von

Prof. Dr. Reiner LauterbachProf. Dr. Roland GuneschProf. Dr. Ian Melbourne

Hamburg, den 22. September 2014

Prof. Dr. Michael HinzeLeiter des Fachbereichs Mathematik

Acknowledgement

I would like to thank my supervisors Prof. Dr. R. Lauterbach and Prof. Dr. R. Gunesch fortheir support during the past years. I am greatly indebted to Prof. Dr. R. Gunesch for fruitfuldiscussions and drawing my attention to the interesting field of combinatorial constructions inSmooth Ergodic Theory. Apart from many helpful discussions I wish to thank Prof. Dr. R.Lauterbach for his great encouragement and the possibility to graduate at the University ofHamburg.Many thanks to my colleagues and fellow PhD students for the years of cooperative studies aswell as the pleasant atmosphere.Last but not least, I want to express my deepest gratitude to my family and friends for theirsupport and sympathy.

Contents

1 Introduction 9

2 Preliminaries 132.1 Short introduction to Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . . . 132.2 C∞-topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Analytic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Reduction to the case S1 × [0, 1]m−1 . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Weak mixing diffeomorphisms preserving a measurable Riemannian metricwith arbitrary Liouvillean rotation number 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 First steps of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Explicit constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.1 Sequences of partial partitions . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 The conjugation map gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.3 The conjugation map φn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 (γ, δ, ε)-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Criterion for weak mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 Convergence of (fn)n∈N in Diff∞ (M) . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6.1 Properties of the conjugation maps φn and Hn . . . . . . . . . . . . . . . 423.6.2 Proof of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.7 Construction of the measurable f -invariant Riemannian metric . . . . . . . . . . 52

4 Weak mixing and uniquely ergodic diffeomorphisms preserving a measurableRiemannian metric with arbitrary Liouvillean rotation number on Tm 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 Definition of strict ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Reduction to Proposition 4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . 564.2.3 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Criterion for unique ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 Explicit constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.1 The trapping map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.2 The trapping regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.3 Sequences of partial partitions . . . . . . . . . . . . . . . . . . . . . . . . 624.4.4 The conjugation map gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 CONTENTS

4.4.5 The conjugation map φn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.5 (γ, ε)-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.6 Criterion for weak mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.7 Convergence of (fn)n∈N in Diff∞ (Tm) . . . . . . . . . . . . . . . . . . . . . . . . 71

4.8 Proof of strict ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.8.1 Trapping property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.8.2 Application of the criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.9 Construction of the f -invariant measurable Riemannian metric . . . . . . . . . . 77

5 Minimal but not uniquely ergodic diffeomorphisms preserving a measurableRiemannian metric with arbitrary Liouvillean rotation number on Tm 795.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 Reduction to Proposition 5.2.1 . . . . . . . . . . . . . . . . . . . . . . . . 805.2.2 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Explicit constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3.1 The trapping map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3.2 Trapping regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.3 Sequences of partial partitions . . . . . . . . . . . . . . . . . . . . . . . . 855.3.4 The conjugation map gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3.5 The conjugation map φn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4 Convergence of (fn)n∈N in Diff∞ (Tm) . . . . . . . . . . . . . . . . . . . . . . . . 895.4.1 Properties of the conjugation maps φn and Hn . . . . . . . . . . . . . . . 895.4.2 Proof of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.5 Proof of minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.5.1 Criterion for minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.5.2 Application of the criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.6 The ergodic invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.6.1 Exactly d ergodic invariant measures . . . . . . . . . . . . . . . . . . . . . 935.6.2 Weak mixing property w.r.t. ξt . . . . . . . . . . . . . . . . . . . . . . . . 98

5.7 Construction of the f -invariant measurable Riemannian metric . . . . . . . . . . 1035.8 The case of countable many ergodic invariant measures . . . . . . . . . . . . . . . 103

5.8.1 Modifications of the explicit constructions . . . . . . . . . . . . . . . . . . 1035.8.2 Norm estimates and convergence of (fn)n∈N . . . . . . . . . . . . . . . . . 1145.8.3 Proof of minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.8.4 The ergodic invariant measures . . . . . . . . . . . . . . . . . . . . . . . . 116

6 Natural measures of diffeomorphisms with arbitrary Liouvillean rotation num-ber 1216.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.2.1 Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2.2 First steps of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.2.3 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.3 Explicit constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.3.1 The trapping map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.3.2 Trapping regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

CONTENTS 7

6.3.3 Sequences of partial partitions . . . . . . . . . . . . . . . . . . . . . . . . 1286.3.4 The conjugation map gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.3.5 The conjugation map φn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.4 Convergence of (fn)n∈N in Diff∞(S1 × [0, 1] , µ

). . . . . . . . . . . . . . . . . . . 132

6.5 The invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.5.1 Trapping property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.5.2 Weak mixing with respect to Lebesgue measure on S1 × [0, 1] . . . . . . . 141

6.6 Construction of the f -invariant measurable Riemannian metric . . . . . . . . . . 144

7 Smooth diffeomorphisms with homogeneous spectrum and disjointness of con-volutions 1457.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2.1 First steps of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.2.2 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.3 Periodic approximation in Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . 1477.4 Spectral theory of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.4.1 Spectral types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.4.2 Spectral multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.4.3 Disjointness of convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.5 Construction of the conjugation maps . . . . . . . . . . . . . . . . . . . . . . . . 1517.5.1 The map φ(i)

λ,ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.5.2 The map ψk,q,~a,ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527.5.3 The conjugation map φn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.6 Convergence of (fn)n∈N in Diff∞ (M) . . . . . . . . . . . . . . . . . . . . . . . . . 1537.6.1 Properties of the conjugation maps φn . . . . . . . . . . . . . . . . . . . . 1537.6.2 Proof of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.7 Proof of (h, h+ 1)-property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.7.1 Towers for approximation of type (h, h+ 1) . . . . . . . . . . . . . . . . . 1567.7.2 Speed of approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.8 Proof of good cyclic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.8.1 Tower for good cyclic approximation . . . . . . . . . . . . . . . . . . . . . 1637.8.2 Speed of approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.9 Reduction to Proposition 7.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Uniform rigidity sequences for weak mixing diffeomorphisms on T2 1698.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.2 Criterion for weak mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.2.1 (γ, δ, ε)-distribution of horizontal intervals . . . . . . . . . . . . . . . . . . 1718.2.2 Statement of the criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.3 Convergence of (fn)n∈N in Diff∞(T2)

. . . . . . . . . . . . . . . . . . . . . . . . 1728.3.1 Properties of the conjugation maps hn and Hn . . . . . . . . . . . . . . . 1728.3.2 Proof of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.4 Convergence of (fn)n∈N in Diffωρ(T2). . . . . . . . . . . . . . . . . . . . . . . . . 177

8.4.1 Properties of the conjugation maps hn and Hn . . . . . . . . . . . . . . . 1778.4.2 Proof of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.5 Proof of weak mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

8 CONTENTS

8.5.1 Choice of the mixing sequence (mn)n∈N . . . . . . . . . . . . . . . . . . . 1828.5.2 Standard partial decomposition ηn . . . . . . . . . . . . . . . . . . . . . . 1828.5.3 Application of the criterion for weak mixing . . . . . . . . . . . . . . . . . 185

8.6 Proof of the Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.6.1 Proof of Corollary C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.6.2 Proof of Corollary D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Bibliography 189

Chapter 1

Introduction

The first breakthrough in Smooth Ergodic Theory as the study of geometric and statistical prop-erties of measures invariant under smooth dynamical systems was the work of E. Hopf on theergodicity of the geodesic flow on negatively curved surfaces in the 1930’s. On the other handKolmogorov, Arnold and Moser developed their famous perturbative KAM-theory in the 1950’s,which produces obstructions to ergodicity in Hamiltonian systems and makes the existence ofergodic diffeomorphisms near an elliptic fixed point unexpected.Until 1970 it was an open question if there exists an ergodic area-preserving smooth diffeomor-phism on the disc D2. This problem was solved by the so-called “Conjugation by approximation”-method developed by D. Anosov and A. Katok in [AK70]. In fact, on every smooth compactconnected manifold M of dimension m ≥ 2 admitting a non-trivial circle action S = Stt∈S1

preserving a smooth volume ν this method enables the construction of smooth diffeomorphismswith specific ergodic properties (e.g. weak mixing ones in [AK70], section 5) or non-standardsmooth realizations of measure-preserving systems (e.g. [AK70], section 6, and [FSW07]).These diffeomorphisms are constructed as limits of conjugates fn = Hn Sαn+1 H−1

n , whereαn+1 = αn + 1

kn·ln·q2n∈ Q, Hn = Hn−1 hn and hn is a measure-preserving diffeomorphism

satisfying S 1qn hn = hn S 1

qn. In each step the conjugation map hn and the parameter kn

are chosen such that the diffeomorphism fn imitates the desired property with a certain preci-sion. Then the parameter ln is chosen large enough to guarantee closeness of fn to fn−1 in theC∞-topology and so the convergence of the sequence (fn)n∈N to a limit diffeomorphism is pro-vided. It is even possible to keep this limit diffeomorphism within any given C∞-neighbourhoodof the initial element Sα1 or, by applying a fixed diffeomorphism g first, of g Sα1 g−1. Sothe construction can be carried out in a neighbourhood of any diffeomorphism conjugate to anelement of the action. Thus, A (M) = h St h−1 : t ∈ S1, h ∈ Diff∞ (M,ν)

is a natu-ral space for the produced diffeomorphisms. Moreover, we will consider the restricted spaceAα (M) = h Sα h−1 : h ∈ Diff∞ (M,ν)

for α ∈ S1.Another feature of the Anosov-Katok-method is the possibility to deduce statements on thegenericity of the constructed properties: For example, the set of volume-preserving diffeomor-phisms with arbitrarily fast cyclic approximation is a residual subset (i.e. it contains a denseGδ-set) in A (M) by [AK70], Theorem 7.1., as is the set of weak mixing diffeomorphisms.

In this thesis we apply the “Conjugation by approximation”-method in various fields of ErgodicTheory and Dynamical Systems: Initially, we develop achievements in Smooth Ergodic Theoryobtained by the Anosov-Katok-method further. Subsequently, we prove a statement concerning

10 Introduction

the spectral theory of smooth dynamical systems in chapter 7. In chapter 8 we explore uniformlyrigid and weak mixing maps, an up-to-date research topic in topological dynamics, in the smoothand beyond that even in the real-analytic category for the first time.At this point we shortly present the results. More detailed comments about their significanceand comprehensive introductions to each topic can be found at the beginning of the respectivechapters.

As mentioned above Anosov and Katok proved that the set of weak mixing diffeomorphismsis generic in A (M) in the C∞ (M)-topology. In extension of it R. Gunesch and A. Katokconstructed weak mixing diffeomorphisms preserving a measurable Riemannian metric on anysmooth compact and connected manifold of dimension at least 2 admitting a non-trivial circleaction preserving a smooth volume in [GK00]. Actually, it follows from the respective proofsthat both results are true in Aα (M) for a Gδ-set of α ∈ S1. However, both proofs do not givea full description of the set of α ∈ S1 for which the particular result holds in Aα (M). Such aninvestigation is started in [FS05]: B. Fayad and M. Saprykina showed that if α ∈ S1 is Liouville,the set of weak mixing diffeomorphisms is generic in the C∞ (M)-topology in Aα (M) in case ofdimension 2. Generalizing the results of [GK00] as well as [FS05] we prove in chapter 3

Theorem A. Let M be a smooth compact and connected manifold of dimension m ≥ 2 witha non-trivial circle action S = Stt∈S1 preserving a smooth volume ν. If α ∈ S1 is Liouville,the set of volume-preserving diffeomorphisms, that are weak mixing and preserve a measurableRiemannian metric, is dense in the C∞-topology in Aα (M).

As a consequence we deduce the statement of [FS05] on manifolds of dimension at least 2:

Corollary A. Let M be a smooth compact and connected manifold of dimension m ≥ 2 with anon-trivial circle action S = Stt∈S1 preserving a smooth volume ν. If α ∈ S1 is Liouville, theset of volume-preserving weak mixing diffeomorphisms is a dense Gδ-set in the C∞-topology inAα (M).

By introducing so-called “trapping maps” and “trapping regions” we modify these constructionsin chapter 4 in order to conclude

Theorem B. Let m ≥ 2. If α ∈ R is Liouville, the set of volume-preserving diffeomorphisms,that are weak mixing and strictly ergodic, is a dense Gδ-set in the C∞-topology in Aα (Tm).

This result is related to [FH77], where Fathi and Herman proved that on any compact andconnected smooth boundaryless m-dimensional manifold M (m ≥ 2) admitting a locally freecircle action the set of strictly ergodic C∞-diffeomorphisms is a dense Gδ-set in A (M) in theC∞ (M)-topology. Another construction of uniquely ergodic diffeomorphisms in Aα (Tm) is ex-hibited in [FSW07]: For every Liouvillean number α ∈ S1 Fayad, Saprykina and A. Windsorconstructed uniquely ergodic smooth diffeomorphisms that are measure-theoretically isomorphicto the rigid rotation Rα on S1. In particular, their diffeomorphisms are not weak mixing.

Concerning this number of ergodic invariant measures for any d ∈ N A. Windsor constructedminimal diffeomorphisms with exactly d ergodic invariant measures in A (M) on any compactand connected smooth boundaryless manifold of dimension at least 2 admitting a free C∞-actionpreserving a smooth volume. In the special case of the torus Tm we can extend his achievementsby constructing such diffeomorphisms in Aα (Tm) preserving a measurable Riemannian metric:

Theorem C. Let m ≥ 2, d ∈ N and α ∈ S1 be Liouville. Then there is a smooth diffeomorphismf ∈ Aα (Tm) preserving a measurable Riemannian metric which is minimal but has exactly d

Introduction 11

ergodic invariant measures. These measures are absolutely continuous with respect to Lebesguemeasure.Furthermore, the set of diffeomorphisms with these properties is dense in the C∞-topology inAα (Tm).

Moreover, we prove that the obtained diffeomorphisms are weak mixing with respect to the dergodic invariant measures. In [Win01] this is mentioned as a possible improvement of the resultbut was not executed. Another indicated variant of the constructions is exhibited in our settingin chapter 5 as well:

Theorem D. Let m ≥ 2 and α ∈ S1 be Liouville. Then there is a smooth diffeomorphismf ∈ Aα (Tm) preserving a measurable Riemannian metric which is minimal but has a countablenumber of ergodic invariant absolutely continuous measures.Furthermore, the set of diffeomorphisms with these properties is dense in the C∞-topology inAα (Tm).

Special cases of manifolds with boundary are treated in chapter 6. As argued there any area- andorientation-preserving diffeomorphism of the disc D2 and the annulus S1× [0, 1] respectively hasat least three ergodic invariant measures, namely the Lebesgue measure µ on the manifold, theDirac-δ-measures at the fixed points of the rotations and the Lebesgue measures on the bound-ary components. These measures are called the natural measures. In [FK04], §3, Fayad andKatok constructed diffeomorphisms with this minimal number of ergodic invariant measures. Infact, they proved that the set of such diffeomorphisms is a residual subset in the closure A′

in the C∞-topology of the conjugates of rotations with conjugacies fixing every point of theboundary and the fixed points of the action by rotations (the boundary points and the fixedpoints of the action are called singularities). We will extend the result of [FK04] by construct-ing diffeomorphisms with the minimal number of ergodic invariant measures in the restrictedspace A′α (M) := H Rα H−1 : H ∈ Diff∞ (M,µ) , H = id on the singularities

for everyLiouvillean number α ∈ S1. In addition our constructed diffeomorphisms are weak mixing withrespect to the area and preserve a measurable Riemannian metric. Finally, we will conclude

Theorem E. Let M be the disc D2 or the annulus S1× [0, 1] and R = Rtt∈S1 be the respectivestandard action by rotations. Then for every Liouvillean number α ∈ S1 the set of smoothdiffeomorphisms f ∈ A′α (M) that have exactly three ergodic invariant measures, namely thenatural measures on M , and are weak mixing with respect to the Lebesgue measure on M andpreserve a measurable Riemannian metric is a dense subset of A′α (M) in the C∞-topology.

as well as

Corollary B. The set of smooth diffeomorphisms f ∈ A′α (M) that have exactly three ergodicinvariant measures, namely the natural measures on M , and are weak mixing with respect to theLebesgue measure on M is a residual set in the C∞-topology in A′α (M) for every Liouvilleannumber α ∈ S1.

In chapter 7 we turn our attention to the spectral theory of dynamical systems. Therefore,we give a short overview of concepts like spectral types and spectral multiplicities of measure-preserving transformations. Moreover, we present the method of periodic approximation inErgodic Theory in general and compile results of it with implications on the spectral propertiesof the approximated transformations. With these we answer a question of Fayad and Katok (see[FK04], Problem 7.11.) about the existence of smooth diffeomorphisms admitting a special typeof periodic approximation and possessing specific spectral properties affirmatively:

12 Introduction

Theorem F. Let M be a smooth compact connected manifold of dimension m ≥ 2 admitting asmooth non-trivial circle action S = Stt∈S1 preserving a smooth volume ν and α a Liouvilleannumber. Then the set of smooth diffeomorphisms, that have a maximal spectral type disjointwith its convolutions, a homogeneous spectrum of multiplicity 2 for f × f and admit a goodapproximation of type (h, h+ 1), is residual in Aα (M) in the Diff∞ (M)-topology.

Finally, we study the notion uniform rigidity in chapter 8. It was introduced in [GM89] asthe topological analogue of rigidity in Ergodic Theory: A measure-preserving homeomorphismT : X → X on a Lebesgue probability space (X,B, µ) with (X, d) being a compact metricspace is called uniformly rigid if there exists an increasing sequence (mn)n∈N of natural num-bers such that du (Tmn , id) → 0 as n → ∞. Here du (S, T ) = d0 (S, T ) + d0

(S−1, T−1

d0 (S, T ) := supx∈X d (S (x) , T (x)) is the uniform metric on the group of measure-preservinghomeomorphisms on X. It is an up-to-date question which sequences can occur as uniform rigid-ity sequences for an ergodic transformation. K. Yancey considered this question in the setting ofhomeomorphisms on T2 (see [Ya]) and given a sufficient growth rate of the sequence she provedthe existence of a weak mixing homeomorphism of T2 that is uniformly rigid with respect to thissequence. In this thesis we start to examine this question in the smooth category:

Theorem G. Let ϕ1 (n) := 2n ·(n+ 1)! ·((n+ 2)!)(n+2)n−2·(n+1) ·(2πn)(n+2)·(n+1)n+1

. If (qn)n∈Nis a sequence of natural numbers satisfying

qn+1 ≥ ϕ1 (n) · q2·((n+2)·(n+1)n+1+1)n ,

there exists a weak mixing C∞-diffeomorphism of T2 that is uniformly rigid with respect to(qn)n∈N.

We want to point out that our condition on the growth rate is less restrictive than Yancey’s.Beyond that we deal with this problem in the real-analytic topology:

Theorem H. Let ρ > 0. If (qn)n∈N is a sequence of natural numbers satisfying q1 ≥ ρ+ 1 and

qn+1 ≥ 2n · 64π2 · n2 · q14n · exp

(4π · n · q6

n · exp(2π · q4

n · (1 + n · qn))),

there exists a weak mixing Diffωρ -diffeomorphism of T2 that is uniformly rigid along the sequence(qn)n∈N.

Chapter 2

Preliminaries

In this chapter we want to introduce some important concepts in Ergodic Theory as well asadvantageous definitions and notations. Moreover, we show how constructions on S1 × [0, 1]m−1

can be transfered to a general compact connected smooth manifold M with a non-trivial circleaction.

2.1 Short introduction to Ergodic Theory

Ergodic Theory has its origin in Boltzmann’s ergodic hypothesis in statistical physics in the late19th century stating that the time mean of a physical variable coincides with the space mean.This motivated Birkhoff and von Neumann to define ergodicity and to prove their foundationalergodic theorems in the 1930’s.

Proposition 2.1.1 (Birkhoff Ergodic Theorem). Let (X,B, µ) be a probability space, T : X → Xbe a measure-preserving transformation and f ∈ L1 (X,µ). Then

1. For µ-almost every x ∈ X

f(x) := limn→∞

n−1∑k=0

f(T kx

)exists.

2. f (Tx) = f (x) almost everywhere.

3. f ∈ L1 (X,µ) and∥∥f∥∥

L1(X,µ)≤ ‖f‖L1(X,µ).

4. If A ∈ B with T−1A = A, then∫Af dµ =

∫Af dµ.

∑n−1k=0 f T k

n→∞−→ f in L1 (X,µ).

Proof. See [Pet83], Theorem 2.2.3.

In number 4 of the precedent proposition we consider invariant sets. Since T−1 (A) = Aimplies T−1 (X \A) = X \A, we can study T in two separate parts, namely T |A and T |X\A. Incase of 0 < µ (A) < 1 this splitting into two parts simplifies the study of T . This motivates aconcept of irreducibility of measure-preserving transformations:

14 Short introduction to Ergodic Theory

Definition 2.1.2. Let (X,B, µ) be a probability space and T : X → X be a measure-preservingtransformation. The transformation T is called ergodic if every measurable invariant set hasmeasure 0 or 1.

There are several equivalent characterisations of ergodicity:

Lemma 2.1.3. Let (X,B, µ) be a probability space and T : X → X be a measure-preservingtransformation. Then the following properties are equivalent:

1. T is ergodic.

2. Every invariant function on X (i.e. f T = f a.e.) is constant a.e..

3. 1 is a simple eigenvalue of the Koopman-operator UT : L2 (X,µ) → L2 (X,µ), f 7→ f Tinduced by T .

4. For every f, g ∈ L2 (X,µ) we have:

limn→∞

n−1∑k=0

⟨UkT f, g

⟩= 〈f, 1〉 · 〈1, g〉 .

Here 〈·, ·〉 denotes the standard scalar product on L2 (X,µ).

5. For each f ∈ L1 (X,µ) the time mean of f equals the space mean of f almost everywhere,i.e.

limn→∞

n−1∑k=0

f(T kx

)=∫X

f dµ a.e..

Proof. See [Pet83], Proposition 2.4.1, Theorem 2.4.2, Theorem 2.4.4 and Proposition 2.4.5.

Considering the eigenfunctions of the induced operator UT on L2 (X,µ) a transformation issaid to have discrete spectrum if the eigenfunctions span L2 (X,µ). At the other extreme liesthe important concept of weak mixing:

Definition 2.1.4. Let (X,B, µ) be a probability space and T : X → X be a measure-preservingtransformation. The transformation T is called weak mixing if T has continuous spectrum, i.e.1 is the only eigenvalue of the associated operator UT : L2 (X,µ) → L2 (X,µ) and the onlyeigenfunctions are the constants.

Remark 2.1.5. Using the characterisation of ergodicity given in Lemma 2.1.3, 2., we see thata weak mixing transformation is ergodic.

Again there are several equivalent characterisations of weak mixing:

Lemma 2.1.6. Let (X,B, µ) be a probability space and T : X → X be a measure-preservingtransformation. Then the following properties are equivalent:

1. T is weak mixing.

2. For all f, g ∈ L2 (X,µ) there exists an increasing sequence (mn)n∈N of natural numberswith density 1 such that

limn→∞

〈UmnT f, g〉 = 〈f, 1〉 · 〈1, g〉 .

C∞-topology 15

3. For every pair of measurable sets A,B ∈ B there exists an increasing sequence (mn)n∈N ofnatural numbers with density 1 such that

limn→∞

µ(A ∩ T−mn (B)

)= µ (A) · µ (B) .

4. There exists an increasing sequence (mn)n∈N of natural numbers such that for any pair ofmeasurable sets A,B ∈ B:

limn→∞

µ(A ∩ T−mn (B)

)= µ (A) · µ (B) .

Proof. The equivalence of 1. and 2. follows from [Pet83], Theorem 2.6.1, with the aid of [Pet83],Lemma 2.6.2. In [Pet83], Theorem 2.6.1, the equivalence of 1. and 3. is proven as well. Theproof that 1. and 4. are equivalent can be found in [Skl67].

Remark 2.1.7. By [Pet83], Proposition 2.5.4, ergodicity as well as weak mixing are isomorphisminvariants. In this connection two measure-preserving dynamical systems (X1,B1, µ1, T1) and(X2,B2, µ2, T2) are metrically isomorphic if there exist B1 ∈ B1, B2 ∈ B2 such that T1B1 ⊆ B1,T2B2 ⊆ B2, µ1 (B1) = 1, µ2 (B2) = 1 and there exists an automorphism φ : B1 → B2 satisfyingφ T1 = T2 φ.

As another important tool in measurable dynamics we introduce the notion of a partialpartition of a measure space (X,µ), which is a pairwise disjoint countable collection of measurablesubsets of X.

Definition 2.1.8. • A sequence of partial partitions νn converges to the decomposition intopoints if and only if for a given measurable set A and for every n ∈ N there exists ameasurable set An, which is a union of elements of νn, such that limn→∞ µ (A4An) = 0.We often denote this by νn → ε.

• A partial partition ν is a refinement of a partial partition η if and only if for every C ∈ νthere exists a set D ∈ η such that C ⊆ D. We write this as η ≤ ν.

Using the notion of a partition we can introduce the weak topology in the space of measure-preserving transformations on a Lebesgue space:

Definition 2.1.9. 1. For two measure-preserving transformations T, S and for a finite parti-tion ξ the weak distance with respect to ξ is defined by d (ξ, T, S) :=

∑c∈ξ µ (T (c)4S (c)).

2. The base of neighbourhoods of T in the weak topology consists of the sets

W (T, ξ, ε) = S : d (ξ, T, S) < ε ,

where ξ is a finite partition and ε is a positive number.

2.2 C∞-topology

In this section we consider smooth diffeomorphisms on smooth compact connected manifolds. Incase of a manifold with boundary by a smooth diffeomorphism we mean infinitely differentiablein the interior and such that all the derivatives can be extended to the boundary continuously.In the general case let M and N be Cr-manifolds of dimension m and n respectively. In thissetting Cr (M,N) denotes the set of Cr-maps from M to N . At first we consider the situation

16 C∞-topology

1 ≤ r <∞. Then Crw (M,N) denotes the weak topolgy on Cr (M,N) generated by the followingsets: Let f ∈ Cr (M,N), (ϕ,U) be a chart on M and (ψ, V ) be a chart on N . Moreover, letK ⊂ U be a compact set with f (K) ⊂ V and 0 < ε ≤ ∞. Then we define the neighbourhoodN r (f ; (ϕ,U) , (ψ, V ) ,K, ε) consisting of these Cr-maps g : M → N satisfying g (K) ⊂ V as wellas

maxi=1,...,n

∣∣D~a ([ψ f ϕ−1]i

)(x)−D~a

([ψ g ϕ−1

)(x)∣∣ < ε

for every x ∈ ϕ (K) and every multiindex ~a with |~a| ≤ r. Here[ψ f ϕ−1

]iis the i-th coordinate

function of ψ f ϕ−1 and we use the subsequent notation:

Definition 2.2.1. For a sufficiently differentiable function f : Rm → R and a multiindex ~a =(a1, ..., am) ∈ Nm0

D~af :=∂|~a|

∂xa11 ...∂xamm

where |~a| =∑mi=1 ai is the order of ~a.

The collection U of these neighbourhoods N r (f ; (ϕ,U) , (ψ, V ) ,K, ε) is a subbase of the weaktopology on Cr (M,N). Accordingly a neighbourhood in the weak topolgy of a function f is aset containing a finite intersection of sets of the form N r (f ; (ϕ,U) , (ψ, V ) ,K, ε). Referring to[Hir76], chapter 2.1., there exists a complete metric on Crw (M,N).Next we consider the space Diffr (M) ⊂ Cr (M,M) of Cr-diffeomorphisms on a compact m-dimensional manifold M . The complete metric on Cr (M,M) is denoted by ρr. With it wedefine the metric

ρr(f, g) := maxρr (f, g) , ρr

(f−1, g−1

)on Diffr (M,N) measuring the distance between the diffeomorphisms as well as their inverses.Then the space Diff∞ (M) of smooth diffeomorphisms on M with the metric

ρ∞ (f, g) =∞∑r=1

ρr (f, g)2r · (1 + ρr (f, g))

is a complete metric space (see [AK70], chapter 7).By Diff∞ (M,ν) we denote the space of ν-preserving diffeomorphisms on M equipped with theC∞-topology induced by the metric ρ∞.

More detailled we discuss topologies on the space of smooth diffeomorphisms on the manifoldS1 × [0, 1]m−1. Note that for diffeomorphisms f = (f1, ..., fm) : S1 × [0, 1]m−1 → S1 × [0, 1]m−1

the coordinate function f1 understood as a map R× [0, 1]m−1 → R has to satisfy the conditionf1 (θ + n, r1, ..., rm−1) = f1 (θ, r1, ..., rm−1)+l for n ∈ Z, where either l = n or l = −n. Moreover,for i ∈ 2, ...,m the coordinate function fi has to be Z-periodic in the first component, i.e.fi (θ + n, r1, ..., rm−1) = fi (θ, r1, ..., rm−1) for every n ∈ Z.For defining explicit metrics on Diffk

(S1 × [0, 1]m−1

)we introduce:

Definition 2.2.2. For a continuous function F : (0, 1)m → R

‖F‖0 := supz∈(0,1)m

|F (z)| .

Diffeomorphisms on S1× [0, 1]m−1 can be regarded as maps from [0, 1]m to Rm. In this spiritthe expressions ‖fi‖0 as well as ‖D~afi‖0 for any multiindex ~a with |~a| ≤ k have to be understood

C∞-topology 17

for f = (f1, ..., fm) ∈ Diffk(S1 × [0, 1]m−1

). Since such a diffeomorphism is a continuous map on

the compact manifold and every partial derivative can be extended continuously to the boundary,all of these expressions are finite. Thus, the subsequent definition makes sense:

Definition 2.2.3. 1. For f, g ∈ Diffk(S1 × [0, 1]m−1

)with coordinate functions fi resp. gi

we defined0 (f, g) = max

i=1,..,m

infp∈Z‖(f − g)i + p‖

as well as

dk (f, g) = maxd0 (f, g) , ‖D~a (f − g)i‖0 : i = 1, ...,m , 1 ≤ |~a| ≤ k

2. Using the definitions from 1. we define for f, g ∈ Diffk(S1 × [0, 1]m−1

dk (f, g) = maxdk (f, g) , dk

(f−1, g−1

Obviously dk describes a metric on Diffk(S1 × [0, 1]m−1

)measuring the distance between

the diffeomorphisms as well as their inverses. As in the case of a general compact manifold thefollowing definition connects to it:

Definition 2.2.4. 1. A sequence of Diff∞(S1 × [0, 1]m−1

)-diffeomorphisms is called conver-

gent in Diff∞(S1 × [0, 1]m−1

)if it converges in Diffk

(S1 × [0, 1]m−1

)for every k ∈ N.

2. On Diff∞(S1 × [0, 1]m−1

)we declare the following metric

d∞ (f, g) =∞∑k=1

dk (f, g)2k · (1 + dk (f, g))

As above Diff∞(S1 × [0, 1]m−1

)is a complete metric space with respect to this metric d∞.

Again considering diffeomorphisms on S1 × [0, 1]m−1 as maps from [0, 1]m to Rm we add theadjacent notation:

Definition 2.2.5. Let f ∈ Diffk(S1 × [0, 1]m−1

)with coordinate functions fi be given. Then

‖Df‖0 := maxi,j∈1,...,m

‖Djfi‖0

|||f |||k := max‖D~afi‖0 ,

∥∥D~a (f−1i

)∥∥0

: i = 1, ...,m, ~a multiindex with 0 ≤ |~a| ≤ k.

Remark 2.2.6. By the above-mentioned observations for every multiindex ~a with |~a| ≥ 1 andevery i ∈ 1, ...,m the derivative D~ahi is Z-periodic in the first variable. Since in case of adiffeomorphism g = (g1, ..., gm) on S1× [0, 1]m−1 regarded as a map [0, 1]m → Rm the coordinatefunctions gj for j ∈ 2, ...,m satisfy gj ([0, 1]m) ⊆ [0, 1], it holds:

supz∈(0,1)m

|(D~ahi) (g (z))| ≤ |||h||||~a|.

Analogously we can define the same expressions in the case of the torus Tm.

18 Reduction to the case S1 × [0, 1]m−1

2.3 Analytic topology

Real-analytic diffeomorphisms of T2 homotopic to the identity have a lift of type

F (θ, r) = (θ + f1 (θ, r) , r + f2 (θ, r)) ,

where the functions fi : R2 → R are real-analytic and Z2-periodic for i = 1, 2. For these functionswe introduce the subsequent definition:

Definition 2.3.1. For any ρ > 0 we consider the set of real-analytic Z2-periodic functions on R2,that can be extended to a holomorphic function on Aρ :=

(θ, r) ∈ C2 : |imθ| < ρ, |imr| < ρ

1. For these functions let ‖f‖ρ := sup(θ,r)∈Aρ |f (θ, r)|.

2. The set of these functions satisfying the condition ‖f‖ρ <∞ is denoted by Cωρ(T2).

Furthermore, we consider the space Diffωρ(T2)of those diffeomorphisms homotopic to the

identity, for whose lift we have fi ∈ Cωρ(T2)for i = 1, 2.

Definition 2.3.2. For f, g ∈ Diffωρ(T2)we define

‖f‖ρ = maxi=1,2

‖fi‖ρ

and the distancedρ (f, g) = max

infp∈Z‖fi − gi − p‖ρ

Remark 2.3.3. Diffωρ(T2)is a Banach space (see [Sa03] or [Ly99] for a more extensive treatment

of these spaces).

Moreover, for a diffeomorphism T with lift T (θ, r) = (T1 (θ, r) , T2 (θ, r)) we define

‖DT‖ρ = max

∥∥∥∥∂T1

∥∥∥∥ρ

∥∥∥∥∂T1

∥∥∥∥ρ

∥∥∥∥∂T2

∥∥∥∥ρ

∥∥∥∥∂T2

∥∥∥∥ρ

and use the advantageous notation

‖T‖ρ = maxi=1,2

infk∈Z

sup(θ,r)∈Aρ

|Ti (θ, r) + k| .

2.4 Reduction to the case S1 × [0, 1]m−1

In chapters 3 and 7 the aimed diffeomorphisms are produced on S1 × [0, 1]m−1. These construc-tions can be transfered to a general compact connected smooth manifold M with a non-trivialcircle action S = Stt∈R, St+1 = St. By [AK70], Proposition 2.1., we can assume that 1 is thesmallest positive number satisfying St = id. Hence, we can assume S to be effective. We denotethe set of fixed points of S by F and for q ∈ N Fq is the set of fixed points of the map S 1

On the other hand, we consider S1 × [0, 1]m−1 with Lebesgue measure µ. Furthermore, letR = Rαα∈S1 be the standard action of S1 on S1 × [0, 1]m−1, where the map Rα is given byRα (θ, r1, ..., rm−1) = (θ + α, r1, ..., rm−1). Hereby, we can formulate the following result (see[FSW07], Proposition 1, as well as [Ka79], Proposition 1.2.):

Reduction to the case S1 × [0, 1]m−1 19

Proposition 2.4.1. Let M be a m-dimensional smooth, compact and connected manifold ad-mitting an effective circle action S = Stt∈R, St+1 = St, preserving a smooth volume ν. Let

B := ∂M ∪ F ∪(⋃

q≥1 Fq

). There exists a continuous surjective map G : S1 × [0, 1]m−1 → M

with the following properties:

1. The restriction of G to S1 × (0, 1)m−1 is a C∞-diffeomorphic embedding.

2. ν(G(∂(S1 × [0, 1]m−1

)))= 0

3. G(∂(S1 × [0, 1]m−1

))⊇ B

4. G∗ (µ) = ν

5. S G = G R

By the same reasoning as in [FSW07], section 2.2., this proposition allows us to carry aconstruction from

(S1 × [0, 1]m−1

,R, µ)to the general case (M,S, ν):

Suppose f : S1 × [0, 1]m−1 → S1 × [0, 1]m−1 is a diffeomorphism sufficiently close to Rα in theC∞-topology obtained by f = limn→∞ fn with fn = Hn Rαn+1 H−1

n , where fn = Rαn+1 in aneighbourhood of the boundary (in Proposition 3.2.1 and Proposition 7.2.1 respectively we willsee that these conditions can be satisfied in the constructions of this thesis). Then we define asequence of diffeomorphisms:

fn : M →M fn (x) =

G fn G−1 (x) if x ∈ G

(S1 × (0, 1)m−1

)Sαn+1 (x) if x ∈ G

(∂(S1 × (0, 1)m−1

))Constituted in [FK04], section 5.1. (which bases upon [Ka79], Proposition 1.1.), this sequence isconvergent in the C∞-topology to the diffeomorphism

f : M →M f (x) =

G f G−1 (x) if x ∈ G

(S1 × (0, 1)m−1

)Sα (x) if x ∈ G

(∂(S1 × (0, 1)m−1

))provided the closeness from f to Rα in the C∞-topology.We observe that f and f are metrically isomorphic.

Chapter 3

Weak mixing diffeomorphismspreserving a measurableRiemannian metric with arbitraryLiouvillean rotation number

3.1 Introduction

In the following let M be a smooth compact connected manifold of dimension m ≥ 2 admittinga non-trivial circle action S = Stt∈S1 preseving a smooth volume ν.Let A(M) be the closure in the C∞ (M)-topology of the set of smooth diffeomorphisms ofthe form h St h−1 with t ∈ S1 and h a measure-preserving smooth diffeomorphism, i.e.A (M) = h St h−1 : t ∈ S1, h ∈ Diff∞ (M,ν)

. For α ∈ S1 we introduce the restrictedspace Aα (M) = h Sα h−1 : h ∈ Diff∞ (M,ν)

.In their influential paper [AK70] Anosov and Katok proved amongst others that in A (M) the setof weak mixing diffeomorphisms is generic (i.e. it is a dense Gδ-set) in the C∞ (M)-topology. Forit they used the “approximation by conjugation”-method. In [GK00] the conjugation maps areconstructed more explicitly such that they can be equipped with the additional structure of beinglocally very close to an isometry. Hereby, it is shown that there exists a weak mixing smoothdiffeomorphism preserving a smooth measure and a measurable Riemannian metric. Actually, itfollows from the respective proofs that both results are true in Aα (M) for a Gδ-set of α ∈ S1.However, both proofs do not give a full description of the set of α ∈ S1 for which the particularresult holds in Aα (M).On the other hand Fayad and Saprykina showed that if α ∈ S1 is Liouville the set of weakmixing diffeomorphisms is generic in the C∞ (M)-topology in Aα (M) on every two-dimensionalmanifold M (see [FS05]). Here an irrational number α is called Liouville if and only if for everyC ∈ R>0 and for every n ∈ N there are infinitely many pairs of coprime integers p, q such that∣∣∣α− p

∣∣∣ < Cqn .

In this article we prove the following theorem generalizing the results of [GK00] as well as [FS05]:

Theorem A. Let M be a smooth compact and connected manifold of dimension m ≥ 2 witha non-trivial circle action S = Stt∈S1 preserving a smooth volume ν. If α ∈ S1 is Liouville,

22 Preliminaries

the set of volume-preserving diffeomorphisms, that are weak mixing and preserve a measurableRiemannian metric, is dense in the C∞-topology in Aα (M).

We want to point out that this result is in some sense the best we can obtain:

• By [FS05], corollary 1.4., whose proof uses Herman’s last geometric result ([FKr09]), wehave the following dichotomy in case ofM = S1×[0, 1]: A number α ∈ S1\Q is Diophantineif and only if there is no ergodic diffeomorphism of M whose rotation number (on at leastone of the boundaries) is equal to α. Since weak mixing diffeomorphisms are ergodic, therecannot be a weak mixing f ∈ Aα

(S1 × [0, 1]

)for α ∈ R \Q Diophantine.

• By a result of A. Furman (appendix to [GK00]) a weak mixing diffeomorphism cannotpreserve a Riemannian metric with L2-distortion (i.e. both the norm and its inverse areL2-functions). Moreover, it is conjectured that a weak mixing diffeomorphism cannotpreserve a Riemannian metric with L1-distortion (see [GK00], Conjecture 3.7.).

Using the standard techniques to prove genericity of the weak mixing-property and TheoremA we conclude in section 3.2.1

Corollary A. Let M be a smooth compact and connected manifold of dimension m ≥ 2 with anon-trivial circle action S = Stt∈S1 preserving a smooth volume ν. If α ∈ R is Liouville, theset of volume-preserving weak mixing diffeomorphisms is a dense Gδ-set in the C∞-topology inAα (M).

Hence, we obtain the result of [FS05] in arbitrary dimension at least 2.

3.2 Preliminaries

3.2.1 First steps of the proof

First of all we show how constructions on S1 × [0, 1]m−1 can be transfered to a general compactconnected manifold M with a non-trivial circle action: Let f be a produced weak mixing dif-feomorphism in Aα

(S1 × [0, 1]m−1

)preserving a measurable Riemannian metric ω and f as in

section 2.4. We observed that f and f are metrically isomorphic. Then f is weak mixing becausethe weak mixing-property is invariant under isomorphisms.Moreover, we can construct a f -invariant measurable Riemannian metric ω out of the f -invariantmetric ω: Since ω only needs to be a measurable metric and ν

(G(∂(S1 × [0, 1]m−1

)))= 0,

we only have to construct it on G(S1 × (0, 1)m−1

). Using the diffeomorphic embedding G

we consider ω|G(S1×(0,1)m−1) :=(G−1

)∗ω|G(S1×(0,1)m−1) and show that it is f -invariant: On

G(S1 × (0, 1)m−1

)we have f = G f G−1 and thus we can compute:

f∗ω =(G f G−1

)∗ ((G−1

)∗ω)

=(G−1

)∗f∗G∗(G−1)∗ω =

(G−1

)∗f∗ω =(G−1

)∗ω = ω

Altogether, the construction done in the case of(S1 × [0, 1]m−1

,R, µ)is transfered to (M,S, ν).

Hence, it suffices to consider constructions on M = S1 × [0, 1]m−1 with circle action R subse-quently. In this case we will prove the following result:

Preliminaries 23

Proposition 3.2.1. For every Liouvillean number α there is a sequence (αn)n∈N of rationalnumbers αn = pn

qnsatisfying limn→∞ |α− αn| = 0 monotonically and sequences (gn)n∈N, (φn)n∈N

of measure-preserving diffeomorphisms satisfying gn R 1qn

= R 1qn gn as well as φn R 1

R 1qn φn such that the diffeomorphisms fn = Hn Rαn+1 H−1

n with Hn = h1 h2 ... hn,where hn = gn φn, coincide with Rαn+1 in a neighbourhood of the boundary, converge in theDiff∞ (M)-topology and the diffeomorphism f = limn→∞ fn is weak mixing, has an invariantmeasurable Riemannian metric and satisfies f ∈ Aα (M).Furthermore, for every ε > 0 the parameters in the construction can be chosen in such a waythat d∞ (f,Rα) < ε.

By this Proposition weak mixing diffeomorphisms preserving a measurable Riemannian met-ric are dense in Aα (M):Because of Aα (M) = h Rα h−1 : h ∈ Diff∞ (M,µ)

it is enough to show that for everydiffeomorphism h ∈ Diff∞ (M,µ) and every ε > 0 there is a weak mixing diffeomorphism f pre-serving a measurable Riemannian metric such that d∞

(f , h Rα h−1

)< ε. For this purpose,

let h ∈ Diff∞ (M,µ) and ε > 0 be arbitrary. By [Om74], p. 3, resp. [KM97], Theorem 43.1.,Diff∞ (M) is a Lie group. In particular, the conjugating map g 7→ h g h−1 is continuouswith respect to the metric d∞. Continuity in the point Rα yields the existence of δ > 0, suchthat d∞ (g,Rα) < δ implies d∞

(h g h−1, h Rα h−1

)< ε. By Proposition 3.2.1 we can

find a weak mixing diffeomorphism f with f -invariant measurable Riemannian metric ω andd∞(f,Rα) < δ. Hence, f := h f h−1 satisfies d∞

(f , h Rα h−1

)< ε. Note that f is weak

mixing and preserves the measurable Riemannian metric ω :=(h−1

)∗ω.

Hence, Theorem A is deduced from Proposition 3.2.1.

Remark 3.2.2. Using the same technique as in [Ha56], section Category, and Proposition 3.2.1we can show that the set of weak mixing diffeomorphisms is generic in Aα (M) (i.e. it is a denseGδ-set). Thereby, we consider a countable dense set ϕnn∈N in L2 (M,µ), which is a separablespace, and define the sets:

O (i, j, k, n) =T ∈ Aα (M) : |〈UnTϕi, ϕj〉 − 〈ϕi, 1〉 · 〈1, ϕj〉| <

Since 〈UTϕ,ψ〉 depends continuously on T , each O (i, j, k, n) is open. Hence

K :=⋂i∈N

⋂j∈N

⋂k∈N

⋃n∈N

O (i, j, k, n)

is a Gδ-set.By one of the equivalent characterisations in Lemma 2.1.6 a measure-preserving transformationT is weak mixing if and only if for every ϕ,ψ ∈ L2 (M,µ) there is a sequence (mn)n∈N of densityone such that limn→∞ 〈UmnT ϕ,ψ〉 = 〈ϕ, 1〉 · 〈1, ψ〉. Thus, every weak mixing diffeomorphism iscontained in K. On the other hand, we show that a transformation, that is not weak mixing,does not belong to K: If T is not weak mixing, UT has a non-trivial eigenfunction. W.l.o.g. wecan assume the existence of f ∈ L2 (M,µ) and c ∈ C of absolute value 1 satisfying UT f = c · f ,‖f‖L2 = 1 and 〈1, f〉 = 0. Since ϕnn∈N is dense in L2 (M,µ), there is an index i such that‖f − ϕi‖L2 < 0.1. Obviously ‖ϕi‖L2 ≤ ‖f‖L2 +‖f − ϕi‖L2 < 1.1 and |〈UnT f, f〉 − 〈f, 1〉 · 〈1, f〉| =

24 Explicit constructions

|〈cn · f, f〉| = |cn| · ‖f‖2L2 = 1. Consequently we can estimate:

1 = |〈UnT f, f〉 − 〈f, 1〉 · 〈1, f〉|≤ |〈UnT f, f〉 − 〈UnT f, ϕi〉|+ |〈UnT f, ϕi〉 − 〈UnTϕi, ϕi〉|+ |〈UnTϕi, ϕi〉 − 〈ϕi, 1〉 · 〈1, ϕi〉|

+ |〈ϕi, 1〉 · 〈1, ϕi〉 − 〈ϕi, 1〉 · 〈1, f〉|+ |〈ϕi, 1〉 · 〈1, f〉 − 〈f, 1〉 · 〈1, f〉|≤ |c|n · ‖f‖L2 · ‖f − ϕi‖L2 + ‖f − ϕi‖L2 · ‖ϕi‖L2 + |〈UnTϕi, ϕi〉 − 〈ϕi, 1〉 · 〈1, ϕi〉|

+ ‖ϕi‖L2 · ‖f − ϕi‖L2

≤ 0.1 + 0.11 + |〈UnTϕi, ϕi〉 − 〈ϕi, 1〉 · 〈1, ϕi〉|+ 0.11< |〈UnTϕi, ϕi〉 − 〈ϕi, 1〉 · 〈1, ϕi〉|+ 0.5

Thus, |〈UnTϕi, ϕi〉 − 〈ϕi, 1〉 · 〈1, ϕi〉| has to be larger than 12 . Hence, T does not belong to

O (i, i, 2, n) for any value of n and accordingly does not belong to K. So K coincides withthe set of weak mixing diffeomorphisms in Aα (M). By the observations above we know thatthis set is dense. In conclusion the set of weak mixing diffeomorphisms is a dense Gδ-set inAα (M). Therefore, Corollary A is proven.

3.2.2 Outline of the proofFirst of all we will define two sequences of partial partitions, which converge to the decompositioninto points in each case. The first type of partial partition, called ηn, will satisfy the requirementsin the proof of the weak mixing-property. On the partition elements of the even more detailedsecond type, called ζn, the conjugation map hn will act as an isometry and this will enableus to construct an invariant measurable Riemannian metric. Afterwards, these conjugatingdiffeomorphisms hn = gn φn, which are composed of two step-by-step defined smooth measure-preserving diffeomorphisms, will be constructed. In this connection the map gn shall introduceshear in the θ-direction as in [FS05]. So g[nqσn] (θ, r1, ..., rm−1) = (θ + [n · qσn] · r1, r1, ..., rm−1) isa nearby candidate. Unfortunately, this map is not an isometry. Therefore, the map gn will beconstructed in such a way that gn is an isometry on the image under φn of any partition elementIn ∈ ζn and gn

)= g[nqσn]

)as well as gn

(Φn(In

))= g[nqσn]

(Φn(In

))for every In ∈ ηn,

where Φn = φn Rmnαn+1 φ−1

n with a specific sequence (mn)n∈N of natural numbers (see section3.4) is important in the proof of the weak mixing property. Likewise the conjugation map φnwill be built such that it acts on the elements of ζn as an isometry and on the elements of ηnin such a way that it satisfies the requirements of the aimed criterion for weak mixing. Thiscriterion is established in section 3.5. It is based on the notion of a (γ, δ, ε)-distribution, whichwill be introduced in section 3.4, and is similar to the criterion in [FS05], but modified in manyplaces because of the new conjugation map gn and the new type of partitions.In section 3.6 we will show convergence of the sequence (fn)n∈N in Aα (M) for a given Liouvillenumber α by the same approach as in [FS05]. Therefore, we have to estimate the norms |||Hn|||kvery carefully. Furthermore, we will see at the end of section 3.6 that the criterion for weakmixing applies to the obtained diffeomorphism f = limn→∞ fn. Finally, we will construct theaimed f -invariant measurable Riemannian metric in section 3.7.

3.3 Explicit constructions

3.3.1 Sequences of partial partitionsIn this subsection we define the two announced sequences of partial partitions (ηn)n∈N and(ζn)n∈N of M = S1 × [0, 1]m−1.

Explicit constructions 25

3.3.1.1 Partial partition ηn

Remark 3.3.1. For convenience we will use the notation∏mi=2 [ai, bi] for [a2, b2]× ...× [am, bm].

Initially, ηn will be constructed on the fundamental sector[0, 1

]×[0, 1]m−1. For this purpose,

we divide the fundamental sector in n sections:

• In case of k ∈ N and 2 ≤ k ≤ n − 1 on[k−1n·qn ,

kn·qn

]× [0, 1]m−1 the partial partition ηn

consists of all multidimensional intervals of the following form:

[k − 1n · qn

n · q2n

+ ...+j((m−1)· (k+1)·k

n · q1+(m−1)· (k+1)·k2

10 · n5 · q1+(m−1)· (k+1)·k2

k − 1n · qn

n · q2n

+ ...+j((m−1)· (k+1)·k

2 )1 + 1

n · q1+(m−1)· (k+1)·k2

10 · n5 · q1+(m−1)· (k+1)·k2

×m∏i=2

qn+ ...+

j(k+1)i

26 · n4 · qk+1n

qn+ ...+

j(k+1)i + 1qk+1n

− 126 · n4 · qk+1

where j(l)1 ∈ Z and

⌉≤ j(l)

1 ≤ qn −⌈qn

⌉− 1 for l = 1, ..., (m− 1) · (k+1)·k

2 as well asj

(l)i ∈ Z and

⌉≤ j(l)

i ≤ qn −⌈qn

⌉− 1 for i = 2, ...,m and l = 1, ..., k + 1.

• On[0, 1

]× [0, 1]m−1 as well as

[n−1n·qn ,

]× [0, 1]m−1 there are no elements of the partial

partition ηn.

As the image under Rl/qn with l ∈ Z this partial partition of[0, 1

]× [0, 1]m−1 is extended

to a partial partition of S1 × [0, 1]m−1.

Remark 3.3.2. By construction this sequence of partial partitions converges to the decomposi-tion into points.

3.3.1.2 Partial partition ζn

As in the previous case we will construct the partial partition ζn on the fundamental sector[0, 1

]× [0, 1]m−1 initially and therefore divide this sector into n sections: In case of k ∈ N and

1 ≤ k ≤ n on[k−1n·qn ,

kn·qn

]× [0, 1]m−1 the partial partition ζn consists of all multidimensional

intervals of the following form:

[k − 1n · qn

n · q2n

+ ...+j((m−1)· k·(k+1)

n · q1+(m−1)· k·(k+1)2

n5 · q1+(m−1)· k·(k+1)2

k − 1n · qn

n · q2n

+ ...+j((m−1)· k·(k+1)

2 )1 + 1

n · q1+(m−1)· k·(k+1)2

n5 · q1+(m−1)· k·(k+1)2

qn+ ...+

j((m−1)· k·(k+1)

2 +1)2

q1+(m−1)· k·(k+1)

+j((m−1)· k·(k+1)

2 +2)2

8n5 · q1+(m−1)· k·(k+1)2

n · [nqσn]+

8n9 · q1+(m−1)· k·(k+1)2

n · [nqσn],

qn+ ...+

j((m−1)· k·(k+1)

2 +1)2

q1+(m−1)· k·(k+1)

+j((m−1)· k·(k+1)

2 +2)2 + 1

8n5 · q1+(m−1)· k·(k+1)2

n · [nqσn]− 1

8n9 · q1+(m−1)· k·(k+1)2

n · [nqσn]

×m∏i=3

qn+ ...+

1n4 · qkn

qn+ ...+

j(k)i + 1qkn

− 1n4 · qkn

where j(l)i ∈ Z and

⌈qnn4

⌉≤ j

(l)i ≤ qn −

⌈qnn4

⌉− 1 for i = 3, ...,m and l = 1, .., k as well as

j(l)1 ∈ Z,

⌈qnn4

⌉≤ j

(l)1 ≤ qn −

⌈qnn4

⌉− 1 for l = 1, ..., (m − 1) · k·(k+1)

2 as well as j(l)2 ∈ Z and⌈

⌉≤ j

(l)2 ≤ qn −

⌈qnn4

⌉− 1 for l = 1, ..., (m − 1) · k·(k+1)

2 + 1 as well as j((m−1)· k·(k+1)

2 +2)2 ∈ Z

and 8 · n · [n · qσn] ≤ j((m−1)· k·(k+1)2 +2)

2 ≤ 8 · n5 · [n · qσn]− 8 · n · [n · qσn]− 1.

Remark 3.3.3. For every n ≥ 3 the partial partition ζn consists of disjoint sets, covers a set ofmeasure at least 1− 4·m

n2 and the sequence (ζn)n∈N converges to the decomposition into points.

3.3.2 The conjugation map gn

As mentioned in the sketch of the proof we aim for a smooth measure-preserving diffeomorphismgn, which satisfies gn

)= g[nqσn]

)as well as gn

(Φn(In

))= g[nqσn]

(Φn(In

))for every

In ∈ ηn and is an isometry on the image under φn of any partition element In ∈ ζn.Let a, b ∈ Z and ε ∈

]such that 1

ε ∈ Z. Moreover, we consider δ > 0 such that 1δ ∈ Z and

a·b·δε ∈ Z. We denote [0, 1]2 by ∆ and [ε, 1− ε]2 by ∆ (ε).

Lemma 3.3.4. For every ε ∈(0, 1

]there exists a smooth measure-preserving diffeomorphism

gε : [0, 1]2 → (x+ ε · y, y) : x, y ∈ [0, 1], that is the identity on ∆ (4ε) and coincides with themap (x, y) 7→ (x+ ε · y, y) on ∆ \∆ (ε).

Proof. First of all let ψε be a smooth diffeomorphism satisfying

ψε (x, y) =

(x, y) on R2 \∆ (2ε)(

12 + 1

5 ·(x− 1

2 + 15 ·(y − 1

))on ∆ (4ε)

Furthermore, let τε be a smooth diffeomorphism with the following properties

τε (x, y) =

(x+ ε · y, y) on(x− 1

)2 +(y − 1

)2 ≥ ( 516

)2(x, y) on

(x− 1

)2 +(y − 1

)2 ≤ 150

We define gε := ψ−1ε τε ψε. Then the diffeomorphism gε coincides with the identity on ∆ (4ε)

and with the map (x, y) 7→ (x+ ε · y, y) on R2 \∆ (ε). From this we conclude that det (Dgε) > 0.Moreover, gε is measure-preserving on Uε :=

(R2 \∆ (ε)

)∪∆ (4ε).

With the aid of “Moser’s trick” we want to construct a diffeomorphism gε, that is measure-preserving on the whole R2 and agrees with gε on Uε. Therefore, we consider the canonical volumeform Ω0 on R2: Ω0 = dx ∧ dy respectively Ω0 = dω0 using the 1-form ω0 = 1

2 · (x · dy − y · dx).Additionally we introduce the volume form Ω1 := g∗εΩ0.At first we note that gε preserves the 1-form ω0 on Uε: Clearly this holds on ∆ (4ε), where gε isthe identity. On R2 \∆ (ε) we have Dgε (x, y) = (x+ εy, y) and so we get

g∗εω0 = ω0 (x+ ε · y, y) =12· ((x+ ε · y) dy − y · d (x+ ε · y)) =

12· (x · dy − y · dx) = ω0 (x, y) .

Furthermore, we introduce Ω′ := Ω1 − Ω0. Since the exterior derivative commutes with thepull-back, it holds Ω′ = d (g∗εω0 − ω0). In addition we consider the volume form Ωt := Ω0 + t ·Ω′and note that Ωt is non-degenerate. Thus, we get a uniquely defined vectorfield Xt such thatΩt (Xt, ·) = (ω0 − g∗εω0) (·). Since ∆ is a compact manifold, the non-autonomous differentialequation d

dtu(t) = Xt (u(t)) with initial values in ∆ has a solution defined on R. Hence, we geta one-parameter family of diffeomorphisms νtt∈[0,1] on ∆ satisfying νt = Xt (νt), ν0 = id.Referring to [Ber98], Lemma 2.2., it holds

dtν∗t Ωt = d (ν∗t (i (Xt) Ωt)) + ν∗t

dtΩt + i (Xt) dΩt

Because of d (ν∗t (i (Xt) Ωt)) = ν∗t (d (i (Xt) Ωt)) and dΩt = d (dω0 + t · (d (g∗εω0)− dω0)) = 0 wecompute:

dtν∗t Ωt = ν∗t (d (i (Xt) Ωt)) + ν∗t

)= ν∗t d (Ωt (Xt, ·)) + ν∗t Ω′

= ν∗t d (ω0 − g∗εω0) + ν∗t Ω′ = ν∗t (Ω0 − Ω1) + ν∗t (Ω1 − Ω0) = 0.

Consequently ν∗1Ω1 = ν∗0Ω0 = Ω0 using ν0 = id in the last step. As we have seen g∗εω0 = ω0 onUε. Therefore, on Uε it holds: Ωt (Xt, ·) = 0. Since Ωt is non-degenerate, we conclude Xt = 0on Uε and hence ν1 = ν0 = id on Uε ∩∆. Now we can extend ν1 smoothly to R2 as the identity.Denote gε := gε ν1. Because we have ν1 = id on Uε, gε coincides with gε on Uε as announced.Furthermore, we have

g∗εΩ0 = (gε ν1)∗ Ω0 = ν∗1 (g∗εΩ0) = ν∗1Ω1 = Ω0.

Using the transformation formula we compute for an arbitrary measurable set A ⊆ R2:

µ (gε (A)) =∫gε(A)

Ω0 =∫A

|det (Dgε)| · Ω0.

We have det (Dν1) > 0 (because ν0 = id and all the maps νt are diffeomorphisms) as well asdet (Dgε) > 0 and thus |det (Dgε)| = det (Dgε). Since g∗εΩ0 = (det (Dgε)) · Ω0 (compare with[HK95], proposition 5.1.3.) we finally conclude:

µ (gε (A)) =∫A

(det (Dgε)) · Ω0 =∫A

g∗εΩ0 =∫A

Ω0 = µ (A) .

Consequently gε is a measure-preserving diffeomorphism on R2 satisfying the aimed properties.

Let gb : S1 × [0, 1]m−1 → S1 × [0, 1]m−1 be the smooth measure-preserving diffeomorphismgb (θ, r1, ..., rm−1) = (θ + b · r1, r1, ..., rm−1) and denote

]×[0, ε

b·a]×[δ, 1− δ]m−2 by ∆a,b,ε,δ.

Using the map Da,b,ε : Rm → Rm, (θ, r1, ..., rm−1) 7→(a · θ, b·aε · r1, r2, ..., rm−1

)and gε from

Lemma 3.3.4 we define the measure-preserving diffeomorphism ga,b,ε,δ : ∆a,b,ε,δ → gb (∆a,b,ε,δ)by ga,b,ε,δ = D−1

a,b,ε (gε, idRm−2) Da,b,ε. Using the fact that a·b·δε ∈ Z we extend it to a smooth

diffeomorphism ga,b,ε,δ :[0, 1

]× [δ, 1− δ]m−1 → gb

([0, 1

]× [δ, 1− δ]m−1

)by the description:

ga,b,ε,δ

(θ, r1 + l · ε

b · a, r2, ..., rm−1

)=(l · εa, l · ε

b · a,~0)

+ ga,b,ε,δ (θ, r1, ..., rm−1)

for r1 ∈[0, ε

b·a]and some l ∈ Z satisfying b·a·δ

ε ≤ l ≤ b·aε −

b·a·δε − 1. Since this map co-

incides with the map gb in a neighbourhood of the boundary, we can proceed it to a mapga,b,ε,δ :

]× [0, 1]m−1 → gb

([0, 1

]× [0, 1]m−1

)by putting it equal to gb.

We initially construct the smooth measure-preserving diffeomorphism gn on the fundamentalsector and for this divide it into n sections: On

[k−1n·qn ,

kn·qn

]× [0, 1]m−1 in case of k ∈ Z and

1 ≤ k ≤ n:gn = g

n·q1+(m−1)· (k+1)·k

2n ,[n·qσn], 1

8n4 ,1

Since gn coincides with the map g[n·qσn] in a neighbourhood of the boundary of the differentsections on the θ-axis, this yields a smooth map and we can extend it to a smooth measure-preserving diffeomorphism on S1× [0, 1]m−1 using the description gn R l

qn= R l

qn gn for l ∈ Z.

Furthermore, we note that the subsequent constructions are done in such a way that 260n4

divides qn (see Lemma 3.6.9) and so the assumption a·b·δε = a·b

4 ∈ Z is satisfied. Indeed, thismap gn satisfies the following aimed property:

Lemma 3.3.5. For every element In ∈ ηn we have: gn(In

)= g[nqσn]

Proof. We consider a partition element In,k ∈ ηn on[k−1n·qn ,

kn·qn

]× [0, 1]m−1 in case of k ∈ Z

and 2 ≤ k ≤ n− 1 and want to examine the effect of gn = gn·q

1+(m−1)· (k+1)·k2

n ,[n·qσn], 18n4 ,

on it.

In the r1-coordinate we use that there is u ∈ Z such that1

26n4qk+1n

= u · ε

b · a= u · 1

8n4 · [nqσn] · nq1+(m−1)· (k+1)·k2

where we exploit the fact that 260n4 divides qn by Lemma 3.6.9. Also with respect to theθ-coordinate the boundary of this element lies in the domain, where ga,b,ε,δ = g[nqσn], because

110·n4 < ε = 1

8·n4 .

3.3.3 The conjugation map φn

)and every i, j ∈ 1, ...,m there exists a smooth measure-

preserving diffeomorphism ϕε,i,j on Rm, which is the rotation in the xi − xj-plane by π/2 aboutthe point

(12 , ...,

)∈ Rm on [2ε, 1− 2ε]m and coincides with the identity outside of [ε, 1− ε]m.

Proof. W.l.o.g. we prove the statement in case of i < j. This time we denote [0, 1]m by ∆ and[ε, 1− ε]m by ∆ (ε). Let ψε be a smooth diffeomorphism satisfying

ψε (x1, .., xm) =

(x1, ..., xm) on Rm \∆ (ε)(

12 + 1

5·√m·(x1 − 1

), .., 1

2 + 15·√m·(xm − 1

))on ∆ (2ε)

τε (x, y) =

(x1, ..., xm) on∑m

(xi − 1

)2 ≥ 116

(x1, ..., xi−1, 1− xj , xi+1, ..., xj−1, xi, xj+1, ..., xm) on

∑mi=1

(xi − 1

)2 ≤ 150

We define ϕε := ψ−1

ε τε ψε. Then the diffeomorphism ϕε coincides with the identity onR2 \∆ (ε) and with the rotation in the xi − xj-plane on ∆ (2ε).Now we use “Moser’s trick” in the same way as in the proof of Lemma 3.3.4.

Furthermore, for λ ∈ N we define the maps Cλ (x1, x2, ..., xm) = (λ · x1, x2, ..., xm) andDλ (x1, ..., xm) = (λ · x1, λ · x2, ..., λ · xm). Let µ ∈ N, 1

δ ∈ N and 1δ divides µ. We construct a

diffeomorphism ψµ,δ,i,j,ε2 in the following way:

• Consider [0, 1− 2 · δ]m: Since 1δ divides µ, we can divide [0, 1− 2 · δ]m in cubes of side

length 1µ .

• Under the map Dµ any of these cubes of the form∏mi=1

[liµ ,

li+1µ

]with li ∈ N is mapped

onto∏mi=1 [li, li + 1].

• On [0, 1]m we will use the diffeomorphism ϕ−1ε2,i,j

constructed in Lemma 3.3.6 . Since thisis the identity outside of ∆ (ε2), we can extend it to a diffeomorphism ϕ−1

ε2,i,jon Rm using

the instruction ϕ−1ε2,i,j

(x1 + k1, x2 + k2, ..., xm + km) = (k1, ..., km) +ϕ−1ε2,i,j

(x1, x2, ..., xm),where ki ∈ Z and xi ∈ [0, 1].

• Now we define the smooth measure-preserving diffeomorphism

ψµ,δ,i,j,ε2 = D−1µ ϕ−1

ε2,i,jDµ : [0, 1− 2δ]m → [0, 1− 2δ]m

• Hereby, we define

ψµ,δ,i,j,ε2 (x1, ..., xm) =([ψµ,δ,i,j,ε2 (x1 − δ, ..., xm − δ)

+ δ, ...,[ψµ,δ,i,j,ε2 (x1 − δ, ..., xm − δ)

on [δ, 1− δ]m

(x1, ..., xm) else

This is a smooth map because ψµ,δ,i,j,ε2 is the identity in a neighbourhood of the boundaryby the construction.

Remark 3.3.7. For every set W =∏mi=1

[liµ + ri,

li+1µ − ri

], where li ∈ Z and ri ∈ R satisfies

|ri · µ| ≤ ε2, we have ψµ,δ,i,j,ε2 (W ) = W .

Using these maps we build the following smooth measure-preserving diffeomorphism

φλ,ε,i,j,µ,δ,ε2 :[0,

]× [0, 1]m−1 →

]× [0, 1]m−1

, φλ,ε,i,j,µ,δ,ε2 = C−1λ ψµ,δ,i,j,ε2 ϕε,i,j Cλ

Afterwards, φλ,ε,i,j,µ,δ,ε2 is extended to a diffeomorphism on S1 × [0, 1]m−1 by the descriptionφλ,ε,i,j,µ,δ,ε2

(x1 + 1

λ , x2, ..., xm)

1λ , 0, ..., 0

)+ φλ,ε,i,j,µ,δ,ε2 (x1, x2, ..., xm).

30 (γ, δ, ε)-distribution

For convenience we will use the notation φ(j)λ,µ = φλ, 1

60n4 ,1,j,µ,1

10n4 ,1

22n4. With this we define

the diffeomorphism φn on the fundamental sector: On[k−1n·qn ,

kn·qn

]× [0, 1]m−1 in case of k ∈ N

and 1 ≤ k ≤ n:

φn = φ(m)

n·q1+(m−1)· k·(k−1)

2 +(m−2)·kn ,qkn

φ(m−1)

n·q1+(m−1)· k·(k−1)

2 +(m−3)·kn ,qkn

... φ(2)

n·q1+(m−1)· k·(k−1)

2n ,qkn

This is a smooth map because φn coincides with the identity in a neighbourhood of the differentsections.Now we extend φn to a diffeomorphism on S1×[0, 1]m−1 using the description φnR 1

qn= R 1

qnφn.

3.4 (γ, δ, ε)-distribution

We introduce the central notion of the criterion for weak mixing deduced in the next section:

Definition 3.4.1. Let Φ : M → M be a diffeomorphism. We say Φ (γ, δ, ε)-distributes anelement I of a partial partition, if the following properties are satisfied:

• π~r(

Φ(I))

is a (m− 1)-dimensional interval J , i.e. J = I1 × ... × Im−1 with intervalsIk ⊆ [0, 1], and 1 − δ ≤ λ (Ik) ≤ 1 for k = 1, ...,m − 1. Here π~r denotes the projection onthe (r1, ..., rm−1)-coordinates.

• Φ(I)is contained in a set of the form [c, c+ γ]× J for some c ∈ S1.

• For every (m− 1)-dimensional interval J ⊆ J it holds:∣∣∣∣∣∣µ(I ∩ Φ−1

(S1 × J

))µ(I) −

µ(m−1)(J)

µ(m−1) (J)

∣∣∣∣∣∣ ≤ ε ·µ(m−1)

µ(m−1) (J),

where µ(m−1) is the Lebesgue measure on [0, 1]m−1.

Remark 3.4.2. Analogous to [FS05] we will call the third property “almost uniform distribution”of I in the r1, .., rm−1-coordinates. In the following we will often write it in the form of∣∣∣µ(I ∩ Φ−1

(S1 × J

))· µ(m−1) (J)− µ

(I)· µ(m−1)

(J)∣∣∣ ≤ ε · µ(I) · µ(m−1)

In the next step we define the sequence of natural numbers (mn)n∈N:

mn = minm ≤ qn+1 : m ∈ N, inf

∣∣∣∣m · pn+1

qn+1− 1n · qn

∣∣∣∣ ≤ 260 · (n+ 1)4

m ≤ qn+1 : m ∈ N, inf

∣∣∣∣m · qn · pn+1

qn+1− 1n

∣∣∣∣ ≤ 260 · (n+ 1)4 · qnqn+1

Lemma 3.4.3. The setm ≤ qn+1 : m ∈ N, infk∈Z

∣∣∣m · qn·pn+1qn+1

− 1n + k

∣∣∣ ≤ 260(n+1)4·qnqn+1

nonempty for every n ∈ N, i.e. mn exists.

(γ, δ, ε)-distribution 31

Proof. In Lemma 3.6.9 we will construct the sequence αn = pnqn

in such a way, that qn = 260n4·qnand pn = 260n4 ·pn with pn, qn relatively prime. Therefore, the set

j · qn·pn+1

qn+1: j = 1, ..., qn+1

contains qn+1

260(n+1)4·gcd(qn,qn+1) different equally distributed points on S1. Hence, there are at leastqn+1

260(n+1)4·qn different such points and so for every x ∈ S1 there is a j ∈ 1, ..., qn+1, such that

infk∈Z

∣∣∣x− j · qn·pn+1qn+1

+ k∣∣∣ ≤ 260(n+1)4·qn

qn+1. In particular, this is true for x = 1

Remark 3.4.4. We define

an =(mn ·

qn+1− 1n · qn

By the above construction of mn it holds that |an| ≤ 260·(n+1)4

qn+1. In Lemma 3.6.9 we will see that

it is possible to choose qn+1 ≥ 64 · 260 · (n+ 1)4 · n11 · q(m−1)·n2+3n . Thus, we get:

|an| ≤1

64 · n11 · q(m−1)·n2+3n

Our constructions are done in such a way that the following property is satisfied:

Lemma 3.4.5. The map Φn := φn Rmnαn+1φ−1

n with the conjugating maps φn defined in section

3.3.3(

1n·qmn

, 1n4 ,

)-distributes the elements of the partition ηn.

Proof. We consider a partition element In,k on[k−1n·qn ,

kn·qn

]× [0, 1]m−1. When applying the map

φ−1n we observe that this element is positioned in such a way that all the occuring maps ϕ−1

ε,1,j

and ϕε2,1,j act as the respective rotations. Then we compute φ−1n

[k − 1n · qn

n · q2n

+ ...+j((m−1)· (k−1)·k

n · q(m−1)· (k−1)·k2 +1

n · q(m−1)· (k−1)·k2 +2

+ ...+j

n · q(m−1)· (k−1)·k2 +k+1

n · q(m−1)· (k−1)·k2 +k+2

+ ...+j

n · q(m−1)· (k+1)·k2 +1

10 · n5 · q(m−1)· (k+1)·k2 +1

k − 1n · qn

n · q2n

+ ...+j

(k)m + 1

n · q(m−1)· (k+1)·k2 +1

10 · n5 · q(m−1)· (k+1)·k2 +1

×m∏i=2

[1− j

((m−1)· (k−1)·k2 +(i−2)·k+1)

qn− ...− j

((m−1)· (k−1)·k2 +(i−1)·k)

1 + 1qkn

(k+1)i

26 · n4 · qk+1n

1− j((m−1)· (k−1)·k

2 +(i−2)·k+1)1

qn− ...− j

((m−1)· (k−1)·k2 +(i−1)·k)

1 + 1qkn

(k+1)i + 1qk+1n

− 126n4 · qk+1

By our choice of the number mn the subsequent application of Rmnαn+1yields modulo 1

32 (γ, δ, ε)-distribution

n · qn+

n · q2n

+ ...+j((m−1)· (k−1)·k

n · q(m−1)· (k−1)·k2 +1

n · q(m−1)· (k−1)·k2 +2

+ ...+j

n · q(m−1)· (k−1)·k2 +k+1

n · q(m−1)· (k−1)·k2 +k+2

+ ...+j

n · q(m−1)· (k+1)·k2 +1

10 · n5 · q(m−1)· (k+1)·k2 +1

n · qn+

n · q2n

+ ...+j

(k)m + 1

n · q(m−1)· (k+1)·k2 +1

10 · n5 · q(m−1)· (k+1)·k2 +1

×m∏i=2

[1− j

((m−1)· (k−1)·k2 +(i−2)·k+1)

qn− ...− j

((m−1)· (k−1)·k2 +(i−1)·k)

1 + 1qkn

(k+1)i

26 · n4 · qk+1n

1− j((m−1)· (k−1)·k

2 +(i−2)·k+1)1

qn− ...− j

((m−1)· (k−1)·k2 +(i−1)·k)

1 + 1qkn

(k+1)i + 1qk+1n

− 126n4 · qk+1

at which an is the “error term” introduced in Remark 3.4.4. Under ϕ 160n4 ,1,2

(m−1)· (k+1)·k2 +1

this is mapped to

n · qn+

n · q2n

+ ...+j

n · q(m−1)· (k+1)·k2 +1

[j((m−1)· (k−1)·k

2 +1)1

qn+ ...+

j((m−1)· (k−1)·k

2 +k)1 + 1

qkn− j

(k+1)2 + 1qk+1n

26 · n4 · qk+1n

j((m−1)· (k−1)·k

2 +1)1

qn+ ...+

j((m−1)· (k−1)·k

2 +k)1 + 1

qkn− j

(k+1)2

− 126 · n4 · qk+1

110 · n4

+ n · q(m−1)· (k+1)·k2 +1

n · an, 1−1

10 · n4+ n · q(m−1)· (k+1)·k

2 +1n · an

×m∏i=3

[1− j

((m−1)· (k−1)·k2 +(i−2)·k+1)

qn− ...− j

((m−1)· (k−1)·k2 +(i−1)·k)

1 + 1qkn

(k+1)i

26 · n4 · qk+1n

1− j((m−1)· (k−1)·k

2 +(i−2)·k+1)1

qn− ...− j

((m−1)· (k−1)·k2 +(i−1)·k)

1 + 1qkn

(k+1)i + 1qk+1n

− 126n4 · qk+1

using the bound on an. With the aid of Remark 3.3.7, the bound on an from Remark 3.4.4and the fact that 10n4 divides qk+1

n by Lemma 3.6.9 we can compute the image of In,k underφ

n·q(m−1)· (k+1)·k

2 +1n ,qk+1

Rmnαn+1 φ−1

(γ, δ, ε)-distribution 33

n · qn+

n · q2n

+ ...+j

n · q(m−1)· (k+1)·k2 +1

+j((m−1)· (k−1)·k

2 +1)1

n · q(m−1)· (k+1)·k2 +2

+j((m−1)· (k−1)·k

2 +k)1 + 1

n · q(m−1)· (k+1)·k2 +k+1

− j(k+1)2 + 1

n · q(m−1)· (k+1)·k2 +k+2

26 · n5 · q(m−1)· (k+1)·k2 +k+2

n · qn+

n · q2n

+ ...− j(k+1)2

n · q(m−1)· (k+1)·k2 +k+2

26 · n5 · q(m−1)· (k+1)·k2 +k+2

110 · n4

+ n · q(m−1)· (k+1)·k2 +1

n · an, 1−1

10 · n4+ n · q(m−1)· (k+1)·k

2 +1n · an

×m∏i=3

[1− j

((m−1)· (k−1)·k2 +(i−2)·k+1)

qn− ...− j

((m−1)· (k−1)·k2 +(i−1)·k)

1 + 1qkn

(k+1)i

26 · n4 · qk+1n

1− j((m−1)· (k−1)·k

2 +(i−2)·k+1)1

qn− ...− j

((m−1)· (k−1)·k2 +(i−1)·k)

1 + 1qkn

(k+1)i + 1qk+1n

− 126n4 · qk+1

Continuing in the same way we obtain that Φn(In,k

)is equal to

n · qn+

n · q2n

+ ...+j((m−1)· (k−1)·k

n · q(m−1)· (k−1)·k2 +1

n · q(m−1)· (k−1)·k2 +2

+ ...+j

n · q(m−1)· (k−1)·k2 +k+1

n · q(m−1)· (k−1)·k2 +k+2

+ ...+j

n · q(m−1)· (k+1)·k2 +1

+j((m−1)· (k−1)·k

2 +1)1

n · q(m−1)· (k+1)·k2 +2

+ ...+j((m−1)· (k−1)·k

2 +k)1 + 1

n · q(m−1)· (k+1)·k2 +k+1

− j(k+1)2 + 1

n · q(m−1)· (k+1)·k2 +k+2

+j((m−1)· (k−1)·k

2 +k+1)1

n · q(m−1)· (k+1)·k2 +k+3

+ ...+j((m−1)· (k−1)·k

2 +2k)1 + 1

n · q(m−1)· (k+1)·k2 +2k+2

− j(k+1)3 + 1

n · q(m−1)· (k+1)·k2 +2k+3

+j((m−1)· (k+1)·k

2 )1 + 1

n · q(m−1)· (k+1)·(k+2)2

− j(k+1)m + 1

n · q(m−1)· (k+1)·(k+2)2 +1

26 · n5 · q(m−1)· (k+1)·(k+2)2 +1

n · qn+

n · q2n

+ ...− j(k+1)m

n · q(m−1)· (k+1)·(k+2)2 +1

26 · n5 · q(m−1)· (k+1)·(k+2)2 +1

110 · n4

+ n · q(m−1)(k+1)·k

2 +1n · an, 1−

110 · n4

+ n · q(m−1)(k+1)·k

2 +1n · an

m∏i=3

26n4, 1− 1

Thus, such a set Φn(In

)with In ∈ ηn has a θ-witdth of at most 1

n·q3m+1n

.Moreover, we see that we can choose ε = 0 in the definition of a (γ, δ, ε)-distribution: With thenotation Aθ := πθ

(Φn(In

))we have Φn

)= Aθ × J and so for every (m − 1)-dimensional

34 Criterion for weak mixing

interval J ⊆ J :

µ(In ∩ Φ−1

(S1 × J

))µ(In

) =µ(

Φn(In

)∩ S1 × J

Φn(In

)) =λ (Aθ) · µ(m−1)

λ (Aθ) · µ(m−1) (J)=µ(m−1)

µ(m−1) (J)

because Φn is measure-preserving.

Furthermore, we show the next property concerning the conjugating map gn constructed insection 3.3.2:

Lemma 3.4.6. For every In ∈ ηn we have: gn(

Φn(In

))= g[nqσn]

(Φn(In

Proof. In the proof of the precedent Lemma 3.4.5 we computed Φn(In,k

)for a partition element

In,k. Now we have to examine the effect of gn = gn·q

1+(m−1)· (k+1)·(k+2)2

n ,[n·qσn], 18n4 ,

on it.

Since 260n4 divides qn by Lemma 3.6.9 there is u ∈ Z such that

= u · ε

b · a= u · 1

8n4 · [nqσn] · nq1+(m−1)· (k+1)·(k+2)2

By 126n4 < ε = 1

8n4 and the bound on an the boundary of Φn(In,k

)lies in the domain where

1+(m−1)· (k+1)·(k+2)2

n ,[n·qσn], 18n4 ,

= g[nqσn].

3.5 Criterion for weak mixing

In this section we will prove a criterion for weak mixing on M = S1 × [0, 1]m−1 in the settingof the beforehand constructions. For the derivation we need a couple of lemmas. The first oneexpresses the weak mixing property on the elements of a partial partition ηn generally:

Lemma 3.5.1. Let f ∈ Diff∞ (M,µ), (mn)n∈N be a sequence of natural numbers and (νn)n∈Nbe a sequence of partial partitions, where νn → ε and for every n ∈ N νn is the image of a partialpartition ηn under a measure-preserving diffeomorphism Fn, satisfying the following property:For every m-dimensional cube A ⊆ S1 × (0, 1)m−1 and for every ε > 0 there exists N ∈ N suchthat for every n ≥ N and for every Γn ∈ νn we have∣∣µ (Γn ∩ f−mn (A)

)− µ (Γn) · µ (A)

∣∣ ≤ 3 · ε · µ (Γn) · µ (A) . (3.1)

Then f is weak mixing.

Proof. A diffeomorphism f is weak mixing if for all measurable sets A,B ⊆M it holds:

limn→∞

∣∣µ (B ∩ f−mn (A))− µ (B) · µ (A)

∣∣ = 0.

Since every measurable set in M = S1 × [0, 1]m−1 can be approximated by a countable disjointunion of m-dimensional cubes in S1× (0, 1)m−1 in arbitrary precision, we only have to prove thestatement in case that A is a m-dimensional cube in S1 × (0, 1)m−1.Hence, we consider an arbitrary m-dimensional cube A ⊂ S1 × (0, 1)m−1. Moreover, let B ⊆Mbe a measurable set. Since νn → ε for every ε ∈ (0, 1] there are n ∈ N and a set B =

⋃i∈Λ Γin,

Criterion for weak mixing 35

where Γin ∈ νn and Λ is a countable set of indices, such that µ(B4B

)< ε · µ (B) · µ (A). We

obtain for sufficiently large n:∣∣µ (B ∩ f−mn (A))− µ (B) · µ (A)

∣∣≤∣∣∣µ (B ∩ f−mn (A)

)− µ

(B ∩ f−mn (A)

)∣∣∣+∣∣∣µ(B ∩ f−mn (A)

)− µ

(B)· µ (A)

∣∣∣+∣∣∣µ(B) · µ (A)− µ (B) · µ (A)

∣∣∣=∣∣∣µ (B ∩ f−mn (A)

)− µ

(B ∩ f−mn (A)

)∣∣∣+

∣∣∣∣∣µ(⋃i∈Λ

(Γin ∩ f−mn (A)

))− µ

(⋃i∈Λ

)· µ (A)

∣∣∣∣∣+ µ (A) ·∣∣∣µ(B)− µ (B)

∣∣∣≤ µ

∣∣∣∣∣∑i∈Λ

µ(Γin ∩ f−mn (A)

)− µ

(Γin)· µ (A)

∣∣∣∣∣+ µ (A) · µ(B4B

)≤ ε · µ(B) · µ(A) +

∑i∈Λ

(∣∣µ (Γin ∩ f−mn(A))− µ

(Γin)· µ(A)

∣∣)+ ε · µ(A)2 · µ(B)

≤∑i∈Λ

(3 · ε · µ

(Γin)· µ(A)

)+ 2 · ε · µ(A) · µ(B) = 3 · ε · µ(A) · µ

(⋃i∈Λ

)+ 2 · ε · µ(A) · µ(B)

= 3 · ε · µ(A) · µ(B)

+ 2 · ε · µ(A) · µ(B) ≤ 3 · ε · µ(A) ·(µ(B) + µ

))+ 2 · ε · µ(A) · µ(B)

≤ 5 · ε · µ(A) · µ(B) + 3 · ε2 · µ(A)2 · µ(B).

This estimate shows limn→∞ |µ (B ∩ f−mn (A))− µ (B) · µ (A)| = 0, because ε can be chosenarbitrarily small.

In property (3.1) we want to replace f by fn:

Lemma 3.5.2. Let f = limn→∞ fn be a diffeomorphism obtained by the constructions in thepreceding sections and (mn)n∈N be a sequence of natural numbers fulfilling d0 (fmn , fmnn ) < 1

2n .Furthermore, let (νn)n∈N be a sequence of partial partitions, where νn → ε and for every n ∈ N νnis the image of a partial partition ηn under a measure-preserving diffeomorphism Fn, satisfyingthe following property: For every m-dimensional cube A ⊆ S1× (0, 1)m−1 and for every ε ∈ (0, 1]there exists N ∈ N such that for every n ≥ N and for every Γn ∈ νn we have∣∣µ (Γn ∩ f−mnn (A)

)− µ (Γn) · µ (A)

∣∣ ≤ ε · µ (Γn) · µ (A) . (3.2)

Proof. We want to show that the requirements of Lemma 3.5.1 are fulfilled. This implies that fis weak mixing. For it let A ⊆ S1 × (0, 1)m−1 be an arbitrary m-dimensional cube and ε ∈ (0, 1].We consider two m-dimensional cubes A1, A2 ⊂ S1 × (0, 1)m−1 with A1 ⊂ A ⊂ A2 as well asµ (A4Ai) < ε · µ (A) and for sufficiently large n: dist(∂A, ∂Ai) > 1

2n for i = 1, 2.If n is sufficiently large, we obtain for Γn ∈ νn and for i = 1, 2 by the assumptions of this

Lemma: ∣∣µ (Γn ∩ f−mnn (Ai))− µ (Γn) · µ (Ai)

∣∣ ≤ ε · µ (Γn) · µ (Ai) .

Herefrom we conclude (1− ε) · µ (Γn) · µ (A1) ≤ µ (Γn ∩ f−mnn (A1)) on the one hand andµ (Γn ∩ f−mnn (A2)) ≤ (1 + ε) · µ (Γn) · µ (A2) on the other hand. Because of d0 (fmn , fmnn ) < 1

the following relations are true:

fmnn (x) ∈ A1 =⇒ fmn(x) ∈ A,fmn(x) ∈ A =⇒ fmnn (x) ∈ A2.

Thus: µ (Γn ∩ f−mnn (A1)) ≤ µ (Γn ∩ f−mn (A)) ≤ µ (Γn ∩ f−mnn (A2)).Altogether, it holds: (1− ε) · µ (Γn) · µ (A1) ≤ µ (Γn ∩ f−mn (A)) ≤ (1 + ε) · µ (Γn) · µ (A2).Therewith, we obtain the following estimate from above:

µ(Γn ∩ f−mn (A)

)− µ (Γn) · µ (A)

≤ (1 + ε) · µ (Γn) · µ (A2)− µ (Γn) · µ (A2) + µ (Γn) · (µ (A2)− µ (A))≤ ε · µ (Γn) · µ (A2) + µ (Γn) · µ (A24A) ≤ ε · µ (Γn) · (µ(A) + µ (A24A)) + ε · µ (Γn) · µ (A)

≤ 2 · ε · µ (Γn) · µ (A) + ε2 · µ (Γn) · µ (A) ≤ 3 · ε · µ (Γn) · µ (A)

Furthermore, we deduce the following estimate from below in an analogous way:

µ(Γn ∩ f−mn (A)

)− µ (Γn) · µ (A) ≥ −3 · ε · µ (Γn) · µ (A) .

Hence, we get: |µ (Γn ∩ f−mn (A))− µ (Γn) · µ (A)| ≤ 3 ·ε ·µ (Γn) ·µ (A), i.e. the requirementsof Lemma 3.5.1 are met.

Now we concentrate on the setting of our explicit constructions:

Lemma 3.5.3. Consider the sequence of partial partitions (ηn)n∈N constructed in section 3.3.1.1and the diffeomorphisms gn from chapter 3.3.2. Furthermore, let (Hn)n∈N be a sequence ofmeasure-preserving smooth diffeomorphisms satisfying ‖DHn−1‖ ≤ ln(qn)

n for every n ∈ N and

define the partial partitions νn =

Γn = Hn−1 gn(In

): In ∈ ηn

Then we get νn → ε.

Proof. By construction ηn =Iin : i ∈ Λn

, where Λn is a countable set of indices. Because of

ηn → ε it holds limn→∞ µ(⋃

i∈ΛnIin

)= 1. Since Hn−1 gn is measure-preserving, we conclude:

limn→∞

( ⋃i∈Λn

)= limn→∞

( ⋃i∈Λn

Hn−1 gn(Iin

))= limn→∞

(Hn−1 gn

( ⋃i∈Λn

))= 1.

For any m-dimensional cube with sidelength ln it holds: diam(Wn) =√m · ln. Because every

element of the partition ηn is contained in a cube of side length 1qn

it follows for every i ∈ Λn:

diam(Iin

)≤√m · 1

qn. Furthermore, we saw in Lemma 3.3.5: gn

)= g[nqσn]

)for every

i ∈ Λn. Hence for every Γin = Hn−1 g[nqσn]

(Iin):

diam(Γin)≤ ‖DHn−1‖0 ·

∥∥Dg[nqσn]

∥∥0· diam

)≤ ln (qn)

n· [n · qσn] ·

qn≤√m · qσ−1

n · ln (qn) .

Because of σ < 1 we conclude limn→∞diam(Γin)

= 0 and consequently νn → ε.

In the following the Lebesgue measures on S1, [0, 1]m−2, [0, 1]m−1 are denoted by λ, µ(m−2)

and µ respectively. The next technical result is needed in the proof of Lemma 3.5.5.

Lemma 3.5.4. Given an interval on the r1-axis of the form K =⋃k∈Z,k1≤k≤k2

[k·εb·a ,

(k+1)·εb·a

where k1, k2 ∈ Z with b·aε · δ ≤ k1 < k2 ≤ b·a

ε −b·aε · δ − 1, and a (m − 2)-dimensional interval

Z in (r2, ..., rm−1) Kc,γ denotes the cuboid [c, c+ γ] ×K × Z for some γ > 0. We consider thediffeomorphism ga,b,ε constructed in subsection 3.3.2 and an interval L = [l1, l2] of S1 satisfyingλ (L) ≥ 4 · 1−2ε

a − γ.If b · λ(K) > 2. then for the set Q := π~r

(Kc,γ ∩ g−1

a,b,ε (L×K × Z))we have:∣∣∣µ (Q)− λ (K) · λ (L) · µ(m−2) (Z)

∣∣∣≤(

2b· λ (L) +

2 · γb

+ γ · λ (K) + 4 · 1− 2εa· λ(K) + 8 · 1− 2ε

b · a

)· µ(m−2) (Z) .

Proof. We consider the diffeomorphism gb : M →M , (θ, r1, ..., rm−1) 7→ (θ + b · r1, r1, ..., rm−1)and the set:

Qb := π~r(Kc,γ ∩ g−1

b (L×K × Z))

= (r1, r2, ..., rm−1) ∈ K × Z : (θ + b · r1, ~r) ∈ L×K × Z, θ ∈ [c, c+ γ]= (r1, r2, ..., rm−1) ∈ K × Z : b · r1 ∈ [l1 − c− γ, l2 − c] mod 1 .

The interval b ·K seen as an interval in R does not intersect more than b · λ(K) + 2 and not lessthan b · λ (K)− 2 intervals of the form [i, i+ 1] with i ∈ Z. By construction of the map ga,b,ε itholds for ∆l :=

[l·εb·a ,

(l+1)·εb·a

]in consideration: π~r (ga,b,ε ([c, c+ γ]×∆l × Z)) = ∆l × Z.

Claim: A resulting interval on the r1-axis of Kc,γ ∩ g−1b (L×K × Z) and the corresponding

r1-projection of Kc,γ ∩ g−1a,b,ε (L×K × Z) can differ by a length of at most 4 · 1−2ε

b·a .Proof: If c×∆l×Z (resp. c+ γ×∆l×Z) are contained in the domain, where ga,b,ε = gb, theleft (resp. the right) boundaries of πθ (ga,b,ε ([c, c+ γ]×∆l × Z)) and πθ (gb ([c, c+ γ]×∆l × Z))coincide. Otherwise, i.e. c ∈

(ka + ε, k+1

a − ε)(resp. c + γ ∈

(ka + ε, k+1

a − ε)) the sets

πθ (ga,b,ε (c ×∆l × Z)) and πθ (gb (c ×∆l × Z)) (resp. πθ (ga,b,ε (c+ γ ×∆l × Z)) andπθ (gb (c+ γ ×∆l × Z))) differ by a length of at most 1−2ε

a . Since πθ (gb (u ×∆l × Z))for arbitrary u ∈ S1 has a length of ε

a on the θ-axis, this discrepancy will be equalised afterat most 1−2ε

a : εa = 1−2ε

ε blocks ∆l on the r1-axis. Thus, the resulting interval on the r1-axisof Kc,γ ∩ g−1

b (L×K × Z) and the corresponding r1-projection of Kc,γ ∩ g−1a,b,ε (L×K × Z) can

differ by a length of at most 4 · 1−2εε · ε

b·a = 4 · (1− 2ε) 1b·a .

Therefore, we compute on the one side:

µ (Q) ≤ (b · λ (K) + 2) ·(l2 − (l1 − γ)

b+ 4 · 1− 2ε

b · a

)· µ(m−2) (Z)

(λ (K) · λ (L) + 2 · λ (L)

b+ λ (K) · γ +

2 · γb

+ 4 · λ(K) · 1− 2εa

+ 8 · 1− 2εb · a

)· µ(m−2) (Z)

and on the other side

µ (Q) ≥ (b · λ (K)− 2) ·(l2 − (l1 − γ)

b− 4 · 1− 2ε

b · a

)· µ(m−2) (Z)

(λ (K) · λ (L)− 2 · λ (L)

b+ λ (K) · γ − 2 · γ

b− 4 · λ(K) · 1− 2ε

a+ 8 · 1− 2ε

b · a

)· µ(m−2) (Z) .

Both equations together yield:∣∣∣∣µ (Q)− λ (K) · λ (L) · µ(m−2) (Z)− γ · λ (K) · µ(m−2) (Z)− 8 · 1− 2εb · a

· µ(m−2) (Z)∣∣∣∣

2b· λ (L) +

2 · γb

+ 4 · λ(K) · 1− 2εa

)· µ(m−2) (Z) .

The claim follows because∣∣∣µ (Q)− λ (K) · λ (L) · µ(m−2) (Z)∣∣∣− γ · λ (K) · µ(m−2) (Z)− 8 · 1− 2ε

b · a· µ(m−2) (Z)

≤∣∣∣∣µ (Q)− λ (K) · λ (L) · µ(m−2) (Z)− γ · λ (K) · µ(m−2) (Z)− 8 · 1− 2ε

b · a· µ(m−2) (Z)

∣∣∣∣ .

Lemma 3.5.5. Let n ≥ 5, gn as in section 3.3.2 and In ∈ ηn, where ηn is the partial partitionconstructed in section 3.3.1.1. For the diffeomorphism φn constructed in section 3.3.3 and mn

as in chapter 3.4 we consider Φn = φn Rmnαn+1 φ−1

n and denote π~r(

Φn(In

))by J.

Then for every m-dimensional cube S of side length q−σn lying in S1 × J we get∣∣∣µ(I ∩ Φ−1n g−1

n (S))· µ (J)− µ

(I)· µ (S)

∣∣∣ ≤ 21n· µ(I)· µ (S) . (3.3)

In other words this Lemma tells us that a partition element is “almost uniformly distributed”under gn Φn on the whole manifold M = S1 × [0, 1]m−1.

Proof. Let S be a m-dimensional cube with sidelength q−σn lying in S1 × J . Furthermore, wedenote:

Sθ = πθ (S) Sr1 = πr1 (S) S~r = π(r2,...,rm−1) (S) Sr = Sr1 × S~r = π~r (S)

Obviously: λ (Sθ) = λ (Sr1) = q−σn and λ (Sθ) · λ (Sr1) · µ(m−2)(S~r)

= µ (S) = q−mσn .

According to Lemma 3.4.5 Φn(

1n·qmn

, 1n4 ,

)-distributes the partition element In ∈ ηn, in par-

ticular Φn(In

)⊆ [c, c+ γ]× J for some c ∈ S1 and some γ ≤ 1

n·qmn. Furthermore, we saw in the

proof of Lemma 3.4.6 that [c, c+ γ]× J is contained in the interior of the step-by-step domainsof the map gn and on its boundary gn = g[nqσn] holds. Particularly it follows γ ≥ 1−2ε

a in case of

gn = ga,b,ε. For l ∈ Z, 0 ≤ l ≤ b·aε − 1 we introduce the set ∆l =

[lεba ,

(l+1)εba

]and therewith we

consider

Sr1 :=⋃

∆l⊆Sr1

∆l; Sr :=⋃

∆l⊆Sr1

∆l × S~r as well as S := Sθ × Sr ⊆ S

Using the triangle inequality we obtain∣∣∣µ(I ∩ Φ−1n

(g−1n (S)

))· µ (J)− µ

(I)· µ (S)

∣∣∣≤∣∣∣µ(I ∩ Φ−1

(g−1n (S)

))− µ

(I ∩ Φ−1

(g−1n

(S)))∣∣∣ · µ (J)

+∣∣∣µ(I ∩ Φ−1

(g−1n

· µ (J)− µ(I)µ(S)∣∣∣+ µ

(I)·∣∣∣µ(S)− µ (S)

∣∣∣

Here∣∣∣µ(S)− µ (S)

∣∣∣ = µ(S \ S

)≤ 2 · ε

b·a · λ (Sθ) · µ(m−2)(S~r)≤ 2 · εa · µ (S), where we used

b = [n · qσn] ≥ qσn in case of n > 4. Since Φn and gn are measure-preserving, we additionallyobtain:

∣∣∣µ(I ∩ Φ−1n

(g−1n (S)

))− µ

(I ∩ Φ−1

(g−1n

(S)))∣∣∣ ≤ µ(S \ S) ≤ 2 · εa · µ (S).

In the proof of Lemma 3.4.6 we observed µ(

Φn(I))

= 1a ·(1− 2

)· µ (J). Hence:

∣∣∣µ(I ∩ Φ−1n

(g−1n (S)

))− µ

(I ∩ Φ−1

(g−1n

(S)))∣∣∣ · µ (J) ≤ 2 · ε

a· µ (S) · µ (J)

= 2 · ε

1− 226n4

· µ (S) · µ(

Φn(I))≤ 4 · ε · µ (S) · µ

(Φn(I))

= 4 · ε · µ (S) · µ(I)

Thus, we obtain:∣∣∣µ(I ∩ Φ−1n

(g−1n (S)

))· µ (J)− µ

(I)· µ (S)

∣∣∣≤∣∣∣µ(I ∩ Φ−1

(g−1n

· µ (J)− µ(I)µ(S)∣∣∣+ 5 · ε · µ (S) · µ

(I) (3.4)

Next, we want to estimate the first summand. By construction of the map gn = ga,b,ε and thedefinition of S it holds: Φn

(I)∩ g−1

(S)⊆ [c, c+ γ] × Sr =: Kc,γ . Considering the proof of

Lemma 3.4.6 again, we obtain gn (Kc,γ) = g[nqσn] (Kc,γ) (since c and c + γ are in the domainwhere gn = g[nqσn] holds).Because of Lemma 3.4.5 2γ ≤ 2

n·qmn< q−σn for n > 2. So we can define a cuboid S1 ⊆ S, where

S1 := [s1 + γ, s2 − γ]× Sr using the notation Sθ = [s1, s2]. We examine the two sets

Q := π~r

(Kc,γ ∩ g−1

(Sθ × Sr

))Q1 := π~r

(Kc,γ ∩ g−1

([s1 + γ, s2 − γ]× Sr

))As seen above Φn

(I)∩ g−1

(S)⊆ Kc,γ . Hence Φn

(I)∩ g−1

(S)⊆ Φn

(I)∩ g−1

(S)∩Kc,γ ,

which implies Φn(I)∩ g−1

(S)⊆ Φn

(I)∩(S1 ×Q

Claim: On the other hand: Φn(I)∩(S1 ×Q1

)⊆ Φn

(I)∩ g−1

Proof of the claim: For (θ, ~r) ∈ Φn(I)∩(S1 ×Q1

)arbitrary it holds (θ, ~r) ∈ Φn

i.e. θ ∈ [c, c+ γ], and ~r ∈ π~r

(Kc,γ ∩ g−1

([s1 + γ, s2 − γ]× Sr

)), i.e. in particular ~r ∈ Sr.

This implies the existence of θ ∈ [c, c+ γ] satisfying(θ, ~r)∈ Kc,γ ∩ g−1

n (S1). Hence, there areβ ∈ [s1 + γ, s2 − γ] and ~r1 ∈ Sr, such that gn

(θ, ~r)

= (β,~r1). Because of θ ∈ [c, c+ γ] and ~r ∈ Srthe point

(θ, ~r)is contained in one cuboid of the form ∆a,b,ε. Since θ ∈ [c, c+ γ] (θ, ~r) is contained

in the same ∆a,b,ε. Thus π~r (gn (θ, ~r)) ∈ Sr. Furthermore, gn (θ, ~r) and gn(θ, ~r)are in a distance

of at most γ on the θ-axis, because θ, θ ∈ [c, c+ γ], i.e.∣∣θ − θ∣∣ ≤ γ, gn (Kc,γ) = g[nqσn] (Kc,γ) and

the map g[nqσn] preserves the distances on the θ-axis. Thus, there are β ∈ [s1, s2] and ~r2 ∈ Srsuch that gn (θ, ~r) =

(β, ~r2

). So (θ, ~r) ∈ Φn

(I)∩ g−1

Altogether, the following inclusions are true:

Φn(I)∩(S1 ×Q1

)⊆ Φn

(I)∩ g−1

(S)⊆ Φn

(I)∩(S1 ×Q

Thus, we obtain: ∣∣∣µ(I ∩ Φ−1n

(g−1n (S)

))· µ (J)− µ

(I)· µ(S)∣∣∣

≤ max

(∣∣∣µ(I ∩ Φ−1n

(S1 ×Q

))· µ (J)− µ

(I)· µ(S)∣∣∣ ,

∣∣∣µ(I ∩ Φ−1n

(S1 ×Q1

))· µ (J)− µ

(I)· µ(S)∣∣∣)

We want to apply Lemma 3.5.4 for K = Sr1 , L = Sθ, Z = S~r and b = [n · qσn] (note that4 · 1−2ε

a − γ ≤ 3 · 1−2εa ≤ 3

n·qmn< 1

qσn= λ (L) because of the mentioned relation γ ≥ 1−2ε

a and forn > 4: b · λ(K) = [nqσn] · q−σn ≥ 1

2nqσn · q−σn > 2):∣∣∣µ (Q)− µ

(S)∣∣∣

2[n · qσn]

· λ (Sθ) +2γ

[n · qσn]+ γ · λ

)+ 4 · 1− 2ε

aλ(Sr1

)+ 8 · 1− 2ε

[nqσn] · a

)· µ(m−2)

4n · qσn

λ (Sθ) +4

n · qσn · qσn+

1n · qσn

λ (Sr1) + 4 · 1− 2εn · qmn

λ (Sr1) +16 · (1− 2ε)n · qσn · n · qmn

)· µ(m−2)

≤ 14n· µ (S) .

In particular, we receive from this estimate: 14n · µ (S) ≥ µ (Q)− µ

(S)≥ µ (Q)− µ (S), hence:

µ (Q) ≤(1 + 14

)· µ (S) ≤ 4 · µ (S).

Analogously we obtain: µ (Q1) ≤ 4 · µ (S) as well as∣∣∣µ (Q1)− µ

)∣∣∣ ≤ 14n · µ (S).

Since Q as well as Q1 are a finite union of disjoint (m− 1)-dimensional intervals contained in Jand Φn

n·qmn, 1n4 ,

)-distributes the interval I we get:∣∣∣µ(I ∩ Φ−1

(S1 ×Q

))· µ (J)− µ

(I)· µ (Q)

∣∣∣ ≤ 1n· µ(I)· µ (Q) ≤ 4

n· µ(I)· µ (S)

as well as∣∣∣µ(I ∩ Φ−1n

(S1 ×Q1

))· µ (J)− µ

(I)· µ (Q1)

∣∣∣ ≤ 1n· µ(I)· µ (Q1) ≤ 4

n· µ(I)· µ (S) .

Now we can proceed∣∣∣µ(I ∩ Φ−1n

(S1 ×Q

))· µ (J)− µ

(I)· µ(S)∣∣∣

≤∣∣∣µ(I ∩ Φ−1

(S1 ×Q

))· µ (J)− µ

(I)· µ (Q)

∣∣∣+ µ(I)·∣∣∣µ (Q)− µ

(S)∣∣∣

≤ 4n· µ(I)· µ (S) + µ

(I)· 14n· µ (S) =

18n· µ(I)· µ (S) .

Noting that µ (S1) = µ(S)− 2γ · µ

)and so µ

(S)− µ (S1) ≤ 2 · 1

n·qσn· µ(Sr

)≤ 2

n · µ (S)we obtain in the same way as above:∣∣∣µ(I ∩ Φ−1

(S1 ×Q1

))· µ (J)− µ

(I)· µ(S)∣∣∣ ≤ 20

n· µ(I)· µ (S) .

Using equation 3.5 this yields:∣∣∣µ(I ∩ Φ−1n

(g−1n

· µ (J)− µ(I)· µ(S)∣∣∣ ≤ 20

n· µ(I)· µ (S) .

Finally, we conclude with the aid of equation 3.4 because of ε = 18n4 :∣∣∣µ(I ∩ Φ−1

(g−1n (S)

))· µ (J)− µ

(I)· µ (S)

∣∣∣ ≤ 21n· µ(I)· µ (S) .

Now we are able to prove the aimed criterion for weak mixing.

Proposition 3.5.6 (Criterion for weak mixing). Let fn = Hn Rαn+1 H−1n and the sequence

(mn)n∈N be constructed as in the previous sections. Suppose additionally that d0 (fmn , fmnn ) < 12n

for every n ∈ N, ‖DHn−1‖0 ≤ln(qn)n and that the limit f = limn→∞ fn exists.

Proof. To apply Lemma 3.5.2 we consider the partial partitions νn := Hn−1gn (ηn). As provenin Lemma 3.5.3 these partial partitions satisfy νn → ε. We have to establish equation 3.2. Forit let ε > 0 and a m-dimensional cube A ⊆ S1 × (0, 1)m−1 be given. There exists N ∈ N suchthat A ⊆ S1 ×

[1n4 , 1− 1

]m−1 for every n ≥ N . Because of Lemma 3.4.5 and the properties

1n·qmn

, 1n4 ,

)-distribution we obtain for every In ∈ ηn: π~r

(Φn(In

))⊇[

1n4 , 1− 1

]m−1.

Furthermore, we note fmnn = Hn Rmnαn+1H−1

n = Hn−1 gn Φn g−1n H−1

n−1.

Let Sn be a m-dimensional cube of side length q−σn contained in S1 ×[

1n4 , 1− 1

]m−1. We lookat Cn := Hn−1 (Sn), Γn ∈ νn, and compute (since gn and Hn−1 are measure-preserving):∣∣µ (Γn ∩ f−mnn (Cn)

)− µ (Γn) · µ (Cn)

∣∣ =∣∣∣µ(In ∩ Φ−1

n g−1n (Sn)

)− µ

)· µ (Sn)

∣∣∣≤ 1µ (J)

·∣∣∣µ(In ∩ Φ−1

n g−1n (Sn)

)· µ (J)− µ

)· µ (Sn)

∣∣∣+1− µ (J)µ (J)

· µ(In

)· µ (Sn)

Bernoulli’s inequality yields: µ(J) ≥(1− 1

)m−1 ≥ 1 + (m− 1) ·(− 1n

)= 1 − m−1

n . Hence weobtain for n > 2 · (m− 1): µ (J) ≥ 1

2 and so: 1−µ(J)µ(J) ≤ 2 · (1− µ (J)) ≤ 2·(m−1)

n . We continue byapplying Lemma 3.5.5:

∣∣µ (Γn ∩ f−mnn (Cn))− µ (Γn) · µ (Cn)

∣∣ ≤ 2 · 21n· µ(In

)· µ (Sn) +

2 · (m− 1)n

· µ(In

)· µ (Sn)

=40 + 2 ·m

n· µ(In

)· µ (Sn)

Moreover, it holds diam(Cn) ≤ ‖DHn−1‖0 · diam (Sn) ≤√m · ln(qn)

qσn, i.e. diam(Cn) → 0 as

n→∞. Thus, we can approximate A by a countable disjoint union of sets Cn = Hn−1 (Sn) withSn ⊆ S1 ×

[1n4 , 1− 1

]m−1 a m-dimensional cube of sidelength q−σn in given precision, when nis chosen big enough. Consequently for n sufficiently large there are sets A1 =

⋃i∈Σ1

nCin and

A2 =⋃i∈Σ2

nCin with countable sets Σ1

n and Σ2n of indices satisfying A1 ⊆ A ⊆ A2 as well as

|µ(A)− µ(Ai)| ≤ ε3 · µ(A) for i = 1, 2.

42 Convergence of (fn)n∈N in Diff∞ (M)

Additionally we choose n such that 40+2·mn < ε

3 holds. It follows:

µ(Γn ∩ f−mnn (A)

)− µ (Γn) · µ (A)

≤ µ(Γn ∩ f−mnn (A2)

)− µ (Γn) · µ (A2) + µ (Γn) · (µ (A2)− µ (A))

≤∑i∈Σ2

(µ(Γn ∩ f−mnn

(Cin))− µ (Γn) · µ

(Cin))

3· µ (Γn) · µ (A)

≤∑i∈Σ2

(40 + 2 ·m

n· µ(In

)· µ(Sin))

3· µ (Γn) · µ (A)

=40 + 2 ·m

n· µ (Γn) · µ

⋃i∈Σ2

3· µ (Γn) · µ (A) ≤ ε

3· µ (Γn) · µ (A2) +

3· µ (Γn) · µ (A)

3· µ (Γn) · µ (A) +

3· µ (Γn) · (µ (A2)− µ (A)) +

3· µ (Γn) · µ (A) ≤ ε · µ (Γn) · µ (A) .

Analogously we estimate: µ (Γn ∩ f−mnn (A))−µ (Γn) ·µ (A) ≥ −ε ·µ (Γn) ·µ (A). Both estimatesenable us to conclude: |µ (Γn ∩ f−mnn (A))− µ (Γn) · µ (A)| ≤ ε · µ (Γn) · µ (A).

3.6 Convergence of (fn)n∈N in Diff∞ (M)

In the following we show that the sequence of constructed measure-preserving smooth diffeo-morphisms fn = Hn Rαn+1 H−1

n converges. For this purpose, we need a couple of resultsconcerning the conjugation maps.

3.6.1 Properties of the conjugation maps φn and Hn

In order to find estimates on the norms |‖Hn‖|k we will need the next technical result which isan application of the chain rule:

Lemma 3.6.1. Let φ := φ(m)λm,µm

... φ(2)λ2,µ2

, j ∈ 1, ...,m and k ∈ N. For any multiindex ~awith |~a| = k the partial derivative D~a [φ]j consists of a sum of products of at most (m − 1) · kterms of the following form

(i)λi,µi

) φ(i−1)

λi−1,µi−1 ... φ(2)

λ2,µ2,

where l ∈ 1, ...,m, i ∈ 2, ...,m and ~b is a multiindex with∣∣∣~b∣∣∣ ≤ k.

We will prove a similar result in Lemma 8.3.1. In the same way we obtain an analoguestatement holding for the inverses:

Lemma 3.6.2. Let ψ :=(φ

(2)λ2,µ2

... (φ

(m)λm,µm

, j ∈ 1, ...,m and k ∈ N. For anymultiindex ~a with |~a| = k the partial derivative D~a [ψ]j consists of a sum of products of at most(m− 1) · k terms of the following form

(i)λi,µi

)−1]l

(i+1)λi+1,µi+1

... (φ

(m)λm,µm

where l ∈ 1, ...,m, i ∈ 2, ...,m and ~b is a multiindex with∣∣∣~b∣∣∣ ≤ k.

Convergence of (fn)n∈N in Diff∞ (M) 43

Remark 3.6.3. In the proof of the following lemmas we will use the formula of Faà di Brunoin several variables. It can be found in the paper “A multivariate Faà di Bruno formula withapplications” ([CS96]) for example.For this we introduce an ordering on Nd0: For multiindices ~µ = (µ1, ..., µd) and ~ν = (ν1, ..., νd) inNd0 we will write ~µ ≺ ~ν, if one of the following properties is satisfied:

1. |~µ| < |~ν|, where |~µ| =∑di=1 µi.

2. |~µ| = |~ν| and µ1 < ν1.

3. |~µ| = |~ν|, µi = νi for 1 ≤ i ≤ k and µk+1 < νk+1 for a 1 ≤ k < d.

Additionally we will use these notations:

• For ~ν = (ν1, ..., νd) ∈ Nd0:

~ν! =d∏i=1

• For ~ν = (ν1, ..., νd) ∈ Nd0 and ~z = (z1, ..., zd) ∈ Rd:

~z ~ν =d∏i=1

Then we get for the composition h (x1, ..., xd) := f(g(1) (x1, ..., xd) , ..., g(m) (x1, ..., xd)

sufficiently differentiable functions f : Rm → R, g(i) : Rd → R and a multiindex ~ν ∈ Nd0 with|~ν| = n:

D~νh =∑

~λ∈Nm0 with 1≤|~λ|≤nD~λf ·

n∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

[D~lj~g

]~kj~kj ! ·

(~lj !)|~kj|

Here[D~lj~g

]denotes

(D~ljg

(1), ..., D~ljg(m))and

(~ν,~λ

~k1, ...,~ks,~l1, ...,~ls

): ~ki ∈ Nm0 ,

∣∣∣~ki∣∣∣ > 0,~li ∈ Nd0, 0 ≺ ~l1 ≺ ... ≺ ~ls,s∑i=1

~ki = ~λ ands∑i=1

∣∣∣~ki∣∣∣ ·~li = ~ν

With the aid of these technical results we can prove an estimate on the norms of the map φn:

Lemma 3.6.4. For every k ∈ N it holds

|||φn|||k ≤ C · q(m−1)2·k·n·(n+1)n ,

where C is a constant depending on m, k and n, but is independent of qn.

Proof. First of all we consider the map φλ,µ := φλ,ε,i,j,µ,δ,ε2 = C−1λ ψµ,δ,i,j,ε2 ϕε,i,j Cλ

introduced in subsection 3.3.3:

φλ,µ (x1, ..., xm) =(1λ

[ψµ ϕε]1 (λx1, x2, ..., xm) , [ψµ ϕε]2 (λx1, x2, ..., xm) , ..., [ψµ ϕε]m (λx1, x2, ..., xm)).

Let k ∈ N. We compute for a multiindex ~a with 0 ≤ |~a| ≤ k:∥∥∥D~a [φλ,µ]

∥∥∥0≤ λk−1 · |||ψµ ϕε|||k

and for r ∈ 2, ...,m:∥∥∥D~a [φλ,µ]

∥∥∥0≤ λk · |||ψµ ϕε|||k.

Therefore, we examine the map ψµ. For any multiindex ~a with 0 ≤ |~a| ≤ k and r ∈ 1, ...,m weobtain:

∥∥D~a [ψµ]r∥∥

0≤ µk−1·|||ϕε2 |||k = Ck,ε2 ·µk−1 and analogously

∥∥∥D~a [ψ−1µ

∥∥∥0≤ Ck,ε2 ·µk−1.

Hence: |||ψµ|||k ≤ C · µk−1.In the next step we use the formula of Faà di Bruno mentioned in remark 3.6.3. With it wecompute for any multiindex ~ν with |~ν| = k:

∥∥∥D~ν [(ψµ ϕε)−1]r

∥∥∥0

=∥∥∥D~ν [ϕ−1

ε ψ−1µ

∥∥∥0

∥∥∥∥∥∥∥∥∑

~λ∈Nm0 ,1≤|~λ|≤kD~λ[ϕ−1ε

]r·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

[D~ljψ

−1µ

]~kj~kj ! ·

(~lj !)|~kj|

∥∥∥∥∥∥∥∥0

∥∥∥∥∥∥∥∥∑

~λ∈Nm0 ,1≤|~λ|≤kD~λ[ϕ−1ε

]r·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

∏mt=1

[ψ−1µ

)~kjt~kj ! ·

(~lj !)|~kj|

∥∥∥∥∥∥∥∥0

≤∑

~λ∈Nm0 ,1≤|~λ|≤k

∥∥D~λ [ϕ−1ε

∥∥0·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

∏mt=1

∥∥∥D~lj [ψ−1µ

∥∥∥~kjt0

~kj ! ·(~lj !)|~kj|

≤∑

~λ∈Nm0 with 1≤|~λ|≤k

∥∥D~λ [ϕ−1ε

∥∥0·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

|||ψ−1µ |||

∑mt=1

~kj ! ·(~lj !)|~kj|

∥∥D~λ [ϕ−1ε

∥∥0·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

|||ψ−1µ ||||~kj||~lj|

~kj ! ·(~lj !)|~kj|

As seen above: |||ψ−1µ ||||~kj||~lj| ≤ C · µ|~kj|·|~lj|. Hereby:

∏sj=1 |||ψ−1

µ ||||~kj||~lj| ≤ C · µ

∑sj=1|~lj|·|~kj|, where

C is independent of µ. By definition of the set ps(~ν,~λ

)we have

∑si=1

∣∣∣~ki∣∣∣ ·~li = ~ν. Hence:

k = |~ν| =

∣∣∣∣∣s∑i=1

∣∣∣~ki∣∣∣ ·~li∣∣∣∣∣ =

m∑t=1

(s∑i=1

∣∣∣~ki∣∣∣ ·~li)t

=m∑t=1

s∑i=1

∣∣∣~ki∣∣∣ ·~lit =s∑i=1

∣∣∣~ki∣∣∣ ·( m∑t=1

s∑i=1

∣∣∣~ki∣∣∣ · ∣∣∣~li∣∣∣This shows

∏sj=1 |||ψ−1

µ ||||~kj||~lj| ≤ C ·µ

k and finally∥∥∥D~ν [(ψµ ϕε)−1

∥∥∥0≤ C ·µk. Analogously we

compute∥∥D~ν [ψµ ϕε]r

∥∥0≤ C · |||ψµ|||k ≤ C ·µk−1. Altogether, we obtain |||ψµ ϕε|||k ≤ C ·µk.

Hereby, we estimate∥∥∥D~a [φλ,µ]

∥∥∥0≤ C ·λk ·µk and analogously

∥∥∥D~a [φ−1λ,µ

∥∥∥0≤ C ·λk ·µk. In

conclusion this yields |||φλ,µ|||k ≤ C · µk · λk.In the next step we consider φ := φ

(m)λm,µm

... φ(2)λ2,µ2

. Let λmax := max λ2, ..., λm as well as

µmax := max µ2, ..., µm. Inductively we will show |||φ|||k ≤ C · λ(m−1)·kmax · µ(m−1)·k

max for everyk ∈ N, where C is a constant independent of λi and µi.Start: k = 1Let l ∈ 1, ...,m be arbitrary. By Lemma 3.6.1 a partial derivative of [φ]l of first order consists ofa sum of products of at most m− 1 first order partial derivatives of functions φ(j)

λj ,µj. Therewith,

we obtain using |||φ(j)λj ,µj|||1 ≤ C ·λmax ·µmax the estimate ‖Di [φ]l‖0 ≤ C1 ·λm−1

max ·µm−1max for every

i ∈ 1, ...,m, where C1 is a constant independent of λ and µ. With the aid of Lemma 3.6.2

we obtain the same statement for φ−1 =(φ

(2)λ2,µ2

... (φ

(m)λm,µm

. Hence, we conclude:

|||φ|||1 ≤ C1 · λm−1max · µm−1

max .Assumption: The claim is true for k ∈ N.Induction step k → k + 1:In the proof of Lemma 3.6.1 one observes that at the transition k → k + 1 in the product ofat most (m − 1) · k terms of the form D~b

(i)λi,µi

) φ(i−1)

λi−1,µi−1 ... φ(2)

λ2,µ2one is replaced

by a product of a term(DjD~b

(i)λi,µi

) φ(i−1)

λi−1,µi−1 ... φ(2)

λ2,µ2with j ∈ 1, ...,m and at

most m − 2 partial derivatives of first order. Because of |||φ(i)λi,µi|||k+1 ≤ C · λk+1

max · µk+1max and

|||φ(j)λj ,µj|||1 ≤ C · λmax · µmax the λmax-exponent as well as the µmax-exponent increase by at

most 1 + (m− 2) · 1 = m− 1.In the same spirit one uses the proof of Lemma 3.6.2 to show that also in case of φ−1 the λmax-exponent as well as the µmax-exponent increase by at most m− 1.Using the assumption we conclude

|||φ|||k+1 ≤ C · λk·(m−1)+m−1max · µk·(m−1)+m−1

max = C · λ(k+1)·(m−1)max · µ(k+1)·(m−1)

So the proof by induction is completed.In the setting of our explicit construction of the map φn in section 3.3.3 we have ε1 = 1

60·n4 ,

ε2 = 122·n4 , λmax = n · q1+(m−1)·n·(n−1)

2 +(m−2)·nn and µmax = qnn . Thus:

|||φn|||k ≤ C (m, k, n) ·(n · q1+(m−1)·n·(n−1)

2 +(m−2)·nn

)(m−1)·k

· (qnn)(m−1)·k

≤ C (m, k, n) · q(m−1)2·k·n·(n+1)n ,

where C (m, k, n) is a constant independent of qn.

In the next step we consider the map hn = gn φn, where gn is constructed in section 3.3.2:

Lemma 3.6.5. For every k ∈ N it holds:

|||hn|||k ≤ C · q3·(m−1)2·k·n·(n+1)n ,

Proof. Outside of S1 × [δ, 1− δ]m−1, i.e. gn = g[nqσn], we have:

hn (x1, ..., xm) = gn φn (x1, ..., xm)= ([φn (x1, ..., xm)]1 + [n · qσn] · [φn (x1, ..., xm)]2 , [φn (x1, ..., xm)]2 , ..., [φn (x1, ..., xm)]m)

h−1n (x1, ..., xm) = φ−1

n g−1n (x1, ..., xm)

=([φ−1n (x1 − [n · qσn] · x2, x2, ..., xm)

]1, ..., [φn (x1 − [n · qσn] · x2, x2, ..., xm)]m

Since σ < 1 we can estimate:

|||hn|||k ≤ 2·[n · qσn]k ·|||φn|||k ≤ C (m, k, n)·qσ·kn ·q(m−1)2·k·n·(n+1)n ≤ C (m, k, n)·q2·(m−1)2·k·n·(n+1)

with a constant C (m, k, n) independent of qn.In the other case we have

gn φn (x1, ..., xm) =([ga,b,ε ([φn]1 , [φn]2)]

1, [ga,b,ε ([φn]1 , [φn]2)]

2, [φn]3 , ..., [φn]m

We will use the formula of Faà di Bruno as above for any multiindex ~ν with |~ν| = k andr ∈ 1, ...,m:

‖D~ν [hn]r‖0 =∥∥D~ν [ga,b,ε φn]r

∥∥0

≤∑

∥∥D~λ [ga,b,ε]r∥∥

0·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

|||φn||||~kj||~lj|

~kj ! ·(~lj !)|~kj|

By Lemma 3.6.4 we have |||φn|||k ≤ C · q(m−1)2·k·n·(n+1)n , where C is a constant independent of

qn. As above we show∏sj=1 |||φn|||

|~kj||~lj| ≤ C · q

(∑sj=1|~lj|·|~kj|)·(m−1)2·n·(n+1)

n = C · q(m−1)2·k·n·(n+1)n ,

where C is a constant independent of qn.Furthermore, we examine the map ga,b,ε = D−1

a,b,ε gε Da,b,ε for a, b ∈ Z and obtain

|||ga,b,ε|||k ≤(b · aε

)k· |||gε|||k = Cε,k · bk · ak.

By our constructions in section 3.3.2 we have b = [n · qσn] ≤ n · qσn, a ≤ n · q1+(m−1)·n·(n+1)2

ε = 18n4 . Hence: |||gn|||k ≤ Cn,k · qσ·kn · qk+k·(m−1)·n·(n+1)

2n ≤ Cn,k · q2·k·(m−1)·n·(n+1)

n . Finally, we

conclude: ‖D~ν [hn]r‖0 ≤ C · q2·k·(m−1)·n·(n+1)n · qk·(m−1)2·n·(n+1)

n ≤ C · q3·k·(m−1)2·n·(n+1)n .

In the next step we consider h−1n = φ−1

n g−1a,b,ε. For r ∈ 1, ...,m and any multiindex ~ν with

|~ν| = k we obtain using the formula of Faà di Bruno again:∥∥D~ν [h−1n

∥∥0

=∥∥D~ν [φ−1

n g−1n

∥∥0

≤∑

∥∥D~λ [φ−1n

∥∥0·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

|||gn||||~kj||~lj|

~kj ! ·(~lj !)|~kj|

As above we show∏sj=1 |||gn|||

|~kj||~lj| ≤ C · q

2·k·(m−1)·n·(n+1)n , where C is a constant independent of

qn. Since |||φn|||k ≤ C · qk·(m−1)2·n·(n+1)n we get∥∥D~ν [h−1

∥∥0≤ C · q2·k·(m−1)·n·(n+1)

n · qk·(m−1)2·n·(n+1)n ≤ C · q3·k·(m−1)2·n·(n+1)

where C is a constant independent of qn.Thus, we finally obtain |||hn|||k ≤ C(n, k,m) · q3·(m−1)2·k·n·(n+1)

Finally, we are able to prove an estimate on the norms of the map Hn:

Lemma 3.6.6. For every k ∈ N we get:

|||Hn|||k ≤ C · q3·(m−1)2·k·n·(n+1)n ,

where C is a constant depending solely on m, k, n and Hn−1. Since Hn−1 is independent of qnin particular, the same is true for C.

Proof. Let k ∈ N, r ∈ 1, ...,m and ~ν ∈ Nm0 be a multiindex with |~ν| = k. As above weestimate:

‖D~ν [Hn]r‖0 = ‖D~ν [Hn−1 hn]r‖0

≤∑

∥∥D~λ [Hn−1]r∥∥

0·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

|||hn||||~kj||~lj|

~kj ! ·(~lj !)|~kj|

and compute using Lemma 3.6.5:∏sj=1 |||hn|||

|~kj||~lj| ≤ C · q

3·(m−1)2·k·n·(n+1)n , where C is a constant

independent of qn. Since Hn−1 is independent of qn we conclude:

‖D~ν [Hn]r‖0 ≤ C · q3·(m−1)2·k·n·(n+1)n ,

where C is a constant independent of qn.In the same way we prove an analogous estimate of

∥∥D~ν [H−1n

∥∥0and verify the claim.

In particular, we see that this norm can be estimated by a power of qn.

3.6.2 Proof of convergence

For the proof of the convergence of the sequence (fn)n∈N in the Diff∞ (M)-topology the nextresult, that can be found in [FSW07], Lemma 4, is very useful.

Lemma 3.6.7. Let k ∈ N0 and h be a C∞-diffeomorphism on M . Then we get for everyα, β ∈ R:

dk(h Rα h−1, h Rβ h−1

)≤ Ck · |||h|||k+1

k+1 · |α− β| ,

where the constant Ck depends solely on k and m. In particular C0 = 1.

Indeed, we prove a more precise statement in Lemma 8.3.4.In the following Lemma we show that under some assumptions on the sequence (αn)n∈N thesequence (fn)n∈N converges to f ∈ Aα (M) in the Diff∞ (M)-topology. Afterwards, we will showthat we can fulfil these conditions (see Lemma 3.6.9).

Lemma 3.6.8. Let ε > 0 be arbitrary and (kn)n∈N be a strictly increasing sequence of naturalnumbers satisfying

∑∞n=1

< ε. Furthermore, we assume that in our constructions the followingconditions are fulfilled:

|α− α1| < ε and |α− αn| ≤ 1

2·kn·Ckn ·|||Hn|||kn+1kn+1

for every n ∈ N,

where Ckn are the constants from Lemma 3.6.7.

1. Then the sequence of diffeomorphisms fn = Hn Rαn+1 H−1n converges in the Diff∞(M)-

topology to a measure-preserving smooth diffeomorphism f , for which d∞ (f,Rα) < 3 · εholds.

2. Also the sequence of diffeomorphisms fn = Hn Rα H−1n ∈ Aα (M) converges to f in the

Diff∞(M)-topology. Hence f ∈ Aα (M).

Proof. 1. According to our construction it holds hn Rαn = Rαn hn and hence

fn−1 = Hn−1 Rαn H−1n−1 = Hn−1 Rαn hn h−1

n H−1n−1

= Hn−1 hn Rαn h−1n H−1

n−1 = Hn Rαn H−1n .

Applying Lemma 3.6.7 we obtain for every k, n ∈ N:

dk (fn, fn−1) = dk(Hn Rαn+1 H−1

n , Hn Rαn H−1n

)≤ Ck · |||Hn|||k+1

k+1 · |αn+1 − αn| .(3.6)

In section 3.2.1 we assumed |α− αn|n→∞−→ 0 monotonically. Using the triangle inequality

we obtain |αn+1 − αn| ≤ |αn+1 − α| + |α− αn| ≤ 2 · |α− αn| and therefore equation 3.6becomes:

dk (fn, fn−1) ≤ Ck · |||Hn|||k+1k+1 · 2 · |αn − α| .

By the assumptions of this Lemma it follows for every k ≤ kn:

dk (fn, fn−1) ≤ dkn (fn, fn−1) ≤ Ckn · |||Hn|||kn+1kn+1 ·2 ·

12 · kn · Ckn · |||Hn|||kn+1

≤ 1kn

In the next step we show that for arbitrary k ∈ N (fn)n∈N is a Cauchy sequence in Diffk (M),i.e. limn,m→∞ dk (fn, fm) = 0. For this purpose, we calculate:

limn→∞

dk (fn, fm) ≤ limn→∞

n∑i=m+1

dk (fi, fi−1) =∞∑

dk (fi, fi−1) . (3.8)

We consider the limit process m → ∞, i.e. we can assume k ≤ km and obtain fromequations 3.7 and 3.8:

limn,m→∞

dk (fn, fm) ≤ limm→∞

∞∑i=m+1

Since Diffk (M) is complete, the sequence (fn)n∈N converges consequently in Diffk (M) forevery k ∈ N. Thus, the sequence converges in Diff∞ (M) by definition.

Furthermore, we estimate:

d∞ (Rα, f) = d∞

(Rα, lim

n→∞fn

)≤ d∞ (Rα, Rα1) +

∞∑n=1

d∞ (fn, fn−1) , (3.9)

where we used the notation f0 = Rα1 .By explicit calculations we obtain dk (Rα, Rα1) = d0 (Rα, Rα1) = |α− α1| for every k ∈ N,hence

d∞ (Rα, Rα1) =∞∑k=1

|α− α1|2k · (1 + dk (Rα, Rα1))

≤ |α− α1| ·∞∑k=1

= |α− α1| .

Additionally it holds:

∞∑n=1

d∞ (fn, fn−1) =∞∑n=1

∞∑k=1

dk (fn, fn−1)2k · (1 + dk (fn, fn−1))

=∞∑n=1

(kn∑k=1

dk (fn, fn−1)2k · (1 + dk (fn, fn−1))

+∞∑

k=kn+1

dk (fn, fn−1)2k · (1 + dk (fn, fn−1))

As seen above dk (fn, fn−1) ≤ 1kn

for every k ≤ kn. Hereby, it follows further:

∞∑n=1

d∞ (fn, fn−1) ≤∞∑n=1

(1kn·kn∑k=1

+∞∑

k=kn+1

dk (fn, fn−1)2k · (1 + dk (fn, fn−1))

≤∞∑n=1

+∞∑n=1

∞∑k=kn+1

Because of∑∞k=kn+1

= 2−∑knk=0

)kn ≤ 1kn

we conclude:

∞∑n=1

d∞ (fn, fn−1) ≤∞∑n=1

+∞∑n=1

< 2 · ε.

Hence, using equation 3.9 we obtain the aimed estimate d∞ (f,Rα) < 3 · ε.

2. We have to show: fn → f in Diff∞ (M).For it we compute with the aid of Lemma 3.6.7 for every n ∈ N and k ≤ kn:

(fn, fn

)≤ dkn

(Hn Rαn+1 H−1

n , Hn Rα H−1n

)≤ Ckn · |||Hn|||kn+1

kn+1 · |αn+1 − α| ≤ Ckn · |||Hn|||kn+1kn+1 · |αn − α|

≤ Ckn · |||Hn|||kn+1kn+1 ·

12 · kn · Ckn · |||Hn|||kn+1

2 · kn≤ 1kn.

Fix some k ∈ N.Claim: ∀δ > 0 ∃N ∀n ≥ N : dk

(f, fn

)< δ, i.e. fn → f in Diffk (M).

Proof: Let δ > 0 be given. Since fn → f in Diff∞ (M) we have fn → f in Diffk (M) inparticular. Hence, there is n1 ∈ N, such that dk (f, fn) < δ

2 for every n ≥ n1. Because ofkn → ∞ we conclude the existence of n2 ∈ N, such that 1

kn< δ

2 for every n ≥ n2, as wellas the existence of n3 ∈ N, such that kn ≥ k for every n ≥ n3. Then we obtain for everyn ≥ max n1, n2, n3:

(f, fn

)≤ dk (f, fn) + dk

(fn, fn

2+ dkn

(fn, fn

)≤ δ

2= δ.

Hence, the claim is proven.

In the next step we show: limn→∞ d∞

(fn, f

)= 0. For this purpose, we examine:

(fn, fn

kn∑k=1

(fn, fn

)2k ·

(1 + dk

(fn, fn

)) +∞∑

k=kn+1

(fn, fn

)2k ·

(1 + dk

(fn, fn

))≤ 1kn·kn∑k=1

+∞∑

k=kn+1

12k≤ 1kn

Consequently limn→∞ d∞

(fn, fn

)= 0. With it we compute:

limn→∞

(f, fn

)= limn→∞

(limm→∞

fm, fn

)= limn→∞

limm→∞

(fm, fn

)≤ limn→∞

limm→∞

d∞ (fi, fi−1) + d∞

(fn, fn

= limn→∞

∞∑i=n+1

d∞ (fi, fi−1) + limn→∞

(fn, fn

As asserted we obtain: limn→∞ d∞

(fn, f

As announced we show that we can satisfy the conditions from Lemma 3.6.8 in our construc-tions:

Lemma 3.6.9. Let (kn)n∈N be a strictly incr. sequence of natural numbers with∑∞n=1

<∞and Ckn be the constants from Lemma 3.6.7. For any Liouvillean number α there exists asequence αn = pn

qnof rational numbers with 260n4 divides qn, such that our conjugation maps

Hn constructed in section 3.3.2 and 3.3.3 fulfil the following conditions:

1. For every n ∈ N:

|α− αn| <1

2 · kn · Ckn · |||Hn|||kn+1kn+1

|α− αn| <1

2n+1 · qn · |||Hn|||1

‖DHn−1‖0 <ln (qn)n

Proof. In Lemma 3.6.6 we saw |||Hn|||kn+1 ≤ Cn · q3·(m−1)2·(kn+1)·n·(n+1)n , where the constant

Cn was independent of qn. Thus, we can choose qn ≥ Cn for every n ∈ N. Hence, we obtain:|||Hn|||kn+1 ≤ q4·(m−1)2·(kn+1)·n·(n+1)

n .Besides qn ≥ Cn we keep the mentioned condition qn ≥ 64 · 260 ·n4 · (n− 1)11 · q(m−1)·(n−1)2+3

n−1 inmind. Furthermore, we can demand ‖DHn−1‖0 <

ln(qn)n from qn, because Hn−1 is independent

of qn. Since α is a Liouvillean number, we find a sequence of rational numbers αn = pnqn, pn, qn

relatively prime, under the above restrictions (formulated for qn) satisfying:

|α− αn| <|α− αn−1|

2n+1 · kn · Ckn · (260n4)1+4·(m−1)2·(kn+1)2·n·(n+1) · q1+4·(m−1)2·(kn+1)2·n·(n+1)n

Put qn := 260n4 · qn and pn := 260n4 · pn. Then we obtain:

|α− αn| <|α− αn−1|

2n+1 · kn · Ckn · q1+4·(m−1)2·(kn+1)2·n·(n+1)n

So we have |α− αn|n→∞−→ 0 monotonically. Because of |||Hn|||kn+1

kn+1 ≤ q4·(m−1)2·(kn+1)2·n·(n+1) thisyields: |α− αn| < 1

2n+1·qn·kn·Ckn ·|||Hn|||kn+1kn+1

. Thus, the first property of this Lemma is fulfilled.

Furthermore, we note kn ≥ 1 and Ckn ≥ 1 by Lemma 3.6.7. Thus qn · kn · Ckn ≥ qn. Moreover,|||Hn|||1 ≥ ‖Hn‖0 = 1, because Hn : S1 × [0, 1]m−1 → S1 × [0, 1]m−1 is a diffeomorphism. Hence|||Hn|||kn+1

kn+1 ≥ |||Hn|||1. Altogether, we conclude 2n+1·qn·kn·Ckn ·|||Hn|||kn+1kn+1 ≥ 2n+1·qn·|||Hn|||1

and so:|α− αn| <

12n+1 · qn · kn · Ckn · |||Hn|||kn+1

≤ 12n+1 · qn · |||Hn|||1

, (3.10)

i.e. we verified the second property.

Remark 3.6.10. Lemma 3.6.9 shows that the conditions of Lemma 3.6.8 are satisfied. There-fore, our sequence of constructed diffeomorphisms fn converges in the Diff∞(M)-topology to adiffeomorphism f ∈ Aα(M).

To apply Proposition 3.5.6 we need another result:

Lemma 3.6.11. Let (αn)n∈N be constructed as in Lemma 3.6.9. Then it holds for every n ∈ Nand for every m ≤ qn+1:

(f m, f mn

)≤ 1

Proof. In the proof of Lemma 3.6.8 we observed fi−1 = Hi Rαi H−1i for every i ∈ N. Hereby

and with the help of Lemma 3.6.7 we compute:

(f mi , f

mi−1

(Hi Rm·αi+1 H−1

i , Hi Rm·αi H−1i

)≤ |||Hi|||1 · m · 2 · |α− αi| .

Since m ≤ qn+1 ≤ qi we conclude for every i > n using equation 3.10 :

(f mi , f

mi−1

)≤ |||Hi|||1 · m · 2 · |α− αi| ≤ |||Hi|||1 · m · 2 ·

12i+1 · qi · |||Hi|||1

qi· 1

2i≤ 1

Thus, for every m ≤ qn+1 we get the claimed result:

(f m, f mn

)= limk→∞

(f mk , f

)≤ limk→∞

k∑i=n+1

(f mi , f

mi−1

∞∑i=n+1

Remark 3.6.12. Note that the sequence (mn)n∈N defined in section 3.4 meets the mentionedcondition mn ≤ qn+1 and hence Lemma 3.6.11 can be applied to it.

Concluding we have checked that all the assumptions of Proposition 3.5.6 are satisfied. Thus,this criterion guarantees that the constructed diffeomorphism f ∈ Aα(M) is weak mixing. Inaddition, for every ε > 0 we can choose the parameters by Lemma 3.6.8 in such a way, thatd∞ (f,Rα) < ε holds.

52 Construction of the measurable f -invariant Riemannian metric

3.7 Construction of the measurable f-invariant Riemannianmetric

Let ω0 denote the standard Riemannian metric on M = S1 × [0, 1]m−1. The following Lemmashows that the conjugation map hn = gn φn constructed in section 3.3 is an isometry withrespect to ω0 on the elements of the partial partition ζn.

Lemma 3.7.1. Let In ∈ ζn. Then hn|In is an isometry with respect to ω0.

This Lemma implies that h−1n |hn(In) is an isometry as well.

Proof. Let In,k ∈ ζn be a partition element on[k−1n·qn ,

kn·qn

]× [0, 1]m−1. This element In,k is

positioned in such a way that all the occurring maps ϕε,1,j and ϕ−1ε2,1,j

in our conjugation mapφn act as rotations on it. Thus, φn|In,k is an isometry and φn

)is equal to

[k − 1n · qn

n · q2n

+ ...+j((m−1)· k·(k−1)

2 )1 + 1

n · q(m−1)· k·(k−1)2 +1

− j(1)2

n · q(m−1)· k·(k−1)2 +2

− ...− j(k)2

n · q(m−1)· k·(k−1)2 +k+1

− j(1)3

n · q(m−1)· k·(k−1)2 +k+2

− ...− j(k)m + 1

n · q(m−1)· k·(k+1)2 +1

n5 · q(m−1)· k·(k+1)2 +1

k − 1n · qn

n · q2n

+ ...− j(k)m

n · q(m−1)· k·(k+1)2 +1

n5 · q(m−1)· k·(k+1)2 +1

[j((m−1)· k·(k−1)

2 +1)1

qn+ ...

j((m−1)· k·(k−1)

2 +k)1

(k+1)2

+ ...+j((m−1)· k·(k+1)

2 +1)2

q1+(m−1)· k·(k+1)

j((m−1)· k·(k+1)

2 +2)2

8n5 · q1+(m−1)· k·(k+1)2

n · [nqσn]+

8n9 · q1+(m−1)· k·(k+1)2

n · [nqσn],

j((m−1)· k·(k−1)

2 +1)1

qn+ ...+

j((m−1)· k·(k+1)

2 +2)2 + 1

8n5 · q1+(m−1)· k·(k+1)2

n · [nqσn]− 1

8n9 · q1+(m−1)· k·(k+1)2

n · [nqσn]

×m∏i=3

[j((m−1)· k·(k−1)

2 +(i−2)·k+1)1

qn+ ...+

j((m−1)· k·(k−1)

2 +(i−1)·k)1

1n4 · qkn

j((m−1)· k·(k−1)

2 +(i−2)·k+1)1

qn+ ...+

j((m−1)· k·(k−1)

2 +(i−1)·k)1 + 1

qkn− 1n4 · qkn

Then we have to examine the application of gn = gn·q

1+(m−1)· (k+1)·k2

n ,[n·qσn], 18n4 ,

. In particular,

we have εb·a = 1

8n4·[n·qσn]·n·q1+(m−1)· (k+1)·k

. Since 4 · ε = 12n4 < 1

n4 , gn acts as translation on

φn(In,k

In the following we construct the f -invariant measurable Riemannian metric. This construc-tion parallels the approach in [GK00], section 4.8.. For it we put ωn :=

(H−1n

)∗ω0. Each ωn

Construction of the measurable f -invariant Riemannian metric 53

is a smooth Riemannian metric because it is the pullback of a smooth metric via a C∞ (M)-diffeomorphism. Since R∗αn+1

ω0 = ω0 the metric ωn is fn-invariant:

f∗nωn =(Hn Rαn+1 H−1

)∗ (H−1n

)∗ω0 =

(H−1n

)∗R∗αn+1

H∗n(H−1n

)∗ω0 =

(H−1n

)∗R∗αn+1

=(H−1n

)∗ω0 = ωn.

With the succeeding Lemmas we show that the limit ω∞ := limn→∞ ωn exists µ-almost every-where and is the aimed f -invariant Riemannian metric.

Lemma 3.7.2. The sequence (ωn)n∈N converges µ-a.e. to a limit ω∞

Proof. For every N ∈ N we have for every k > 0:

ωN+k =(H−1N+k

)∗ω0 =

(h−1N+k ... h

−1N+1 H

)∗ω0 =

(H−1N

)∗ (h−1N+k ... h

−1N+1

)∗ω0.

Since the elements of the partition ζn cover M except a set of measure at most 4mn2 by Remark

3.3.3, Lemma 3.7.1 shows that ωN+k coincides with ωN =(H−1N

)∗ω0 on a set of measure at least

1−∑∞n=N+1

4mn2 . As this measure approaches 1 for N →∞, the sequence (ωn)n∈N converges on

a set of full measure.

Lemma 3.7.3. The limit ω∞ is a measurable Riemannian metric.

Proof. The limit ω∞ is a measurable map because it is the pointwise limit of the smooth metricsωn, which in particular are measurable. By the same reasoning ω∞|p is symmetric for µ-almostevery p ∈M . Furthermore, ω∞ is positive definite because ωn is positive definite for every n ∈ Nand ω∞ coincides with ωN on T1M ⊗ T1M minus a set of measure at most

∑∞n=N+1

4mn2 . Since

this is true for every N ∈ N, ω∞ is positive definite on a set of full measure.

Remark 3.7.4. In the proof of the subsequent Lemma we will need Egoroff’s theorem (forexample [Ha65], §21, Theorem A): Let (N, d) denote a separable metric space. Given a sequence(ϕn)n∈N of N -valued measurable functions on a measure space (X,Σ, µ) and a measurable subsetA ⊆ X, µ (A) < ∞, such that (ϕn)n∈N converges µ-a.e. on A to a limit function ϕ. Then forevery ε > 0 there exists a measurable subset B ⊂ A such that µ (B) < ε and (ϕn)n∈N convergesto ϕ uniformly on A \B.

Lemma 3.7.5. ω∞ is f -invariant, i.e. f∗ω∞ = ω∞ µ-a.e..

Proof. By Lemma 3.7.2 the sequence (ωn)n∈N converges in the C∞-topology pointwise almosteverywhere. Hence, we obtain using Egoroff’s theorem: For every δ > 0 there is a set Cδ ⊆ Msuch that µ (M \ Cδ) < δ and the convergence ωn → ω∞ is uniform on Cδ.The function f was constructed as the limit of the sequence (fn)n∈N in the C∞-topology. Thus,fn := f−1

n f → id in the C∞-topology. Since M is compact, this convergence is uniform, too.Furthermore, the smoothness of f implies f∗ω∞ = f∗ limn→∞ ωn = limn→∞ f∗ωn. Therewith wecompute on Cδ: f∗ω∞ = limn→∞

((fnfn

)∗ωn

)= limn→∞

(f∗nf

∗nωn

)= limn→∞ f∗nωn = ω∞,

where we used the uniform convergence on Cδ in the last step. As this holds on every set Cδwith δ > 0, it also holds on the set

⋃δ>0 Cδ. This is a set of full measure and therefore the claim

follows.

Hence, the aimed f -invariant measurable Riemannian metric ω∞ is constructed and thusProposition 3.2.1 is proven.

Chapter 4

Weak mixing and uniquely ergodicdiffeomorphisms preserving ameasurable Riemannian metric witharbitrary Liouvillean rotationnumber on Tm

4.1 Introduction

In [FH77] Fathi and Herman proved that on any compact and connected smooth boundarylessmanifold M of dimension m ≥ 2 admitting a locally free circle action S = Stt∈S1 preserving asmooth volume ν the set of strictly ergodic C∞-diffeomorphisms is a dense Gδ-set in the closure ofconjugates A (M) := h Rt h−1 : h ∈ Diff∞ (M,ν) , t ∈ S1

in the C∞ (M)-topology. Weaim at such a result in the restricted spaces Aα (M) := h Rα h−1 : h ∈ Diff∞ (M,ν)

.In [FSW07] for every Liouville number α this restricted space Aα (Tm) on the torus Tm is con-sidered: The authors construct a smooth diffeomorphism that is uniquely ergodic and measure-theoretically isomorphic to the rigid rotation Rα on S1. In particular, their diffeomorphism isnot weak mixing. In this chapter we construct C∞-diffeomorphisms in Aα (Tm) that are strictlyergodic and weak mixing. Hereby, we will prove the following theorem:

Theorem B. Let m ≥ 2. If α ∈ R is Liouville, the set of volume-preserving smooth diffeo-morphisms, that are weak mixing and strictly ergodic, is a dense Gδ-set in the C∞-topology inAα (Tm).

Furthermore, we are able to construct such a diffeomorphism in a way that it admits aninvariant measurable Riemannian metric. The first construction of a weak mixing diffeomorphismpreserving a measurable Riemannian metric was exhibited in [GK00]. In chapter 3 this finding isextended by proving that on every smooth compact connected manifold M of dimension m ≥ 2with a non-trivial circle action preserving a smooth volume and for every Liovillean numberα ∈ S1 the set of weak mixing diffeomorphisms admitting an invariant measurable Riemannianmetric is dense in the C∞-topology in Aα (M). Thus, we can supplement Theorem A in the case

56 Preliminaries

of the torus Tm in the following way:

Theorem 4.1.1. Let m ≥ 2. If α ∈ R is Liouville, the set of volume-preserving smooth diffeo-morphisms, that are strictly ergodic, weak mixing and preserve a measurable Riemannian metric,is dense in the C∞-topology in Aα (Tm).

4.2 Preliminaries

4.2.1 Definition of strict ergodicity

Definition 4.2.1. Let X be a compact metric space and T : X → X be a continuous transfor-mation. If there is a unique T -invariant probability measure, then T is uniquely ergodic.

Recall that a map is called minimal if every orbit is dense. Hereby, we define the followinggeneralisation:

Definition 4.2.2. Let X be a compact metric space and T : X → X be a continuous transfor-mation. T is called strictly ergodic if it is minimal and uniquely ergodic.

An equivalent characterisation is that T is uniquely ergodic and the invariant probabilitymeasure ν on X has full topological support (see [Wa75], Theorem 5.16), i.e. supp(ν) = X, wherethe support of a measure is defined to be the set of points for which every open neighbourhoodhas positive measure.

4.2.2 Reduction to Proposition 4.2.3

We consider the torus Tm equipped with Lebesgue measure µ and the standard circle actionR = Rαα∈S1 , where the map Rα is given by Rα (θ, r1, ..., rm−1) = (θ + α, r1, ..., rm−1). In thissetting we will prove the following result:

Proposition 4.2.3. For every Liouvillean number α there are a sequence (αn)n∈N of rationalnumbers αn = pn

qnsatisfying limn→∞ |α− αn| = 0 monotonically and a sequence (hn)n∈N of

measure-preserving diffeomorphisms satisfying hnR 1qn

= R 1qnhn such that the diffeomorphisms

fn = Hn Rαn+1 H−1n with Hn = h1 h2 ... hn converge in the Diff∞ (Tm)-topology and

the diffeomorphism f = limn→∞ fn is strictly ergodic, weak mixing, has an invariant measurableRiemannian metric and satisfies f ∈ Aα (Tm).Furthermore, for every ε > 0 the parameters in the construction can be chosen in such a waythat d∞ (f,Rα) < ε.

In fact, Theorem 4.1.1 can be deduced directly from this result:Because of Aα (Tm) = h Rα h−1 : h ∈ Diff∞ (Tm, µ)

the denseness follows if we showthat for every h ∈ Diff∞ (Tm, µ) and every ε > 0 there is a strictly ergodic, weak mixing dif-feomorphism f preserving a measurable Riemannian metric such that d∞

(f , h Rα h−1

)< ε.

For this purpose, let h ∈ Diff∞ (Tm, µ) and ε > 0 be arbitrary. By [Om74], p. 3, resp. [KM97],Theorem 43.1., Diff∞ (Tm) is a Lie group. In particular, the conjugating map g 7→ h g h−1

is continuous with respect to the metric d∞. Continuity in the point Rα yields the existence ofδ > 0, such that d∞ (g,Rα) < δ implies d∞

(h g h−1, h Rα h−1

)< ε. By Proposition 4.2.3

we can find a strictly ergodic, weak mixing diffeomorphism f with f -invariant measurable Rie-mannian metric ω and d∞(f,Rα) < δ. Hence f := h f h−1 satisfies d∞

(f , h Rα h−1

)< ε.

Preliminaries 57

Note that f is strictly ergodic, weak mixing and preserves the measurable metric ω :=(h−1

)∗ω.

Moreover, we can prove Theorem B with the aid of Proposition 4.2.3:For ρ ∈ C (Tm,R) and ε > 0 we define the open set Uρ,ε byT ∈ Aα (Tm) : ∃N ∈ N :

∣∣∣∣∣∣ 1nn−1∑j=0

ρ(T j(x)

)−∫ρ dµ

∣∣∣∣∣∣ < ε for every x ∈ Tm and n ≥ N

Let Ξ = ρii∈N be a countable set of continuous functions that is dense in C (Tm,R). ByOxtoby’s ergodic theorem (see Lemma 4.3.1)

⋂j∈N

⋂l∈N Uρj , 1l coincides with the set of uniquely

ergodic diffeomorphisms with unique invariant probability measure the Lebesgue measure µ.Since the Lebesgue measure µ has full topological support, these diffeomorphisms are strictlyergodic.By Corollary A we know that the set of weak mixing diffeomorphisms is a dense Gδ-set inAα (Tm). Altogether, we obtain that the set of weak mixing and strictly ergodic diffeomorphismscan be written as a countable intersection of open sets. Since the set of weak mixing and strictlyergodic diffeomorphisms is dense in Aα (Tm) by the above observations, it is a dense Gδ-set.

4.2.3 Outline of the proof

First of all we will deduce a criterion for unique ergodicity. In order to apply it we have to gaincontrol over most of the orbit

Hn Riαn+1

(x)i∈N

for every x ∈ Tm. Therefore, we introducea map Dψn,γn and so-called trapping regions which guarantee that a large portion of the orbitRiαn+1

(x)i∈N

is contained in these regions for every point x ∈ Tm. Under the conjugatingdiffeomorphism Hn each trapping region is mapped into a set of small diameter and these setsare equally met by the iterates of the orbit

Hn Riαn+1

(x)i∈N

for every x ∈ Tm. Finally, thiswill enable us to prove uniform convergence of Birkoff sums in section 4.8.2.As in chapter 3 we define two sequences of partial partitions which converge to the decompositioninto points. Again, the first type of partial partition, called ηn, will satisfy the requirements inthe proof of the weak mixing-property. On the partition elements of the even finer second type,called ζn, the conjugation map hn will act as an isometry, and this will enable us to constructan invariant measurable Riemannian metric. Afterwards, these conjugating diffeomorphismshn = gn φn Dψn,γn , which are composed of the mentioned trapping map Dψn,γn and two step-by-step defined smooth measure-preserving diffeomorphisms, will be constructed. These mapsgn and φn are constructed analogously to the previous chapter.Unfortunately, the partition elements of ηn are not (γ, δ, ε)-distributed under the resulting diffeo-morphism Φn = φn Dψn,γn Rmnαn+1

D−1ψn,γn

φ−1n with a specific sequence (mn)n∈N of natural

numbers (see Remark 4.5.6). Thus, we have to introduce the similar notion of a (γ, ε)-distributionin section 4.5. Then we can establish a modified criterion for the weak mixing property in section4.6.In section 4.7 we will show convergence of the sequence (fn)n∈N in Aα (Tm) for a given Liouvillenumber α by the same approach as in section 3.6. Furthermore, we will see at the end of section4.7 that the criterion for weak mixing applies to the obtained diffeomorphism f = limn→∞ fn.Finally, we can use the same techniques as in chapter 3.7 in order to construct the aimed f -invariant measurable Riemannian metric.

58 Criterion for unique ergodicity

4.3 Criterion for unique ergodicity

By Oxtoby’s Ergodic Theorem unique ergodicity has the following strong convergence property(see for example [Wa75], Theorem 5.17):

Lemma 4.3.1. Let X be a compact metric space and T : X → X be a continuous transformation.The following are equivalent:

1. T is uniquely ergodic.

2. For each f ∈ C (X; R) there exists a constant c (f) such that

n−1∑j=0

f(T j (x)

)→ c (f) (4.1)

uniformly for x ∈ X.

Remark 4.3.2. If µ is the unique T -invariant probability measure, then the constant c (f) inequation 4.1 is

∫fdµ.

Proof. At first we prove that 1. implies 2.. Let µ be the unique T -invariant probability measure.If 2. is true, the constant c (f) must be equal to

∫fdµ. Hence, the convergence in 2. means that

for every f ∈ C (X; R) and every ε > 0 there is N ∈ N such that for every n ≥ N and for everyx ∈M we have: ∣∣∣∣∣∣ 1n

n−1∑j=0

f(T j (x)

)−∫fdµ

∣∣∣∣∣∣ < ε.

Suppose that 2. does not hold. Then there is f ∈ C (X; R), ε > 0 such that there exists asequence nk →∞ monotonically and points xk ∈M satisfying:∣∣∣∣∣∣ 1

nk−1∑j=0

f(T j (xk)

)−∫fdµ

∣∣∣∣∣∣ ≥ ε. (4.2)

In this connection, we define the probability measure µk := 1nk

∑nk−1j=0 T j∗ δxk , where δx is the

Dirac measure centered on point x. Now we can write equation 4.2 in the following form:∣∣∫ fdµk − ∫ fdµ∣∣ ≥ ε. Since the space of probability measures M (X) is weak*-compact (seee.g. [Wa75], Theorem 5.13) and µk ∈M (X) there is a weak*-convergent subsequence with limitν. Because of∣∣∣∣∫ g T dν −

∫g dν

∣∣∣∣= limk→∞

∣∣∣∣∣∣∫

(g T − g) d

nk−1∑j=0

T j∗ δxk

∣∣∣∣∣∣ = limk→∞

∣∣∣∣∣∣ 1nk

∫ nk−1∑j=0

(g T j+1 − g T j

)dδxk

∣∣∣∣∣∣= limk→∞

∣∣∣∣ 1nk

∫(g Tnk − g) dδxk

∣∣∣∣ ≤ limk→∞

2 · ‖g‖0nk

for every g ∈ C (X; R) this limit ν is T -invariant. It satisfies∣∣∫ fdν − ∫ fdµ∣∣ ≥ ε. Hence, ν 6= µ

contradicting unique ergodicity.

Criterion for unique ergodicity 59

Now we show that 2. implies 1.. Suppose that µ, ν are T -invariant probability measures. Inte-grating the expression 4.1 we obtain by the fact that the convergence is uniform∫

fdµ = limn→∞

n−1∑j=0

∫f T jdµ =

∫limn→∞

n−1∑j=0

f T jdµ =∫c (f) dµ = c (f)

and by the same argument∫fdν = c (f). Therefore, we have∫fdµ =

∫fdν for every f ∈ C (X; R)

and thus µ = ν by the Riesz Represantation Theorem.

Next, we find the subsequent criterion for unique ergodicity, which is the same as in [FSW07],Lemma 5:

Lemma 4.3.3. Consider a compact metric space (M,d), a Borel probability measure µ on Mand a dense set Φ ⊆ C (M ; R) of continuous functions. Let (qn)n∈N be an increasing sequenceof natural numbers and (fn)n∈N be a sequence of continuous transformations, which convergesuniformly to a map f . Suppose that

d(qn+1) (fn, f) := maxx∈M

maxi=0,...,qn+1−1

d(f in (x) , f i (x)

) n→∞−→ 0

and for each ϕ ∈ Φ

qn+1−1∑j=0

ϕ(f jn (x)

) n→∞−→ ∫ϕ dµ uniformly

Then f is uniquely ergodic with unique invariant measure µ.

Proof. Since every continuous function on the compact metric space is uniformly continuous,we have for every continuous function ϕ ∈ Φ:

∣∣ϕ (f i (x))− ϕ

(f in (x)

)∣∣ n→∞−→ 0 uniformly for

i = 0, ..., qn+1 − 1. Thus, we get:∥∥∥ 1qn+1

∑qn+1−1j=0 ϕ f j − 1

∑qn+1−1j=0 ϕ f jn

∥∥∥0

n→∞−→ 0. Hence,1

∑qn+1−1j=0 ϕ

(f j (x)

) n→∞−→ ∫ϕ dµ uniformly.

Assume that there is another ergodic invariant measure ξ. By the Birkhoff Ergodic Theorem forevery ϕi ∈ Φ there is a set Ai with ξ (Ai) = 1, such that for every x ∈ Ai:

limn→∞

n−1∑j=0

ϕi(f j (x)

)=∫ϕi dξ.

Then A := ∩i∈NAi satisfies ξ (A) = 1 and for every x ∈ A, ϕ ∈ Φ we have

∫ϕ dξ = lim

n→∞

n−1∑j=0

ϕ(f j (x)

)= limn→∞

qn+1−1∑j=0

ϕ(f j (x)

)=∫ϕ dµ.

By an approximation argument this equality holds true for every ϕ ∈ C (M ; R). With the aid ofthe Riesz Representation Theorem we conclude µ = ξ and obtain a contradiction. Hence, f isuniquely ergodic with unique invariant measure µ.

We fix an arbitrary countable set Ξ = ρ1, ρ2, ... of Lipschitz continuous functions ρi : Tm → R,that is dense in C(Tm,R). Since C(Tm,R) is separable and Lipschitz continuous functions aredense in C(Tm,R), this is possible.

4.4.1 The trapping map

To prove unique ergodicity we have to gain control over most of the orbitHn Riαn+1

(x)i∈N

for every x ∈ Tm. For this purpose, we use for every n ∈ N a smooth map ψn : [0, 1] → Rsatisfying

• ψn is nondecreasing on[0, 1

]and nonincreasing on

[12 , 1].

• ψn is equal to kn4 on

[kn2 + 1

n4 ,k+1n2 − 1

]for 0 ≤ k ≤

⌋− 1 and ψn is equal to k

on[n2−k−1n2 + 1

n4 ,n2−kn2 − 1

]for 0 ≤ k ≤

⌋− 1. On

[⌊n22

⌋n2 ,

n2−⌊n22

]it is put to(⌊

⌋− 1)· 1n4 .

Figure 4.1: Qualitative shape of ψn in case of n = 3

Therewith, we define the map Dψn : [0, 1]m → Rm by: (θ, r1, ..., rm−1) is mapped to(θ, r1 +

+ ...+1

q3+n−1n

)· ψn (θ) , ..., rm−1 +

+ ...+1

q3+n−1n

)· ψn (θ)

Using the maps Cγn (θ, r1, ...rm−1) = (γn · θ, r1, ..., rm−1) for γn ∈ Z we construct the measure-preserving map Dψn,γn := C−1

γn Dψn Cγn :[0, 1

]× Tm−1 →

]× Tm−1. Since this map

coincides with the identity in a neighbourhood of the boundary of the sector on the θ-axis, we canextend it to a smooth map Dψn,γn : Tm → Tm using the description Dψn,γn R l

γn= R l

γnDψn,γn

for any l ∈ Z.In our construction we use

γn = n · q2+3·(m−1)+4·(m−1)+...+(3+n−1)·(m−1)n = n · q2+(m−1)·(3·n+

n·(n−1)2 )

4.4.2 The trapping regionsIn this subsection we define the set of so-called trapping regions. For this purpose, we consideron[lqn

+ kn·qn ,

+ k+1n·qn

]×Tm−1 for l ∈ Z and k = 0, ..., n− 1 the following sets S

l,k,j(1)1 , ~j2,..., ~jm

Sl,k,j

(1)1 , ~j2,..., ~jm

n · qn+

n · q2n

+t(1)1

n · q3n

+ ...+t((m−1)·(3·k+

k·(k−1)2 ))

n · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)n

n · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+1n

+ ...+j

(3+k)2

n · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+3+kn

n · q2+3(m−1)+...+(3+k−1)(m−1)+3+k+1n

+ ...+j

(3+k)3

n · q2+3(m−1)+...+(3+k−1)(m−1)+2·(3+k)n

(3+k)m

n · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)n

+t((m−1)·(3·k+

k·(k−1)2 )+1)

n · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)+1n

+ ...+t((m−1)·(3·(n−1)+

n·(n−1)2 −k))

1n4 · γn

n · qn+ ...+

t((m−1)·(3·(n−1)+

n·(n−1)2 −k))

1 + 1γn

− 1n4 · γn

×m∏i=2

[t(1)i

qn+ ...+

t(3+k)i

4 · n4 · q3+kn

,t(1)i

qn+ ...+

t(3+k)i + 1q3+kn

− 14 · n4 · q3+k

where the union is taken over all t(1)i ∈ Z,

⌈qn4n4

⌉≤ t(1)

i ≤ qn−⌈qn4n4

⌉−1 for i = 2, ...,m as well as

t(l)i ∈ Z, 0 ≤ t(l)i ≤ qn− 1, for l = 2, ..., 3 +k and i = 2, ...,m as well as t(j)1 ∈ Z, 0 ≤ t(j)1 ≤ qn− 1,

for j = 1, ..., (m − 1) ·(

3 · (n− 1) + n·(n−1)2 − k

)apart from t

((m−1)·(3·k+k·(k−1)

2 )+1)1 satisfying⌈

⌉≤ t((m−1)·(3·k+

k·(k−1)2 )+1)

1 ≤ qn −⌈qn4n4

⌉− 1.

Then the set of trapping regions consists of all sets D−1ψn,γn

(Sl,k,j

(1)1 ,~j2,..., ~jm

), where all j(1)

i ∈ Z

satisfy⌈qn4n4

⌉≤ j(1)

i ≤ qn −⌈qn4n4

⌉− 1 and j(t)

i ∈ Z, 0 ≤ j(t)i ≤ qn − 1 for t = 2, ..., 3 + k.

Remark 4.4.1. Let x = (θ, r1, ..., rm−1) ∈ Tm be arbitrary. By the construction of the mapDψn , for every coordinate ri there are at most four sections

[kn2 + 1

n4 ,k+1n2 − 1

]on the domain

[0, 1] such that ri /∈ D−1ψn

([kn2 + 1

n4 ,k+1n2 − 1

14n4 , 1− 1

]m−1).

We have to bear the gaps of our trapping region in the ~r-coordinates in mind. Therefore, we notethat

(1 + 1

+ 1q4n

+ ...+ 1

q3+k−1n

)· 1n4 is a multiple of 1

by our assumptions on qn in Lemma

4.7.6 and hence there are at most four further sections[

kn2γn

+ 1n4γn

, k+1n2γn

− 1n4γn

]on[0, 1

]such that ri does not belong to the image of[

n2 · γn+

1n4 · γn

,k + 1n2 · γn

− 1n4 · γn

m∏i=2

[t(1)i

qn+ ...+

t(3+k)i

4n4 · q3+kn

,t(1)i

qn+ ...+

t(3+k)i + 1q3+kn

− 14n4 · q3+k

under D−1ψn,γn

. Since i · αn+1i=0,...,qn+1−1 is equidistributed, the number of iterates i, such that

the orbit Riαn+1(x) meets an arbitrary trapping region D−1

ψn,γn

(Sl,k,j

(1)1 ,~j2,..., ~jm

), is at least

q3(m−1)+...+(3+n−1)(m−1)−(m−1)(3+k)−1n ·

(qn − 2

⌈ qn4n4

⌉)·(n2 − 8(m− 1)

)·(⌊

qn+1 ·1− 2

n2 · γn

⌋− 2).

For n sufficiently large we estimate this quantity by⌊qn+1 ·

1− 8mn2

n·q(m−1)·(3+k)n ·q2

4.4.3 Sequences of partial partitions

In this subsection we define the two announced sequences of partial partitions (ηn)n∈N and(ζn)n∈N of M = Tm.

]×Tm−1. For this purpose, we

divide the fundamental sector into n sections:

• On[

kn·qn ,

k+1n·qn

]×Tm−1 in case of k ∈ N and 0 ≤ k ≤ n−2 the partial partition ηn consists

of all multidimensional intervals of the following form:[k

n · qn+

n · q2n

+ ...+j(1+(m−1)·(3·(k+1)+

k·(k+1)2 ))

nq2+(m−1)·(3·(k+1)+

k·(k+1)2 )

10 · n5 · q2+(m−1)·(3·(k+1)+k·(k+1)

n · qn+j

+ ...+j(1+(m−1)·(3·(k+1)+

k·(k+1)2 ))

nq2+(m−1)·(3(k+1)+

k·(k+1)2 )

10n5 · q2+(m−1)·(3(k+1)+k·(k+1)

×m∏i=2

qn+ ...+

j(3+k+1)i

q3+k+1n

26 · n4 · q3+k+1n

qn+ ...+

j(3+k+1)i + 1q3+k+1n

− 126 · n4 · q3+k+1

where j(l)1 ∈ Z,

⌉≤ j(l)

1 ≤ qn−⌈qn

⌉−1 for l = 1, ..., 1+(m−1)·

(3 · (k + 1) + k·(k+1)

)as well as j(l)

i ∈ Z and⌈qn

⌉≤ j(l)

i ≤ qn−⌈qn

⌉−1 for i = 2, ...,m and l = 1, ..., 3+k+1.

• On[n−1n·qn ,

]× Tm−1 there are no elements of the partial partition ηn.

]× Tm−1 is extended to

a partial partition of Tm.

]× Tm−1 initially and, therefore, divide this sector into n sections:

kn·qn ,

k+1n·qn

]× Tm−1 in case of k ∈ N and 0 ≤ k ≤ n − 1 the partial partition ζn consists of

all sets D−1ψn,γn

(In), where In is a multidimensional interval of the following form:[

n · qn+

n · q2n

+ ...+j(1+(m−1)·(3·n+

n·(n−1)2 ))

n2 · γn+

1n4 · γn

n · qn+

n · q2n

+ ...+j(1+(m−1)·(3·n+

n·(n−1)2 ))

s+ 1n2 · γn

− 1n4 · γn

qn+ ...+

j(3+k+1)2

q3+k+1n

+ ...+j(2+(m−1)·(3·(k+1)+

k·(k+1)2 ))

q2+(m−1)·(3·(k+1)+

k·(k+1)2 )

j(3+(m−1)·(3·(k+1)+

k·(k+1)2 ))

8n5 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

8n9 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

qn+ ...+

j(3+(m−1)·(3·(k+1)+

k·(k+1)2 ))

8n5 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

8n9 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

×m∏i=3

qn+ ...+

j(3+k)i

n4 · q3+kn

qn+ ...+

j(3+k)i + 1q3+kn

− 1n4 · q3+k

⌈qnn4

⌉≤ j

(l)1 ≤ qn −

⌈qnn4

⌉− 1 for l = 1, ..., 1 + (m− 1) ·

(3 · n+ n·(n−1)

j(l)2 ∈ Z and

⌈qnn4

⌉≤ j

(l)2 ≤ qn −

⌈qnn4

⌉− 1 for l = 1, ..., 2 + (m− 1) ·

(3 · (k + 1) + k·(k+1)

j(3+(m−1)·(3(k+1)+

k(k+1)2 ))

2 ∈ Z, 8n · [nqσn] ≤ j(3+(m−1)·(3(k+1)+k(k+1)

2 ))2 ≤ 8n5 · [nqσn]−8n · [nqσn]−1,

j(l)i ∈ Z and

⌈qnn4

⌉≤ j(l)

i ≤ qn −⌈qnn4

⌉− 1 for i = 3, ...,m and l = 1, .., 3 + k as well as s ∈ N and

0 ≤ s ≤ n2 − 1.

Remark 4.4.3. For every n the partial partition ζn consists of disjoint sets, covers a set ofmeasure at least 1− 5·m

n2 in case of n ≥ 5 and the sequence (ζn)n∈N converges to the decompositioninto points.

Remark 4.4.4. Note that Dψn,γn acts as an isometry on all partition elements D−1ψn,γn

(In)∈ ζn.

Let a, b ∈ Z and ε ∈(0, 1

]such that 1

ε ∈ Z. Moreover, we consider δ > 0 such that1δ ∈ Z and a·b·δ

ε ∈ Z. In subsection 3.3.2 we constructed the measure-preserving smooth dif-

feomorphism ga,b,ε,δ :[0, 1

]× [0, 1]m−1 → gb

([0, 1

]× [0, 1]m−1

), which was equal to gb in

a neighbourhood of the boundary. Hence, we can extend it smoothly to a diffeomorphismga,b,ε,δ :

]× Tm−1 → gb

([0, 1

]× Tm−1

As above we construct the smooth measure-preserving diffeomorphism gn on the fundamentalsector

]× Tm−1 initially and for this divide it into n sections:

kn·qn ,

k+1n·qn

]× Tm−1 in case of k ∈ Z and 0 ≤ k ≤ n− 1:

gn = gn·q

2+(m−1)·(3·(k+1)+ (k+1)·k2 )

n ,[n·qσn], 18n4 ,

Since gn coincides with the map g[n·qσn] in a neighbourhood of the boundary of the different sec-tions on the θ-axis, this yields a smooth map and we can extend it to a smooth measure-preservingdiffeomorphism on Tm using the description gn R l

qn= R l

qn gn for l ∈ Z. Furthermore, we

note that the subsequent constructions are done in such a way that 260n4 divides qn (see Lemma4.7.6) and so the assumption a·b·δ

ε = a·b4 ∈ Z is satisfied.

Indeed, this map gn satisfies the following aimed property:

Lemma 4.4.5. For every element In ∈ ηn we have: gn(In

)= g[nqσn]

Proof. We consider a partition element In,k ∈ ηn on[

kn·qn ,

k+1n·qn

]× [0, 1]m−1 in case of k ∈ Z and

0 ≤ k ≤ n− 2 and want to examine the effect of gn = gn·q

2+(m−1)·(3·(k+1)+ (k+1)·k2 )

n ,[n·qσn], 18n4 ,

it.In the r1-coordinate we use that there is u ∈ Z such that

126n4 · q3+k+1

= u · ε

b · a= u · 1

8n4 · [nqσn] · n · q2+(m−1)·(3·(k+1)+(k+1)·k

where we exploit the fact that 260n4 divides qn by Lemma 4.7.6. Also with respect to theθ-coordinate the boundary of this element lies in the domain, where ga,b,ε,δ = g[nqσn], because

110·n4 < ε = 1

8·n4 .

We recall Lemma 3.3.6

(12 , ...,

)∈ Rm on [2ε, 1− 2ε]m and coincides with the identity outside of [ε, 1− ε]m.

Let λ, µ ∈ N and δ > 0 with 1δ ∈ N as well as 1

δ divides µ. We construct a diffeomorphismψµ,δ,i,j,ε2 as in section 3.3.3; then we build the following smooth measure-preserving diffeomor-phism φλ,ε,i,j,µ,δ,ε2 :

]× [0, 1]m−1 →

]× [0, 1]m−1:

φλ,ε,i,j,µ,δ,ε2 = C−1λ ψµ,δ,i,j,ε2 ϕε,i,j Cλ.

Afterwards, φλ,ε,i,j,µ,δ,ε2 is extended to a diffeomorphism on Tm by the description

φλ,ε,i,j,µ,δ,ε2

λ, x2 + k2, ..., xm + km

λ, k2, ..., km

)+ φλ,ε,i,j,µ,δ,ε2 (x1, x2, ..., xm)

for ki ∈ Z.For the sake of convenience we will use the notation φ

(j)λ,µ = φλ, 1

60n4 ,1,j,µ,1

10n4 ,1

22n4. With this

we define the diffeomorphism φn on the fundamental sector: On[

kn·qn ,

k+1n·qn

]× Tm−1 in case of

k ∈ N and 0 ≤ k ≤ n− 1

n·q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)+(3+k)·(m−2)n ,q3+k

... φ(2)

n·q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)n ,q3+k

=φ(m)

n·q2+(m−1)·(3k+ k·(k−1)

2 )+(3+k)·(m−2)n ,q3+k

... φ(2)

n·q2+(m−1)·(3k+ k·(k−1)

2 )n ,q3+k

This is a smooth map because φn coincides with the identity in a neighbourhood of the differentsections.Now we extend φn to a diffeomorphism on Tm using the description φn R 1

qn= R 1

qn φn.

(γ, ε)-distribution 65

4.5 (γ, ε)-distribution

We introduce the central notion in the proof of the criterion for weak mixing deduced in the nextsection:

Definition 4.5.1. Let Φ : Tm → Tm be a diffeomorphism and J be a (m− 1)-dimensionalinterval in Tm−1. We say that an element I of a partial partition is (γ, ε)-distributed on J underΦ, if the following properties are satisfied:

• [c, c+ γ′]× J ⊆ Φ(I)⊆ [c, c+ γ]× Tm−1 for some c ∈ S1 and 0 < γ′ ≤ γ ≤ γ.

• For every (m− 1)-dimensional interval J ⊆ J it holds:∣∣∣∣∣∣µ(I ∩ Φ−1

(S1 × J

))µ(I) −

µ(m−1)(J)

µ(m−1) (J)

∣∣∣∣∣∣ ≤ ε ·µ(m−1)

µ(m−1) (J),

at which µ(m−1) is the Lebesgue measure on Tm−1.

Remark 4.5.2. Analogous to Remark 3.4.2 we will call the second property “almost uniformdistribution” of I on J . In the following we will often write it in the form∣∣∣µ(I ∩ Φ−1

(S1 × J

))· µ(m−1) (J)− µ

(I)· µ(m−1)

(J)∣∣∣ ≤ ε · µ(I) · µ(m−1)

As in section 3.4 we define the sequence of natural numbers (mn)n∈N:

∣∣∣∣m · pn+1

qn+1− 1n · qn

∣∣∣∣ ≤ 260 · (n+ 1)4

m ≤ qn+1 : m ∈ N, inf

∣∣∣∣m · qn · pn+1

qn+1− 1n

∣∣∣∣ ≤ 260 · (n+ 1)4 · qnqn+1

Lemma 4.5.3. The set

m ≤ qn+1 : m ∈ N, infk∈Z

∣∣∣m · qn·pn+1qn+1

− 1n + k

∣∣∣ ≤ 260(n+1)4·qnqn+1

Proof. In Lemma 4.7.6 we will construct the sequence αn = pnqn

in such a way that qn = 260n4 ·qnand pn = 260n4 · pn with pn, qn relatively prime. Thus, the statement is proven in Lemma3.4.3

an =(mn ·

qn+1− 1n · qn

By the above construction of mn it holds: |an| ≤ 260·(n+1)4

qn+1. Below, we will see that it is possible

to choose qn+1 ≥ 260 · (n+ 1)4 · 320 · n9 · |||ψn|||1 · γ2n. Thus, we get:

|an| ≤1

320 · n9 · |||ψn|||1 · γ2n

66 (γ, ε)-distribution

Lemma 4.5.5. We consider the (m− 1)-dimensional interval J :=[

1n4 , 1− 1

]m−1 ⊂ Tm−1 aswell as the diffeomorphism Φn := φn Dψn,γn Rmnαn+1

D−1ψn,γn

φ−1n with the conjugating maps

φn defined in section 4.4.5. Then the elements of the partition ηn are(

1n·qmn

)-distributed on

J under Φn for n ≥ 3√

4 · (m− 1).

Proof. We consider a partition element In,k on[

kn·qn ,

k+1n·qn

]× [0, 1]m−1. When applying the map

φ−1n we observe that this element is positioned in such a way that all the occurring maps ϕ−1

ε,1,j

and ϕε2,1,j act as the respective rotations. Then we compute φ−1n

n · qn+

n · q2n

+ ...+j(1+(m−1)·(3k+

(k−1)·k2 ))

n · q2+(m−1)·(3k+(k−1)·k

2 )+1n

(3+k)2

n · q2+(m−1)·(3k+(k−1)·k

2 )+3+kn

n · q2+(m−1)·(3k+(k−1)·k

2 )+3+k+1n

(3+k)m

n · q2+(m−1)·(3(k+1)+(k+1)·k

10 · n5 · q2+(m−1)·(3(k+1)+(k+1)·k

n · qn+

n · q2n

+ ...+j(1+(m−1)·(3k+

(k−1)·k2 ))

n · q2+(m−1)·(3k+(k−1)·k

2 )+1n

(3+k)2

n · q2+(m−1)·(3k+(k−1)·k

2 )+3+kn

n · q2+(m−1)·(3k+(k−1)·k

2 )+3+k+1n

(3+k)m + 1

n · q2+(m−1)·(3(k+1)+(k+1)·k

10 · n5 · q2+(m−1)·(3(k+1)+(k+1)·k

×m∏i=2

[1− j

(2+(m−1)·(3k+(k−1)·k

2 )+(i−2)·(3+k))1

qn− ...− j

(1+(m−1)·(3k+(k−1)·k

2 )+(i−1)·(3+k))1 + 1

(3+k+1)i

q3+k+1n

26 · n4 · q3+k+1n

1− j(2+(m−1)·(3k+

(k−1)·k2 )+(i−2)·(3+k))

qn− ...− j

(1+(m−1)·(3k+(k−1)·k

2 )+(i−1)·(3+k))1 + 1

(3+k+1)i + 1q3+k+1n

− 126 · n4 · q3+k+1

By our choice of the number mn the subsequent application of Dψn,γn Rmnαn+1 D−1

ψn,γnyields

modulo 1qn:

[k + 1n · qn

n · q2n

+ ...+j(1+(m−1)·(3k+

(k−1)·k2 ))

n · q2+(m−1)·(3k+(k−1)·k

2 )+1n

(3+k)2

n · q2+(m−1)·(3k+(k−1)·k

2 )+3+kn

n · q2+(m−1)·(3k+(k−1)·k

2 )+3+k+1n

(3+k)m

n · q2+(m−1)·(3(k+1)+(k+1)·k

10 · n5 · q2+(m−1)·(3(k+1)+(k+1)·k

k + 1n · qn

n · q2n

+ ...+j(1+(m−1)·(3k+

(k−1)·k2 ))

n · q2+(m−1)·(3k+(k−1)·k

2 )+1n

(3+k)2

n · q2+(m−1)·(3k+(k−1)·k

2 )+3+kn

n · q2+(m−1)·(3k+(k−1)·k

2 )+3+k+1n

(3+k)m + 1

n · q2+(m−1)·(3(k+1)+(k+1)·k

10 · n5 · q2+(m−1)·(3(k+1)+(k+1)·k

×m∏i=2

[1− j

(2+(m−1)·(3k+(k−1)·k

2 )+(i−2)·(3+k))1

qn− ...− j

(1+(m−1)·(3k+(k−1)·k

2 )+(i−1)·(3+k))1 + 1

(3+k+1)i

q3+k+1n

26 · n4 · q3+k+1n

+ bn (θ) ,

1− j(2+(m−1)·(3k+

(k−1)·k2 )+(i−2)·(3+k))

qn− ...− j

(1+(m−1)·(3k+(k−1)·k

2 )+(i−1)·(3+k))1 + 1

(3+k+1)i + 1q3+k+1n

− 126 · n4 · q3+k+1

+ bn (θ)

at which an is introduced in Remark 4.5.4 and bn (θ) := ψn (γn · (θ + an)) − ψn (γn · θ), whichcan be estimated with the aid of the mean value theorem and Remark 4.5.4:

|bn (θ)| ≤ |||ψn|||1 · γn · an ≤1

320 · n9 · γn.

Under ϕ 160n4 ,1,2

2+(m−1)·(3(k+1)+ (k+1)·k2 )

this is mapped to

68 (γ, ε)-distribution

k + 1n · qn

n · q2n

+ ...+j

(3+k)m

n · q2+(m−1)·(3(k+1)+(k+1)·k

[j(2+(m−1)·(3k+

(k−1)·k2 ))

qn+ ...+

j(1+(m−1)·(3k+

(k−1)·k2 )+3+k)

1 + 1q3+kn

− j(3+k+1)2 + 1q3+k+1n

26 · n4 · q3+k+1n

− bn (θ) ,

j(2+(m−1)·(3k+

(k−1)·k2 ))

qn+ ...− j

(3+k+1)2

q3+k+1n

− 126 · n4 · q3+k+1

− bn (θ)

110 · n4

+ n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )n · an, 1−

110 · n4

+ n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )n · an

×m∏i=3

[1− j

(2+(m−1)·(3k+(k−1)·k

2 )+(i−2)·(3+k))1

qn− ...− j

(1+(m−1)·(3k+(k−1)·k

2 )+(i−1)·(3+k))1 + 1

(3+k+1)i

q3+k+1n

26 · n4 · q3+k+1n

+ bn (θ) ,

1− j(2+(m−1)·(3k+

(k−1)·k2 )+(i−2)·(3+k))

qn− ...+ j

(3+k+1)i + 1q3+k+1n

− 126 · n4 · q3+k+1

+ bn (θ)

]using the bounds on an and bn (θ). With the aid of Remark 3.3.7, the bounds on an, bn (θ)and the fact that 10n4 divides q3+k+1

n by Lemma 4.7.6 we can compute the image of In,k underφ

n·q2+(m−1)·(3(k+1)+ (k+1)·k

2 )n ,q3+k+1

Rmnαn+1 φ−1

[k + 1n · qn

n · q2n

+ ...+j(1+(m−1)·(3k+

(k−1)·k2 )+3+k)

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+3+kn

− j(3+k+1)2 + 1

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+3+k+1n

26 · n5 · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+3+k+1n

− bn (θ)

n · q2+(m−1)·(3(k+1)+(k+1)·k

k + 1n · qn

+ ...− j(3+k+1)2

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+3+k+1n

26 · n5 · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+3+k+1n

− bn (θ)

n · q2+(m−1)·(3(k+1)+(k+1)·k

110 · n4

+ n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )n · an, 1−

110 · n4

+ n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )n · an

×m∏i=3

[1− j

(2+(m−1)·(3k+(k−1)·k

2 )+(i−2)·(3+k))1

qn− ...+ j

(3+k+1)i

q3+k+1n

26 · n4 · q3+k+1n

+ bn (θ) ,

1− j(2+(m−1)·(3k+

(k−1)·k2 )+(i−2)·(3+k))

qn− ...+ j

(3+k+1)i + 1q3+k+1n

− 126 · n4 · q3+k+1

+ bn (θ)

Continuing in the same way we obtain that Φn(In,k

)is equal to

[k + 1n · qn

n · q2n

+ ...+j

(3+k)m

n · q2+(m−1)·(3(k+1)+(k+1)·k

+j(2+(m−1)·(3k+

(k−1)·k2 ))

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+1n

+j(1+(m−1)·(3k+

(k−1)·k2 )+3+k)

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+3+kn

− j(3+k+1)2 + 1

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+3+k+1n

+j(2+(m−1)·(3k+

(k−1)·k2 )+3+k)

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+3+k+2n

+ ...+j(1+(m−1)·(3k+

(k−1)·k2 )+2·(3+k))

n · q1+(m−1)·(3(k+1)+(k+1)·k

2 )+2·(3+k+1)n

− j(3+k+1)3 + 1

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+2·(3+k+1)n

+ ...− j(3+k+1)m + 1

n · q2+(m−1)·(3(k+2)+(k+1)·(k+2)

26 · n5 · q2+(m−1)·(3(k+2)+(k+1)·(k+2)

− bn (θ)

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+(m−2)·(3+k+1)n

k + 1n · qn

+ ...− j(3+k+1)m

n · q2+(m−1)·(3(k+2)+(k+1)·(k+2)

26 · n5 · q2+(m−1)·(3(k+2)+(k+1)·(k+2)

− bn (θ)

n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )+(m−2)·(3+k+1)n

110 · n4

+ n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )n · an, 1−

110 · n4

+ n · q2+(m−1)·(3(k+1)+(k+1)·k

2 )n · an

m∏i=3

26n4− q3+k+1

n · bn (θ) , 1− 126n4

− q3+k+1n · bn (θ)

Thus, such a set Φn(In,k

)with In,k ∈ ηn has a θ-width θn,k of at most 1

n·q7m−5n

≤ 1n·q2m

contains the (m−1)-dimensional interval J in the ~r-coordinates due to our bounds on an as wellas bn (θ). Hence, we have for every (m− 1)-dimensional interval J ⊆ J :

Φn(In,k

)∩ S1 × J

Φn(In,k

)∩ S1 × J

) =θn,k · µ(m−1)

θn,k · µ(m−1) (J)=µ(m−1)

µ(m−1) (J).

Let ∆ := Φn(In,k

Φn(In,k

)∩ S1 × J

). Then we have 0 < µ (∆) ≤ θn,k · 2·(m−1)

n4 and:

Φn(In,k

)∩ S1 × J

)=µ(m−1)

µ(m−1) (J)· µ(

Φn(In,k

)∩ S1 × J

)=µ(m−1)

µ(m−1) (J)·(µ(

Φn(In,k

))− µ (∆)

Since Φn is measure-preserving, we can conclude:

0 ≥µ(In,k ∩ Φ−1

(S1 × J

))µ(In,k

) −µ(m−1)

µ(m−1) (J)= − µ (∆)

Φn(In,k

)) · µ(m−1)(J)

µ(m−1) (J)

≥ −θn,k · 2·(m−1)

θn,k ·(1− 2

)m−1 ·µ(m−1)

µ(m−1) (J)≥ − 1

n·µ(m−1)

µ(m−1) (J),

i.e. we can choose ε = 1n in the definition of a (γ, ε)-distribution.

Remark 4.5.6. Note that we need the new notion of a (γ, ε)-distribution due to the appearanceof the θ-dependent “error terms” bn (θ).

Furthermore, we show the next property concerning the conjugating map gn constructed insection 4.4.4:

Lemma 4.5.7. For every In ∈ ηn we have: gn(

Φn(In

))= g[nqσn]

(Φn(In

Proof. In the proof of the precedent Lemma 4.5.5 we computed Φn(In,k

)for a partition element

In,k. Now we have to examine the effect of gn = gn·q

2+(m−1)·(3·(k+2)+· (k+1)·(k+2)2 )

n ,[n·qσn], 18n4 ,

it.Since 260n4 divides qn by Lemma 4.7.6, there is u ∈ Z such that

= u · ε

b · a= u · 1

8n4 · [nqσn] · n · q2+(m−1)·(3·(k+2)+(k+1)·(k+2)

By 126n4 + q3+k+1

n · |bn (θ)| < ε = 18n4 and the bounds on an the boundary of Φn

)lies in

the domain, where gn·q

2+(m−1)·(3·(k+2)+· (k+1)·(k+2)2 )

n ,[n·qσn], 18n4 ,

= g[nqσn].

In this section we will prove a criterion for weak mixing on M = Tm in the setting of thebeforehand constructions. The derivation is similar to the considerations in section 3.5, but it isbased on the different notion of a (γ, ε)-distribution on J :=

[1n4 , 1− 1

]m−1.

Lemma 4.6.1. Let n ≥ 3√

4 · (m− 1), gn as in section 4.4.4 and In ∈ ηn, where ηn is thepartial partition constructed in section 4.4.3.1. For the diffeomorphism φn constructed in section4.4.5 and mn as in chapter 4.5 we consider Φn = φn Dψn,γn Rmnαn+1

D−1ψn,γn

φ−1n and

1n4 , 1− 1

]m−1.Then for every m-dimensional cube S of side length q−σn lying in S1 × J we get∣∣∣µ(I ∩ Φ−1

n g−1n (S)

)· µ (J)− µ

(I)· µ (S)

∣∣∣ ≤ 21n· µ(I)· µ (S) (4.3)

Proof. Let S be a m-dimensional cube with sidelength q−σn lying in S1 × J .According to Lemma 4.5.5, Φn

n·qmn, 1n

)-distributes the partition element In ∈ ηn on J , in

Convergence of (fn)n∈N in Diff∞ (Tm) 71

particular Φn(In

)⊆ [c, c+ γ] × Tm−1 for some c ∈ S1 and some γ ≤ 1

n·qmn, and we observed

Φn(I))≥ 1

a ·(1− 2

)· µ (J). Furthermore, we saw in the proof of Lemma 4.5.7 that

Φn(In

)is contained in the interior of the step-by-step domains of the map gn and on its boundary

gn = g[nqσn] holds.Then we can prove the stament in the same way as in Lemma 3.5.5.

Now we are able to prove the aimed criterion for weak mixing.

Proposition 4.6.2 (Criterion for weak mixing). Let fn = Hn Rαn+1 H−1n and the sequence

(mn)n∈N be constructed as in the previous sections. Suppose additionally that d0 (fmn , fmnn ) < 12n

for every n ∈ N, ‖DHn−1‖0 ≤ln(qn)n and that the limit f = limn→∞ fn exists.

Proof. To apply Lemma 3.5.2 we consider the partial partitions νn := Hn−1 gn (ηn). By thesame arguments as in the proof of Lemma 3.5.3 these partial partitions satisfy νn → ε. We haveto establish equation 3.2. For it let ε > 0 and a m-dimensional cube A ⊆ S1× (0, 1)m−1 be given.There exists N ∈ N such that A ⊆ S1 ×

[1n4 , 1− 1

]m−1 for every n ≥ N . Because of Lemma

4.5.5 we obtain for every In ∈ ηn: Φn(In

)⊇ [c, c+ γ] ×

[1n4 , 1− 1

]m−1 for some γ ≤ 1n·qmn

Furthermore, we note fmnn = Hn Rmnαn+1H−1

n = Hn−1 gn Φn g−1n H−1

n−1.

Let Sn be a m-dimensional cube of side length q−σn contained in S1 ×[

1n4 , 1− 1

]m−1 = S1 × J .Then we can prove this criterion for weak mixing analogously to Proposition 3.5.6 with the aidof Lemma 4.6.1.

4.7 Convergence of (fn)n∈N in Diff∞ (Tm)

By the same approach as in section 3.6 we show that the sequence of constructed measure-preserving smooth diffeomorphisms fn = Hn Rαn+1 H−1

n converges. For this purpose, we needa couple of results concerning the conjugation maps.

In this subsection we want to find estimates on the norms |||Hn|||k.

|||Dψn,γn |||k ≤ C · γkn,

where C is a constant depending on n and k, but is independent of qn.

Proof. By the construction of the map Dψn,γn = C−1γn Dψn Cγn we observe using the abbre-

viation dn := 1 + 1q3n

+ ...+ 1q3+n−1n

Dψn,γn (θ, r1, ..., rm−1) = (θ, r1 + dn · ψn (γn · θ) , ..., rm−1 + dn · ψn (γn · θ))

as well as

D−1ψn,γn

(θ, r1, ..., rm−1) = (θ, r1 − dn · ψn (γn · θ) , ..., rm−1 − dn · ψn (γn · θ)) .

Since dn ≤ 2 we obtain: |||Dψn,γn |||k ≤ C · dn · γkn ≤ C · qk·(2+(m−1)·(3·n+

n·(n−1)2 ))

72 Convergence of (fn)n∈N in Diff∞ (Tm)

Next, we examine the map φn:

|||φn|||k ≤ C · γ(m−1)·kn ,

Proof. We recall the proof of Lemma 3.6.4. In the setting of our explicit construction of the map

φn in section 4.4.5 we have λmax = n · q2+(m−1)·(3·(n−1)+(n−1)·(n−2)

2 )+(3+n−1)·(m−2)n , ε1 = 1

60·n4 ,ε2 = 1

22·n4 , and µmax = q3+n−1n . Thus:

|||φn|||k ≤ C (m, k, n) ·(n · q2+(m−1)·(3(n−1)+

(n−1)(n−2)2 )+(3+n−1)(m−2)

)(m−1)·k

·(q3+n−1n

)(m−1)·k

≤ C (m, k, n) · γ(m−1)·kn ,

where C (m, k, n) is a constant independent of qn.

In the next step we consider the map gn φn, where gn is constructed in section 4.4.4:

Lemma 4.7.3. For every k ∈ N we have:

|||gn φn|||k ≤ C · qkn · γk·mn ,

Proof. In the proof of Lemma 3.6.5 we saw: |||ga,b,ε|||k ≤ Cε,k · bk · ak. By our choice ofparameters in section 4.4.4 we have b = [n · qσn] ≤ n · qσn, a ≤ γn and ε = 1

8n4 . Hence:

|||gn|||k ≤ Cn,k · qσ·kn · γkn ≤ Cn,k · qkn · γkn.

Then we continue as in the proof of Lemma 3.6.5 with the result of Lemma 4.7.2 and obtain thestatement.

Combining the results of Lemma 4.7.3 and Lemma 4.7.1 with the help of the formula of Faàdi Bruno, we conclude for the conjugating map hn = gn φn Dψn,γn :

|||hn|||k ≤ C · qkn · γk·(m+1)n ,

Finally, we are able to prove an estimate on the norms of the map Hn as in Lemma 3.6.6:

|||Hn|||k ≤ C · qk·(m+1)·m·n·(n+5)

where C is a constant depending solely on m, k, n and Hn−1. Since Hn−1 is independent of qnin particular, the same is true for C.

Proof. By Lemma 4.7.4 and γn = n · q2+(m−1)·(3·n+n·(n−1)

2 )n = n · q2+(m−1)·n·(n+5)

2n we have

|||hn|||k ≤ C · qk·(1+(m+1)·(2+(m−1)·n·(n+5)

2 ))n ≤ C · qk·(m+1)·m·n·(n+5)

Then the proof follows along the lines of Lemma 3.6.6.

4.7.2 Proof of convergence

In Lemma 3.6.8 we proved convergence of the sequence (fn)n∈N to f ∈ Aα (Tm) in the Diff∞ (Tm)-topology under some assumptions on the sequence (αn)n∈N. We show that we can satisfy theseconditions in our constructions:

Lemma 4.7.6. Let (kn)n∈N be a strictly increasing seq. of natural numbers with∑∞n=1

qnof rational numbers with 260n4 divides qn and qn > n2 ·maxi=1,...,n Li (where

Li denotes the Lipschitz constant of ρi ∈ Ξ), such that our conjugation maps Hn fulfil thefollowing conditions:

|α− αn| <1

2 · kn · Ckn · |||Hn|||kn+1kn+1

|α− αn| <1

2n+1 · qn · |||Hn|||1.

Proof. In Lemma 4.7.5 we deduced the estimate |||Hn|||kn+1 ≤ Cn · q(kn+1)·(m+1)·m·n·(n+5)

where the constant Cn was independent of qn. Thus, we can choose qn ≥ Cn for every n ∈ N.

Hence, we obtain: |||Hn|||kn+1 ≤ q2·(m+1)·m·n·(n+5)

2 ·(kn+1)n . Besides qn ≥ Cn we keep the before

mentioned condition qn ≥ 260 · n4 · 320 · (n − 1)11 · |||ψn−1|||1 · q2·(2+(m−1)·(3·(n−1)+

(n−1)·(n−2)2 ))

in mind. Furthermore, we can demand qn > n2 · maxi=1,...,n Li and ‖DHn−1‖0 <ln(qn)n from

qn, because Hn−1 is independent of qn. Since α is a Liouvillean number, we find a sequence ofrational numbers αn = pn

qn, pn, qn relatively prime, converging to α under the above restrictions

(formulated for qn) satisfying:

|α− αn| <|α− αn−1|

2n+1 · kn · Ckn · (260n4)1+(m+1)·m·n·(n+5)·(kn+1)2

· q1+(m+1)·m·n·(n+5)·(kn+1)2

|α− αn| <|α− αn−1|

2n+1 · kn · Ckn · q1+(m+1)·m·n·(n+5)·(kn+1)2

Hence, we have |α− αn|n→∞→ 0 monotonically. Because of |||Hn|||kn+1

kn+1 ≤ q(m+1)·m·n·(n+5)·(kn+1)2

this yields: |α− αn| < 1

. Thus, the first property of this Lemma is

fulfilled.Then we can verify the second property in the same way as in Lemma 3.6.9.

Remark 4.7.7. Lemma 4.7.6 shows that the conditions of Lemma 3.6.8 are satisfied. There-fore, our sequence of constructed diffeomorphisms fn converges in the Diff∞ (Tm)-topology to adiffeomorphism f ∈ Aα (Tm).

74 Proof of strict ergodicity

Analogous to section 3.6 we observe:

(f m, f mn

)≤ 1

Remark 4.7.9. Note that the sequence (mn)n∈N defined in section 4.5 meets the mentionedcondition mn ≤ qn+1 and hence Lemma 4.7.8 can be applied to it.

Concluding we have checked that all the assumptions of Proposition 4.6.2 are satisfied. Thus,this criterion guarantees that the constructed diffeomorphism f ∈ Aα (Tm) is weak mixing. Inaddition, for every ε > 0 we can choose the parameters in such a way by Lemma 3.6.8 thatd∞ (f,Rα) < ε holds.

4.8 Proof of strict ergodicity

In this section we will use the trapping map and the trapping regions to gain control over a largeproportion of every Rt-orbit. Then we are able to proof uniform convergence of the Birkhoffsums and apply our criterion 4.3.3 of unique ergodicity.

4.8.1 Trapping property

In case of 0 ≤ l ≤ qn − 1, 0 ≤ k ≤ n− 1, j(1)i ∈ Z,

⌈qn4n4

⌉≤ j(1)

⌉− 1 for i = 1, ...,m

as well as j(t)i ∈ Z, 0 ≤ j(t)

i ≤ qn − 1 for i = 2, ...,m and t = 2, 3, we introduce the sets

∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

(1)1 + 1nq2n

m∏i=2

(3)i + 1q3n

Note that there are qn · n ·(qn − 2 ·

⌈qn4n4

⌉)m · q(m−1)·2n such sets ∆

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,...,j

(1)m ,j

(2)m ,j

and denote the union of these sets by Tn and the family of these sets by Tn. Then

µ(Tm \ Tn

)= 1−nqn ·

(qn − 2

⌈ qn4n4

⌉)m· q(m−1)·2n · 1

n · q2+3·(m−1)n

≤ 1−(

1− 23n4

)m≤ 2 ·m

We observe that gn = gn on ∆l,k,j

(1)1 ,...,j

and diam(gn

(∆l,k,j

(1)1 ,j

(1)2 ,...,j

(1)m ,j

(2)m ,j

q2n. Since

‖DHn−1‖ < ln(qn)n we conclude that diam

(Hn−1 gn

(∆l,k,j

(1)1 ,...,j

qnfor n sufficiently

large.By the requirements on the number qn in Lemma 4.7.6, we obtain

|ρi (Hn−1 gn (x))− ρi (Hn−1 gn (y))| ≤ Lip (ρi) · diam(Hn−1 gn

(∆l,k,j

(1)1 ,j

(1)2 ,...,j

))≤ Lip (ρi) ·

Proof of strict ergodicity 75

for every x, y ∈ ∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

and the function ρi ∈ Ξ in case ofi = 1, ..., n. Averaging over all y ∈ ∆

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

we obtain:∣∣∣∣∣∣ρi (Hn−1 gn (x))− 1

∆l,k,j

(1)1 ,j

(1)2 ,...,j

) ∫Hn−1gn

(∆l,k,j

(1)1 ,j

(1)2 ,...,j

) ρi dµ∣∣∣∣∣∣ < 1

n2(4.4)

Furthermore, we calculate that the trapping region D−1ψn,γn

(Sl,k,j

(1)1 , ~j2,..., ~jm

)defined in section

4.4.2 is mapped under φn Dψn,γn onto

n · qn+

n · q2n

+t(1)1

n · q3n

+ ...+t((m−1)·(3·k+

k·(k−1)2 ))

n · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)n

− t(1)2

n · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+1n

− ...− t(3+k)2

n · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+3+kn

− t(1)3

n · q2+3(m−1)+...+(3+k−1)(m−1)+3+k+1n

− ...− t(3+k)3

n · q2+3(m−1)+...+(3+k−1)(m−1)+2·(3+k)n

− ...

− t(3+k)m + 1

n · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)n

+t((m−1)·(3·k+

k·(k−1)2 )+1)

n · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)+1n

+ ...+t((m−1)·(3·(n−1)+

n·(n−1)2 −k))

1n4 · γn

n · qn+ ...+

t((m−1)·(3·(n−1)+

n·(n−1)2 −k))

1 + 1γn

− 1n4 · γn

×m∏i=2

qn+ ...+

j(3+k)i

4 · n4 · q3+kn

qn+ ...+

j(3+k)i + 1q3+kn

− 14 · n4 · q3+k

where the union is taken over all t(l)i ∈ Z, 0 ≤ t(l)i ≤ qn − 1, for l = 2, ..., 3 + k and i = 2, ...,m aswell as t(1)

i ∈ Z,⌈qn4n4

⌉≤ t(1)

i ≤ qn−⌈qn4n4

⌉−1 for i = 2, ...,m as well as t(j)1 ∈ Z, 0 ≤ t(j)1 ≤ qn−1,

for j = 1, ..., (m − 1) ·(

3 · (n− 1) + n·(n−1)2 − k

)apart from t

((m−1)·(3·k+k·(k−1)

⌉≤ t((m−1)·(3·k+

k·(k−1)2 )+1)

1 ≤ qn −⌈qn4n4

⌉− 1.

In particular, φn Dψn,γn

(D−1ψn,γn

(Sl,k,j

(1)1 , ~j2,..., ~jm

))is contained in ∆

l,k,j(1)1 ,...,j

(1)m ,j

(2)m ,j

same is true for the other allowed values of j(4)i , ..., j

(3+k)i . Thus, there are qk·(m−1)

n trapping re-gions that are mapped into ∆

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

under φn Dψn,γn . Hence,we can estimate the number of i ∈ 0, ..., qn+1 − 1 such that φn Dψn,γn Riαn+1

(x) is contained

in ∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

by qk·(m−1)n ·

⌊qn+1 ·

1− 8mn2

n·q(m−1)·(3+k)n ·q2

⌋for arbitrary

x ∈ Tm using Remark 4.4.1.

4.8.2 Application of the criterionUsing the preparatory work of the precedent section we can prove the following result on theBirkhoff sums:

76 Proof of strict ergodicity

Lemma 4.8.1. For ρi ∈ Ξ and i = 1, ..., n we have∣∣∣∣∣∣ 1qn+1

qn+1−1∑j=0

ρi(f jn (y)

)−∫ρi dµ

∣∣∣∣∣∣ ≤ 1n2

+20mn2· ‖ρi‖0

for every y ∈ Tm.

Proof. Let x ∈ Tm be arbitrary. In the subsequent estimate we denote the set of iteratesj ∈ 0, ..., qn+1 − 1 such that φn Dψn,γn Rjαn+1

(x) is contained in ∆ ∈ Tn by I∆.∣∣∣∣∣∣ 1qn+1

qn+1−1∑j=0

(Hn−1 gn φn Dψn,γn Rjαn+1

(x))−∫ρi dµ

∣∣∣∣∣∣=

∣∣∣∣∣ 1qn+1

qn+1−1∑j=0

(x))−∑

∆∈Tn

∫Hn−1gn(∆)

ρi dµ

−∫Hn−1gn(Tm\Tn)

ρi dµ

∣∣∣∣∣≤

∣∣∣∣∣∣∑

∆∈Tn

∑j∈I∆

(x))−∫Hn−1gn(∆)

ρidµ

∣∣∣∣∣∣+ µ

(Tm \ Tn

)· ‖ρi‖0

+qn+1 − n ·

(qn − 2 ·

⌈qn4n4

⌉)m · q1+(m−1)·2n · qk·(m−1)

n ·⌊qn+1 ·

1− 8mn2

n·q(m−1)·(3+k)n ·q2

⌋qn+1

‖ρi‖0

The third summand is deduced from the following observations: We noticed that there are exactlyn ·(qn − 2 ·

⌈qn4n4

⌉)m · q(m−1)·2+1n sets ∆ ∈ Tn and qk·(m−1)

n trapping regions Sl,k,j

(1)1 ,~j2,...,~jm

mapped into one such ∆ under the map φnDψn,γn . Since there are at least⌊qn+1

1− 8mn2

nq(m−1)·(3+k)n ·q2

⌋iterates such that Dψn,γn Rjαn+1

(x) is contained in a trapping region Sl,k,j

(1)1 ,~j2,...,~jm

, we canestimate the number of iterates, which are not mapped into a set ∆ ∈ Tn under φn Dψn,γn , by

qn+1 − n ·(qn − 2 ·

⌈ qn4n4

⌉)m· q1+(m−1)·2n · qk·(m−1)

⌊qn+1 ·

1− 8mn2

n · q(m−1)·(3+k)n · q2

≤ qn+1 − n · qmn ·(

1− 1n4

)m· q1+(m−1)·2n · qk·(m−1)

n · qn+1 ·1− 9m

n · q(m−1)·(3+k)n · q2

≤ qn+1 ·(

1− m

1− 9mn2

))≤ qn+1 ·

for n sufficiently large.By the same reasoning the number of points φn Dψn,γn Rjαn+1

(x) contained in an arbitrary set

∆ ∈ Tn is at least qk·(m−1)n · qn+1 ·

1− 9mn2

n·q(m−1)·(3+k)n ·q2

= qn+1 ·1− 9m

n·q2+3·(m−1)n

= qn+1 ·(1− 9m

)· µ (∆)

Construction of the f -invariant measurable Riemannian metric 77

and at most qn+1 · µ (∆). Thus, we obtain using equation 4.4:∣∣∣∣∣∣ 1qn+1

∑j∈I∆

(x))−∫Hn−1gn(∆)

ρidµ

∣∣∣∣∣∣≤µ (∆)

9mn2·∫Hn−1gn(∆)

|ρi| dµ.

Hereby, we can estimate the first summand in the following way:∣∣∣∣∣∣∑

∆∈Tn

∑j∈I∆

(x))−∫Hn−1gn(∆)

ρi dµ

∣∣∣∣∣∣≤∑

∆∈Tn

(µ (∆)n2

+9mn2·∫Hn−1gn(∆)

|ρi| dµ

≤ n ·(qn − 2 ·

⌈ qn4n4

⌉)m· q(m−1)·2+1n · 1

n3 · q(m−1)·3+2n

+9mn2· ‖ρi‖0

≤ n · qmn ·(

1− 12n4

)m· q(m−1)·2+1n · 1

n3 · q(m−1)·3+2n

+9mn2· ‖ρi‖0

≤ 1n2

+9mn2· ‖ρi‖0 .

Altogether, we conclude∣∣∣∣∣∣ 1qn+1

qn+1−1∑j=0

(x))−∫ρi dµ

∣∣∣∣∣∣≤ 1n2

+9mn2· ‖ρi‖0 +m · 2

3n4· ‖ρi‖0 +

10mn2· ‖ρi‖0 ≤

+20mn2· ‖ρi‖0 .

With x = H−1n (y) we obtain the statement of the Lemma.

This Lemma ensures that the Birkhoff sums 1qn+1

∑qn+1−1j=0 ρi

(f jn (x)

)converge uniformly to∫

ρidµ as n → ∞ for every ρi ∈ Ξ, where Ξ was a countable dense set in C(Tm,R). More-over, we have d0

(f mn , f mnn

2n for all mn ∈ 0, 1, ..., qn+1 − 1 by Lemma 4.7.8 and thusd(qn+1) (f, fn) → 0 as n → ∞. Then we can apply our criterion 4.3.3 and conclude that f isuniquely ergodic. By Remark 4.3.2 and the Riesz Representation Theorem the unique invariantprobability measure is the Lebesgue measure µ. Since supp(µ) = Tm we have proven that theconstructed diffeomorphism f is strictly ergodic.

4.9 Construction of the f-invariant measurable Riemannianmetric

Let ω0 denote the standard Riemannian metric on M = S1 × [0, 1]m−1. As in section 3.7 thefollowing Lemma shows that the conjugation map hn = gn φn Dψn,γn constructed in section4.4 is an isometry with respect to ω0 on the elements of the partial partition ζn.

78 Construction of the f -invariant measurable Riemannian metric

Lemma 4.9.1. Let D−1ψn,γn

)∈ ζn. Then hn|D−1

ψn,γn(In) is an isometry with respect to ω0.

Proof. Let Γn,k := D−1ψn,γn

)be a partition element of ζn on

, k+1nqn

]× [0, 1]m−1. By

Remark 4.4.4 Dψn,γn is an isometry on it.The element In,k is positioned in such a way that all the occurring maps ϕε,1,j and ϕ−1

ε2,1,jact

as rotations on it. Hence, φn|In,k is an isometry and φn(In,k

)is equal to

n · qn+

n · q2n

+ ...+j(1+(m−1)·(3k+

k·(k−1)2 ))

n · q2+(m−1)·(3k+k·(k−1)

− j(1)2

n · q2+(m−1)·(3k+k·(k−1)

2 )+1n

− ...

− j(3+k)2

n · q2+(m−1)·(3k+k·(k−1)

2 )+3+kn

− j(1)3

n · q2+(m−1)·(3k+k·(k−1)

2 )+3+k+1n

− ...

− j(3+k)m + 1

n · q2+(m−1)·(3·(k+1)+k·(k+1)

+j(2+(m−1)·(3(k+1)+

k·(k+1)2 ))

n · q2+(m−1)·(3(k+1)+k·(k+1)

2 )+1n

+ ...+s

n2 · γn+

1n4 · γn

n · qn+

n · q2n

+ ...+s+ 1n2 · γn

− 1n4 · γn

[j(2+(m−1)·(3k+

k·(k−1)2 ))

qn+ ...+

j(1+(m−1)·(3k+

k·(k−1)2 )+3+k)

(3+k+1)2

q3+k+1n

+j(3+(m−1)·(3·(k+1)+

k·(k+1)2 ))

8n5 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

8n9 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

j(2+(m−1)·(3k+

k·(k−1)2 ))

qn+ ...+

j(1+(m−1)·(3k+

k·(k−1)2 )+3+k)

(3+k+1)2

q3+k+1n

+j(3+(m−1)·(3·(k+1)+

k·(k+1)2 ))

8n5 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

8n9 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

×m∏i=3

[j(2+(m−1)(3k+

k·(k−1)2 )+(i−2)·(3+k))

qn+ ...+

j(1+(m−1)(3k+

k·(k−1)2 )+(i−1)·(3+k))

n4 · q3+kn

j(2+(m−1)(3k+

k·(k−1)2 )+(i−2)·(3+k))

qn+ ...+

j(1+(m−1)(3k+

k·(k−1)2 )+(i−1)·(3+k))

1 + 1q3+kn

− 1n4 · q3+k

Then we have to examine the application of gn = gn·q

2+(m−1)·(3·(k+1)+ (k+1)·k2 )

n ,[n·qσn], 18n4 ,

particular, we have εb·a = 1

8n4·[n·qσn]·n·q2+(m−1)·(3·(k+1)+ (k+1)·k

. Since 4 · ε = 12n4 <

1n4 , gn acts as

translation on φn(In,k

This Lemma enables us to construct the f -invariant measurable Riemannian metric as insection 3.7. Thus, Proposition 4.2.3 is proven.

Chapter 5

Minimal but not uniquely ergodicdiffeomorphisms preserving ameasurable Riemannian metric witharbitrary Liouvillean rotationnumber on Tm

5.1 Introduction

In Theorem B we showed that for a Liouvillean number α ∈ S1 a generic measure-preservingdiffeomorphism in Aα (Tm), m ≥ 2, is strictly ergodic (i.e. minimal and uniquely ergodic) andweak mixing. On the contrary, in [Win01] A. Windsor proved the existence of minimal but notuniquely ergodic diffeomorphisms in A (M) := h St h−1 : h ∈ Diff∞ (M,ν) , t ∈ S1

onany compact and connected smooth boundaryless manifold of dimension m ≥ 2 admitting a freeC∞-action S = Stt∈S1 preserving a smooth volume ν.While conversely a uniquely ergodic transformation on a compact metric space preserving a Borelmeasure is minimal on the support of the measure (e.g. [HK95], Proposition 4.1.18), the firstexample that minimality does not imply unique ergodicity is due to Markov (see [NS60], section9.35.). In the smooth category H. Furstenberg constructed analytic skew-products admittinguncountably many ergodic measures ([Fu61] or see [HK95], Corollary 12.6.4.). In fact, thesecounterexamples bear a great meaning in the history of Ergodic Theory: They showed that theso-called quasi-ergodic hypothesis (i.e. each orbit is dense in each surface of constant energy)does not imply the equality of space means and time means and so helped to find the right notionof ergodicity.S. Williams’ constructions of minimal transformations with any given finite number, a countablenumber or a continuum of ergodic invariant measures in symbolic dynamics ([Wil84]) serve asa very general counterexample. Windsor provided such a result in the smooth category. Inthe special case of the torus Tm we can generalize Windsor’s achievement by constructing aminimal diffeomorphism f ∈ Aα (Tm) with exactly d ergodic invariant measures for prescribedd ∈ N and Liouvillean number α ∈ S1, that admits a f -invariant measurable Riemannian metric.

80 Preliminaries

Therefore, this result is in line with the findings of the chapters 3 and 4, where in extension of[GK00] constructions of diffeomorphisms in Aα (Tm) with ergodic properties (in particular weakmixing) that preserve a measurable Riemannian metric are exhibited.In conclusion we will obtain the subsequent Theorem:

Theorem C. Let m ≥ 2, d ∈ N and α ∈ S1 be Liouville. Then there is a smooth diffeomorphismf ∈ Aα (Tm) preserving a measurable Riemannian metric which is minimal but has exactly dergodic invariant measures. These measures are absolutely continuous with respect to Lebesguemeasure.Furthermore, the set of diffeomorphisms with these properties is dense in the C∞-topology inAα (Tm).

Moreover, in section 5.6.2 we prove that the obtained diffeomorphism is weak mixing withrespect to the d ergodic invariant measures. In [Win01] this is mentioned as a possible improve-ment of the result but was not executed.So in case of d = 1 we give another proof of Theorem 4.1.1.In [Win01], chapter 8, it is presented how the used method produces a diffeomorphism with acountable number of ergodic components. In section 5.8 we will prove such a statement in oursetting, namely:

Theorem D. Let m ≥ 2 and α ∈ S1 be Liouville. Then there is a smooth diffeomorphismf ∈ Aα (Tm) preserving a measurable Riemannian metric which is minimal but has a countablenumber of ergodic invariant absolutely continuous measures.Furthermore, the set of diffeomorphisms with these properties is dense in the C∞-topology inAα (Tm).

For the sake of completeness we mention the remark in [Win01] that the construction of adiffeomorphism which is minimal but has a power of continuum ergodic measures is the simplestof the arguments, because the measures on the fibers S1 × ~r can be made to converge to therequired ergodic measures by choosing the conjugation map as the identity except on small partsneeded to produce a minimal map.

5.2 Preliminaries

5.2.1 Reduction to Proposition 5.2.1

We fix d ∈ N and consider the torus Tm equipped with Lebesgue measure µ and the standard cir-cle actionR = Rαα∈S1 , where the mapRα is given byRα (θ, r1, ..., rm−1) = (θ + α, r1, ..., rm−1).In this setting we will prove the following result:

measure-preserving diffeomorphisms satisfying hnR 1qn

= R 1qnhn such that the diffeomorphisms

fn = Hn Rαn+1 H−1n with Hn = h1 h2 ... hn converge in the Diff∞ (Tm)-topology and the

diffeomorphism f = limn→∞ fn is minimal, has an invariant measurable Riemannian metric,satisfies f ∈ Aα (Tm) and has exactly d invariant ergodic measures ξt, t = 1, ..., d. Thesemeasures are absolutely continuous with respect to Lebesgue measure and f is weak mixing withrespect to every measure ξt.Furthermore, for every ε > 0 the parameters in the construction can be chosen in such a waythat d∞ (f,Rα) < ε.

In fact, Theorem C can be deduced directly from this result by the same arguments as insection 3.2.1.

In the same way we reduce the proof of Theorem D to the construction of a diffeomorphismwith the aimed properties arbitrarily close to the rotation Rα.

5.2.2 Sketch of the proof

As in the previous chapter the conjugation map is made up of three maps constructed in section5.3: hn = gn φn Dψn,γn . This time there are different parts of the torus Tm introduced withdistinct aims. On the one hand, we will divide it into d sets Nt by requirements on the r1-coordinate. Each set naturally supports an absolutely continuous probability measure µt givenby the normalized restriction of the Lebesgue measure µ. These will enable us to build theergodic invariant measures as the limits ξt of the sequence ξnt :=

(H−1n

)∗µt.

On the other hand, we will use stripes corresponding to small parts of the θ-axis on which theconjugation map φn will intermingle the sets Nt to prove minimality of the limit diffeomorphismf . They are measure theoretically insignificant because the measure of these sets will convergeto zero as n→∞.In order to achieve these aims we need the so-called trapping map Dψn,γn introduced in subsec-tion 5.3.1. On the “minimality”-part this map Dψn,γn enables us to capture parts of every orbitRkαn+1

H−1n (x)

k=0,...,qn+1−1

so that the conjugation map φn can spread it over the almost

whole manifold. Then we can prove minimality in chapter 5.5 by argueing that every elementin a family of sufficiently small cubes covering almost the whole manifold is met by the orbitφn Dψn,γn Rkαn+1

H−1n (x)

k=0,...,qn+1−1

and the image of any cube under Hn−1 gn has

a small diameter, which converges to 0 as n → ∞. In addition the trapping map is used togain control of almost everything of every orbit

Hn Rkαn+1

(x)k=0,...,qn+1−1

. This allows us

to prove a convergence result on Birkhoff sums (see Lemma 5.6.1), which in turn enables us toexclude the existence of further ergodic invariant measures besides the previously mentioned ξt.Finally, we will construct the invariant measurable Riemannian metric by the same approach asin section 3.7: The conjugation maps are constructed in such a way that they act as isometrieson elements of a partial partition ζn with respect to the standard metric ω0. Since these par-tial partitions converge to the decomposition into points, we can prove the convergence of theRiemannian metrics ωn :=

(H−1n

)∗ω0 to a f -invariant measurable metric.

We fix an arbitrary countable set Ξ = ρ1, ρ2, ... of Lipschitz continuous functions ρi : Tm → Rthat is dense in C (Tm,R). This set Ξ will be used in section 5.6.1 to prove that there are exactlyd ergodic invariant measures.

5.3.1 The trapping map

To prove the existence of a prescribed number of ergodic measures we have to gain control overmost of the orbit

Hn Riαn+1

(x)i∈N

for every x ∈ Tm. Analogous to subsection 4.4.1 we use

for every n ∈ N a smooth map ψn : [0, 1]→ R satisfying

• ψn is non-decreasing on[0, 1

]and non-increasing on

[12 , 1].

• ψn is equal to kn6 on

[kn2 + 1

n6 ,k+1n2 − 1

]for 0 ≤ k ≤

⌋− 1 and ψn is equal to k

on[n2−k−1n2 + 1

n6 ,n2−kn2 − 1

]for 0 ≤ k ≤

⌋− 1. On

[⌊n22

⌋n2 ,

n2−⌊n22

]it is put to(⌊

⌋− 1)· 1n6 .

With it we define the map Dψn : [0, 1]m → Rm by: (θ, r1, ..., rm−1) is mapped to(θ, r1 −

+ ...+1

q3+n2−1n

)· ψn (θ) , ..., rm−1 −

+ ...+1

q3+n2−1n

)· ψn (θ)

Using the maps Cγn (θ, r1, ...rm−1) = (γn · θ, r1, ..., rm−1) we construct the map

Dψn,γn := C−1γn Dψn Cγn :

]× Tm−1 →

]× Tm−1.

Since this map coincides with the identity in a neighbourhood of the boundary of the sector onthe θ-axis, we can extend it to a map Dψn,γn : Tm → Tm using the description Dψn,γn R l

R lγnDψn,γn for any l ∈ Z. In our construction we use

γn = n2 · q2+3·(m−1)+4·(m−1)+...+(3+n2−1)·(m−1)n = n2 · q

2+(m−1)·(

3·n2+n2·(n2−1)

5.3.2 Trapping regions

There are different kinds of trapping regions introduced with distinct aims.In order to prove minimality we consider on

[lqn, lqn

+ 1n2·qn

]×Tm−1 for l ∈ Z the following sets

Sl,j2,...,jm :[l

jmn2q2

+jm−1

+ ...+j2

n2qmn,l

jmn2q2

+jm−1

+ ...+j2 + 1n2qmn

m∏i=2

4n6, 1− 1

Then the set of trapping regions of this first kind consists of all sets D−1ψn,γn

(Sl,j2,...,jm), whereall ji ∈ Z satisfy

⌈qn4n6

⌉≤ ji ≤ qn −

⌈qn4n6

⌉− 1 for i = 2, ...,m.

Remark 5.3.1. Let x ∈ Tm be arbitrary. The construction of the regions with the aid of thetrapping map D−1

ψn,γncauses that the orbit

Rkαn+1

H−1n (x)

k=0,...,qn+1−1

meets every trapping

region of the first kind.

Besides these trapping regions introduced in order to prove minimality we will need furtherregions to show that there are exactly d ergodic invariant measures. These measures will beconstructed in section 5.6.1 with the aid of the respective normalized restriction of the Lebesguemeasure on the sets

Nt := S1 ×Nt × Tm−2, at which Nt :=

[t−1∑k=1

t∑k=1

]⊆ R

for t = 1, ..., d, where the r1-lengths lt satisfy 1lt∈ Z as well as

∑dt=1 lt = 1. On such a set Nt

and for l ∈ Z as well as k = 1, ..., n2 − 1 we consider the following sets:

Stl,k,j

(1)1 , ~j2,..., ~jm

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−1)·(3·k+

k·(k−1)2 ))

n2 · q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)n

n2 · q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)+1n

+ ...+j

(3+k)2

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+3+kn

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+3+k+1n

+ ...+j

(3+k)3

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+2·(3+k)n

(3+k)m

n2 · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)(3+k)n

+t((m−1)·(3·k+

k·(k−1)2 )+1)

n2 · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)(3+k)+1n

+ ...+t

((m−1)·

(3·(n2−1)+

n2·(n2−1)2 −k

1n6 · γn

n2 · qn+ ...+

((m−1)·

(3·(n2−1)+

n2·(n2−1)2 −k

))1 + 1

γn− 1n6 · γn

[t−1∑s=1

ls + lt ·

(t(1)2

qn+ ...+

t(3+k)2

4n6 · q3+kn

t−1∑s=1

ls + lt ·

(t(1)2

qn+ ...+

t(3+k)2 + 1q3+kn

− 14n6 · q3+k

×m∏i=3

[t(1)i

qn+ ...+

t(3+k)i

4 · n6 · q3+kn

,t(1)i

qn+ ...+

t(3+k)i + 1q3+kn

− 14 · n6 · q3+k

⌈qn4n6

⌉≤ t(1)

i ≤ qn−⌈qn4n6

⌉−1 for i = 2, ...,m as well as

t(l)i ∈ Z, 0 ≤ t(l)i ≤ qn− 1, for l = 2, ..., 3 +k and i = 2, ...,m as well as t(j)1 ∈ Z, 0 ≤ t(j)1 ≤ qn− 1,

for j = 1, ..., (m− 1) ·(

3 · (n2 − 1) + n2·(n2−1)2 − k

)apart from t

((m−1)·(3·k+k·(k−1)

⌉≤ t((m−1)·(3·k+

k·(k−1)2 )+1)

1 ≤ qn −⌈qn4n6

⌉− 1.

Then the set of trapping regions consists of all sets D−1ψn,γn

(Stl,k,j

(1)1 ,~j2,..., ~jm

), where all j(1)

i ∈ Z

satisfy⌈qn4n6

⌉≤ j

(1)i ≤ qn −

⌈qn4n6

⌉− 1 for i = 1, ...,m and j

(s)i ∈ Z, 0 ≤ j

(s)i ≤ qn − 1 for

s = 2, ..., 3 + k and i = 2, ...,m.

Remark 5.3.2. Let x = (θ, r1, ..., rm−1) ∈ Tm be arbitrary. If we have n > 1lmin

, at whichlmin := mini=1,...,d li, then by the construction of the map Dψn for every coordinate ri there areat most four sections

[kn2 + 1

n6 ,k+1n2 − 1

]on the domain [0, 1] such that ri does not belong to

ψ−1n

([kn2 + 1

n6 ,k+1n2 − 1

]×[∑t−1

s=1 ls + 14n6 · lt,

∑ts=1 ls −

14n6 · lt

14n6 , 1− 1

]m−2)

for all t ∈ 1, ..., d.We have to bear the gaps of our trapping region in the ~r-coordinates in mind. Therefore, wenote that

(1 + 1

+ ...+ 1

q3+k−1n

)· 1n6 is a multiple of 1

and since 1lt∈ Z this translates by full

ltq3+kn

-blocks in the r1-coordinate resp. 1

-blocks in the r2, ..., rm−1-coordinates. Hence, there

are at most four further sections[

kn2·γn + 1

n6·γn ,k+1n2·γn −

1n6·γn

]on every 1

γn-section belonging to

Stl,k,j

(1)1 , ~j2,..., ~jm

such that ri does not belong to D−1ψn,γn

(Stl,k,j

(1)1 , ~j2,..., ~jm

)for any t ∈ 1, ..., d.

A trapping region on[lqn

+ kn2·qn ,

+ k+1n2·qn

]×Nt × Tm−2 consists of at least(

1− 1n6

)· q3·(m−1)+4·(m−1)+...+(3+n2−1)·(m−1)−(m−1)·(3+k)n

many 1γn

-sections for every t ∈ 1, ..., d, l = 0, ..., qn−1, k = 1, ..., n2−1. We fix l, k, j(1)1 , ~j2, ..., ~jm.

Since i · αn+1i=0,...,qn+1−1 is equidistributed on S1 the number of iterates i, such that the orbitRiαn+1

(x)i=0,...,qn+1−1

is captured by one of the d trapping regions D−1ψn,γn

(Stl,k,j

(1)1 , ~j2,..., ~jm

)is at least(

1− 1n6

)·q3·(m−1)+4·(m−1)+...+(3+n2−1)·(m−1)−(m−1)·(3+k)n ·

(n2 − 8 · (m− 1)

)·⌊qn+1 ·

1− 2n4

n2 · γn

Depending on the point x ∈ Tm there is a portion $nt (x) of these iterates spent in trapping

regions of the second kind belonging to the specific Nt. This portion does not depend on theindices l, k, j(1)

1 , ~j2, ..., ~jm. The number of iterates i, such that the orbitRiαn+1

(x)i=0,...,qn+1−1

meets an arbitrary trapping region D−1ψn,γn

(Stl,k,j

(1)1 , ~j2,..., ~jm

), is not less than

q3·(m−1)+4·(m−1)+...+(3+n2−1)·(m−1)−(m−1)·(3+k)n ·$n

t (x) ·(n2 − 8 · (m− 1)

)· qn+1 ·

1− 4n4

n2 · γn

≥$nt (x) · qn+1 ·

(n2 − 8 · (m− 1)

1− 4n4

n4 · q2+(m−1)·(3+k)n

≥$nt (x) · qn+1 ·

(1− 8 ·m

n2 · q2+(m−1)·(3+k)n

iterates. Moreover, for every t ∈ 1, ..., d there are(qn − 2 ·

⌈qn4n6

⌉)m·(q2+kn

)m−1 trapping regions

of the second kind on[lqn

+ kn2·qn ,

+ k+1n2·qn

]×Nt ×Tm−2 for l = 0, ..., qn − 1, k = 1, ..., n2 − 1

and so not less than(qn − 2 ·

⌈ qn4n6

⌉)m·(q2+kn

)m−1 · qn+1 ·(

1− 8 ·mn2

n2 · q2+(m−1)·(3+k)n

≥qn+1 ·(

1− 1n6

1− 8mn2

)· 1n2 · qn

≥ qn+1 ·(

1− 9mn2

)· 1n2 · qn

iterates are trapped here. Altogether, at least qn+1 ·(1− 1

)·(1− 9m

)≥ qn+1 ·

(1− 9m+1

)iterates are captured.

Remark 5.3.3. On the contrary, at most 9m+1n2 ·qn+1 iterates are not captured by these trapping

regions of the second kind.

5.3.3.1 Partial partitions ηtn

For every t ∈ 1, ..., d the partial partition ηtn will be constructed on the fundamental sector[0, 1

]×Nt × Tm−2 initially. For this purpose, we divide the fundamental sector in sections:

• On[

kn2·qn ,

k+1n2·qn

]×Nt×Tm−2 in case of k, t ∈ N, 1 ≤ t ≤ d and 1 ≤ k ≤ n2− 2 the partial

partition ηtn consists of all multidimensional intervals of the following form:

n2 · qn+

+ ...+j(1+(m−1)·(3·(k+1)+

k·(k+1)2 ))

n2 · q2+(m−1)·(3(k+1)+k·(k+1)

10n8 · q2+(m−1)·(3(k+1)+k·(k+1)

n2 · qn+

+ ...+j(1+(m−1)·(3(k+1)+

k·(k+1)2 ))

n2 · q2+(m−1)·(3(k+1)+k·(k+1)

10n8 · q2+(m−1)·(3(k+1)+k·(k+1)

[t−1∑s=1

ls + lt ·

qn+ ...+

j(3+k+1)2

q3+k+1n

26 · n6 · q3+k+1n

t−1∑s=1

ls + lt ·

qn+ ...+

j(3+k+1)2 + 1q3+k+1n

− 126 · n6 · q3+k+1

×m∏i=3

qn+ ...+

j(3+k+1)i

q3+k+1n

26n6 · q3+k+1n

qn+ ...+

j(3+k+1)i + 1q3+k+1n

− 126n6 · q3+k+1

⌉≤ j(l)

1 ≤ qn−⌈qn

⌉−1 for l = 1, ..., 1+(m−1)·

(3 (k + 1) + k(k+1)

)as well as j(l)

i ∈ Z and⌈qn

⌉≤ j(l)

i ≤ qn−⌈qn

⌉−1 for i = 2, ...,m and l = 1, ..., 3+k+1.

• On[0, 1

n2·qn

]× Tm−1 as well as

[n2−1n2·qn ,

]× Tm−1 there are no elements of the partial

partition ηtn.

]×Nt×Tm−2 is extended

to a partial partition of Nt.

Remark 5.3.4. By construction the diameter of such a partition element is bounded by√mq5n

and this sequence of partial partitions converges to the decomposition into points on Nt.

]× Tm−1 initially and therefore divide this sector into sections:

kn2·qn ,

k+1n2·qn

]×Nt×Tm−2 in case of k, t ∈ N, 1 ≤ t ≤ d and 1 ≤ k ≤ n2−1 the partial partition

ζn consists of all sets D−1ψn,γn

(In), where In is a multidimensional interval of the following form:

n2 · qn+

n2 · q2n

+ ...+j

(1+(m−1)·

(3·n2+

n2·(n2−1)2

n2 · γn+

1n6 · γn

n2 · qn+

n2 · q2n

+ ...+j

(1+(m−1)·

(3·n2+

n2·(n2−1)2

s+ 1n2 · γn

− 1n6 · γn

[t−1∑s=1

ls + lt ·

qn+ ...+

j(3+k+1)2

q3+k+1n

+ ...+j(2+(m−1)·(3·(k+1)+

k·(k+1)2 ))

q2+(m−1)·(3·(k+1)+

k·(k+1)2 )

+j(3+(m−1)·(3·(k+1)+

k·(k+1)2 ))

8n8 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

8n14 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

t−1∑s=1

ls + lt ·

qn+ ...+

j(3+k+1)2

q3+k+1n

+ ...+j(2+(m−1)·(3·(k+1)+

k·(k+1)2 ))

q2+(m−1)·(3·(k+1)+

k·(k+1)2 )

+j(3+(m−1)·(3·(k+1)+

k·(k+1)2 ))

8n8 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

− lt

8n14 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσn]

×m∏i=3

qn+ ...+

j(3+k)i

n6 · q3+kn

qn+ ...+

j(3+k)i + 1q3+kn

− 1n6 · q3+k

⌈qnn6

⌉≤ j

(l)1 ≤ qn −

⌈qnn6

⌉− 1 for l = 1, ..., 1 + (m− 1) ·

(3n2 +

n2·(n2−1)2

j(l)2 ∈ Z and

⌈qnn6

⌉≤ j

(l)2 ≤ qn −

⌈qnn6

⌉− 1 for l = 1, ..., 2 + (m− 1) ·

(3 · (k + 1) + k·(k+1)

j(3+(m−1)·(3(k+1)+

k(k+1)2 ))

2 ∈ Z, 8n2·[nqσn] ≤ j(3+(m−1)·(3(k+1)+k(k+1)

2 ))2 ≤ 8n8·[nqσn]−8n2·[nqσn]−1,

j(l)i ∈ Z and

⌈qnn6

⌉≤ j(l)

i ≤ qn −⌈qnn6

⌉− 1 for i = 3, ...,m and l = 1, .., 3 + k as well as s ∈ N and

0 ≤ s ≤ n2 − 1.

Remark 5.3.5. For every n ≥ 4 the partial partition ζn consists of disjoint sets, covers a set ofmeasure at least 1− 3·m+1

n2 and the sequence (ζn)n∈N converges to the decomposition into points.

Remark 5.3.6. Note that Dψn,γn acts as an isometry on all the partition elements Γn ∈ ζn.

Once again we aim for a diffeomorphism gn which satisfies gn(In

)= g[nqσn]

)as well as

(Φn(In

))= g[nqσn]

(Φn(In

))for every In ∈ ηtn. In order to construct an f -invariant mea-

surable Riemannian metric gn has to be an isometry on the image under φn Dψn,γn of anypartition element D−1

ψn,γn

(In)∈ ζn.

]such that 1

ε ∈ Z. Moreover, we consider δ > 0 such that1δ ∈ Z and a·b·δ

ε ∈ Z. We denote[0, 1

]×[0, ε

b·a]× [δ, 1− δ]m−2 by ∆a,b,ε,δ. Using the

maps Da,b,ε : Rm → Rm, (θ, r1, ..., rm−1) 7→(a · θ, b·aε · r1, r2, ..., rm−1

)and gε produced in

Lemma 3.3.4 we define the measure-preserving diffeomorphism ga,b,ε,δ : ∆a,b,ε,δ → gb (∆a,b,ε,δ)by ga,b,ε,δ = D−1

a,b,ε (gε, idRm−2) Da,b,ε. Using the fact that a·b·δε ∈ Z we extend it to a

smooth diffeomorphism gt,a,b,ε·lt,δ :[0, 1

]×[∑t−1

s=1 ls + δ · lt,∑ts=1 ls − δ · lt

]× [δ, 1− δ]m−2 →

([0, 1

]×[∑t−1

s=1 ls + δ · lt,∑ts=1 ls − δ · lt

]× [δ, 1− δ]m−2

gt,a,b,ε·lt,δ

t−1∑s=1

ls + δ · lt + k · ε · ltb · a

+ r1, r2, ..., rm−1

t−1∑s=1

ls + b · δ · lt + k · ε · lta

t−1∑s=1

ls + δ · lt + k · ε · ltb · a

)+ ga,b,ε·lt,δ (θ, r1, ..., rm−1)

for r1 ∈[0, ε·ltb·a

]and some k ∈ Z satisfying 0 ≤ k ≤ 1−2·δ

ε · ba− 1. Since this map coincides withthe map gb in a neighbourhood of the boundary, we can proceed it to a map

gt,a,b,ε·lt,δ :[0,

]×Nt × [0, 1]m−2 → gb

]×Nt × [0, 1]m−2

)smoothly by putting it equal to gb. Then we can extend it smoothly to a diffeomorphismgt,a,b,ε·lt,δ :

]×Nt × Tm−2 → gb

([0, 1

]×Nt × Tm−2

We construct the smooth measure-preserving diffeomorphism gn on the fundamental sector[0, 1

]× Tm−1 initially and for this divide it into n2 sections:

• On[0, 1

n2·qn

]× Tm−1: gn = g[n·qσn].

• On[

kn2·qn ,

k+1n2·qn

]×Nt × Tm−2 in case of k, t ∈ Z, 1 ≤ t ≤ d and 1 ≤ k ≤ n2 − 1:

gn = gt,n2·q

2+(m−1)·(3·(k+1)+ (k+1)·k2 )

n ,[n·qσn],lt

8n6 ,1

Since gn coincides with the map g[n·qσn] in a neighbourhood of the boundary of the differentsections, this yields a smooth map and we can extend it to a smooth measure-preserving diffeo-morphism on Tm using the description gn R l

qn= R l

qngn for l ∈ Z. Furthermore, we note that

the subsequent constructions are done in such a way that 260n6 divides qn (see Lemma 5.4.6)and so the assumption a·b·δ

ε = a·b4 ∈ Z is satisfied for every t ∈ 1, ..., d.

Remark 5.3.7. We will call the parts of the domains ∆a,b,ε,δ corresponding to ∆ (4ε) of gε the“good area” of gn.

Indeed, this map gn satisfies the following aimed property:

Lemma 5.3.8. For every element In ∈ ηtn we have: gn(In

)= g[n·qσn]

Proof. The statement follows from the positioning of the partition elements by the same argu-ments as in the proof of Lemma 4.4.5.

We recall the smooth measure-preserving diffeomorphism ϕε,i,j from Lemma 3.3.6 and the mapsCλ (x1, x2, ..., xm) = (λ · x1, x2, ..., xm) for λ ∈ N. With these we define the measure-preserving

diffeomorphism φλ,ε,i,j :[0, 1

]× [0, 1]m−1 →

]× [0, 1]m−1 by C−1

λ ϕi,j,ε Cλ. Afterwards,φλ,ε,i,j is extended to a diffeomorphism on Tm by the description

φλ,ε,i,j

λ, x2 + k2, ..., xm + km

λ, k2, ..., km

)+ φλ,ε,i,j (x1, x2, ..., xm)

for ki ∈ Z. For convenience we will use the following notation:

φ(j)λ = φλ, 1

60n6 ,1,j.

Moreover, let Dλ (x1, ..., xm) = (λ · x1, λ · x2, ..., λ · xm), µ ∈ N, 1δ ∈ N and 1

δ divides µ. Weconstruct a diffeomorphism ψµ,δ,i,j,ε2 as in subsection 3.3.3.Furthermore, we will use the maps Bt,λ,lt :

]× [0, lt]× [0, 1]m−2 → [0, 1]m defined by

Bt,λ,lt (x1, x2, x3, ..., xm) =(λ · x1,

1lt· x2, x3, ..., xm

With these we construct the measure-preserving diffeomorphisms

φt,lt,i,j,λ,ε1,µ,ε2,δ :[0,

]×Nt × [0, 1]m−2 →

]×Nt × [0, 1]m−2

by the description (with x2 ∈ [0, lt])

φt,lt,i,j,λ,ε1,µ,ε2,δ

t−1∑s=1

ls + x2, x3, ..., xm

0,t−1∑s=1

ls, 0, ..., 0

)+B−1

t,λ,lt ψµ,δ,i,j,ε2 ϕε1,i,j Bt,λ,lt (x1, x2, x3, ..., xm)

As above we extend it to a smooth diffeomorphism on S1×Nt×Tm−2 and introduce advantegeousnotations:

φ(t,lt,j)λ,µ := φt,lt,1,j,λ, 1

60n6 ,µ,1

22n6 ,1

Now we can define the conjugation map φn on the fundamental sector[0, 1

]× [0, 1]× Tm−2:

• On[0, 1

n2·qn

]× [0, 1]× Tm−2:

φ(m)n2·qn ... φ

n2·qm−2n φ(2)

n2·qm−1n

• On[

kn2·qn ,

k+1n2·qn

]×Nt×Tm−2 in case of t ∈ N, 1 ≤ t ≤ d as well as k ∈ N and 1 ≤ k ≤ n2−1:

φ(t,lt,m)

n2·q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)+(3+k)·(m−2)n ,q3+k

... φ(t,lt,2)

n2·q2+3·(m−1)+...+(3+k−1)·(m−1)n ,q3+k

=φ(t,lt,m)

n2·q2+(m−1)·(3k+ k·(k−1)

2 )+(3+k)·(m−2)n ,q3+k

... φ(t,lt,2)

n2·q2+(m−1)·(3k+ k·(k−1)

2 )n ,q3+k

Since the map φn coincides with the identity on a neighbourhood of the different sections, we canpiece it together smoothly in this way and obtain a diffeomorphism on

]×Tm−1. Moreover,

we extend it to a diffeomorphism on Tm using the rule φn R uqn

= R uqn φn, where u ∈ Z.

5.4 Convergence of (fn)n∈N in Diff∞ (Tm)

By the same methods as in section 3.6 we show that the sequence of constructed measure-preserving smooth diffeomorphisms fn = Hn Rαn+1 H−1

n converges.

We want to find estimates on the norms |||Hn|||k. For this purpose, we have to estimate thenorms of the occurrent maps.

|||Dψn,γn |||k ≤ C · γkn,

Proof. See Lemma 4.7.1.

Next we estimate the norms of the map φn:

|||φn|||k ≤ C · γ(m−1)·kn ,

where C is a constant depending on m, k, lmin and n, but is independent of qn.

Proof. First of all we consider the map φt,λ,µ := B−1t,λ,ltψµ,δ,i,j,ε2ϕε,i,jBt,λ,lt on Nt introduced

in subsection 5.3.5:

φt,λ,µ (x1, ..., xm) =(1λ· [ψµ ϕε]1

(λ · x1,

1ltx2, x3, ..., xm

), lt · [ψµ ϕε]2

(λ · x1,

1ltx2, x3, ..., xm

[ψµ ϕε]3

(λ · x1,

1ltx2, x3, ..., xm

), ..., [ψµ ϕε]m

(λ · x1,

1ltx2, x3, ..., xm

Let k ∈ N. We compute for a multiindex ~a with 0 ≤ |~a| ≤ k:∥∥D~a [φt,λ,µ]1∥∥0≤(

)k· λk−1 · |||ψµ ϕε|||k,

∥∥D~a [φt,λ,µ]2∥∥0≤(

)k−1

· λk · |||ψµ ϕε|||k and

for r ∈ 3, ...,m:∥∥D~a [φt,λ,µ]r∥∥0

)k· λk · |||ψµ ϕε|||k.

Then we examine the expression |||ψµ ϕε|||k as in the proof of Lemma 3.6.4 and obtain:|||φt,λ,µ|||k ≤ C · µk · λk, at which the constant depends on m, k, lt, ε1 and ε2.Considering the composition φ := φ

(m)t,λm,µm

... φ(2)t,λ2,µ2

we show |||φ|||k ≤ C ·λ(m−1)·kmax ·µ(m−1)·k

for every k ∈ N inductively as in Lemma 3.6.4, where C is a constant depending on m, k, lt, ε1

and ε2.In the setting of our explicit construction of the map φn in section 5.3.5 we have ε1 = 1

60·n6 ,

ε2 = 122·n6 , λmax = n2 · q

2+(m−1)·(

3·(n2−1)+(n2−1)·(n2−2)

)+(3+n2−1)·(m−2)

n and µmax = q3+n2−1n .

90 Convergence of (fn)n∈N in Diff∞ (Tm)

|||φn|||k ≤ C ·

n2 · q2+(m−1)·

(3(n2−1)+

(n2−1)(n2−2)2

)+(3+n2−1)(m−2)

(m−1)·k

·(q3+n2−1n

)(m−1)·k

≤ C (m, k, n, lmin) · γ(m−1)·kn ,

where C (m, k, n, lmin) is a constant independent of qn.On the domains, where φn is defined with the aid of the maps φ(j)

λ , the estimates follow by thesame arguments.

In the next step we consider the map gn φn, where gn is constructed in section 5.3.4:

|||gn φn|||k ≤ C · qkn · γk·mn ,

Proof. In the proof of Lemma 3.6.5 we saw: |||ga,b,ε|||k ≤ Cε,k · bk · ak. By our choice ofparameters in section 5.3.4 we have b = [n · qσn] ≤ n · qσn, a ≤ γn and ε = lt

8n4 . Hence:

|||gn|||k ≤ Cn,k,lmin · qσ·kn · γkn ≤ Cn,k,lmin · qkn · γkn.

Then we continue as in the proof of Lemma 3.6.5 with the result of Lemma 5.4.2 and obtain thestatement.

Combining the results of Lemma 5.4.3 and Lemma 5.4.1 with the help of the formula of Faàdi Bruno we obtain for the conjugating map hn = gn φn Dψn,γn :

|||hn|||k ≤ C · qkn · γk·(m+1)n ,

Finally, we are able to prove an estimate on the norms of the map Hn as in Lemma 4.7.5:

|||Hn|||k ≤ C · qk·(m+1)·m·

n2·(n2+5)2

where C is a constant depending solely on m, k, lmin, n and Hn−1. Since Hn−1 is independentof qn in particular, the same is true for C.

5.4.2 Proof of convergenceWe check that we can satisfy the requirements of Lemma 3.6.8 under some assumptions on thesequence (αn)n∈N:

qnof rational numbers with 260n6 divides qn and qn > n2 ·maxi=1,...,n Li (where

Li denotes the Lipschitz constant of ρi ∈ Ξ), such that our conjugation maps Hn constructed insection 5.3 fulfil the following conditions:

|α− αn| <1

2 · kn · Ckn · |||Hn|||kn+1kn+1

|α− αn| <1

2n+1 · qn · |||Hn|||1.

Proof. In Lemma 5.4.5 we deduced the estimate |||Hn|||kn+1 ≤ Cn · q(kn+1)·(m+1)·m·

n2·(n2+5)2

n ,where the constant Cn was independent of qn. Thus, we can choose qn ≥ Cn for every n ∈ N.

Hence, we obtain: |||Hn|||kn+1 ≤ q(m+1)·m·n2·(n2+5)·(kn+1)n . Besides qn ≥ Cn we require

qn ≥ 260 · n6 · 320 · (n− 1)17 · |||ψn−1|||1 ·1lmin

· q2·(

2+(m−1)·(

3·(n−1)2+(n−1)2·((n−1)2−1)

))n−1 ,

at which lmin := mint=1,...,d lt as above. Furthermore, we can demand qn > n2 · maxi=1,...,n Liand ‖DHn−1‖0 <

ln(qn)n from qn because Hn−1 is independent of qn. Since α is a Liouvillean

number, we find a sequence of rational numbers αn = pnqn, pn, qn relatively prime, converging to

α under the above restrictions (formulated for qn) satisfying:∣∣∣∣α− pnqn

∣∣∣∣ < |α− αn−1|2n+1 · kn · Ckn · (260n6)1+(m+1)·m·n2·(n2+5)·(kn+1)2

· q1+(m+1)·m·n2·(n2+5)·(kn+1)2

|α− αn| <|α− αn−1|

2n+1 · kn · Ckn · q1+(m+1)·m·n2·(n2+5)·(kn+1)2

So we have |α− αn|n→∞−→ 0 monotonically. Because of |||Hn|||kn+1

kn+1 ≤ q(m+1)·m·n2·(n2+5)·(kn+1)2

this yields: |α− αn| < 1

. Thus, the first property of this Lemma is

fulfilled.Then we can verify the second property in the same way as in the proof of Lemma 3.6.9.

Remark 5.4.7. Lemma 5.4.6 shows that the conditions of Lemma 3.6.8 are satisfied. There-fore, our sequence of constructed diffeomorphisms fn converges in the Diff∞ (Tm)-topology to adiffeomorphism f ∈ Aα (Tm).

In our criterion for minimality in the next section we will need a result on the proximity ofiterates:

(f m, f mn

)≤ 1

Proof. Analogous to the proof of Lemma 3.6.11.

92 Proof of minimality

5.5 Proof of minimality

5.5.1 Criterion for minimalityWe recall the notion of a minimal dynamical system:

Definition 5.5.1. Let X be a topological space and f : X → X be a continuous transformation.The map f is called minimal if for every x ∈ X the orbit

f i (x)

i∈N is dense in X.

Equivalently f is minimal if for every x ∈ X and every non-empty open set U ⊆ X there isi ∈ N such that f i (x) ∈ U . In the case of X being a metric space every open set contains anε-ball for ε sufficiently small. Thus, f is minimal if for every x ∈ X, every ε > 0 and for everyε-ball Bε there is i ∈ N such that f i (x) ∈ Bε. Hereby, we can deduce the subsequent criterionof minimality in the setting of our constructions from section 5.3:

Lemma 5.5.2. Suppose that the set of iteratesφn Dψn,γn Riαn+1

H−1n (x)

i=0,...,qn+1−1

meets every cube∏mk=1

[jkqn, jk+1qn

], where

⌈qn4n6

⌉≤ jk ≤ qn −

⌈qn4n6

⌉− 1, for every x ∈ Tm.

Moreover, we assume that the sequence (fn)n∈N constructed as in section 5.3 converges to adiffeomorphism f in the C∞-topology and satisfies d0

(f i, f in

2n for all i = 0, ..., qn+1 − 1.Then f = limn→∞ fn is minimal.

Proof. At first we observe that

(Hn−1 gn

(m∏k=1

[jkqn,jk + 1qn

]))≤ ‖DHn−1‖0 · [nq

σn] ·√m

which converges to 0 as n → ∞ (because of ‖DHn−1‖0 <ln(qn)n ), and that the measure of the

union of all the sets Hn−1gn(∏m

[jkqn, jk+1qn

]), where

⌈qn4n6

⌉≤ jk ≤ qn−

⌈qn4n6

⌉−1, approaches

1 as n → ∞. Hence, for every ε > 0 and y ∈ Tm there is M1 ∈ N such that for every n ≥ M1

there exists a set Hn−1 gn(∏m

[jkqn, jk+1qn

])⊆ B ε

2(y), at which

⌈qn4n6

⌉≤ jk ≤ qn −

⌈qn4n6

⌉− 1.

Let x ∈ N, ε > 0 and an ε-ball Bε (y), at which y ∈ Tm, be arbitrary. Since d0

(f i, f in

2n forall i = 0, ..., qn+1 − 1 there is M2 ∈ N such that d0

(f i, f in

2 for all i = 0, ..., qn+1 − 1 andn ≥M2.We consider n ≥ N := max M1,M2. Then there is a setHn−1gn

(∏mk=1

[jkqn, jk+1qn

])⊆ B ε

where⌈qn4n6

⌉≤ jk ≤ qn−

⌈qn4n6

⌉− 1, and by the assumption of this Lemma an i < qn+1 such that

f in (x) ∈ Hn−1 gn(∏m

[jkqn, jk+1qn

])⊆ B ε

2(y). By the triangle inequality we obtain

d(f i (x) , y

)≤ d

(f i (x) , f in (x)

(f in (x) , y

)≤ d0

(f i, f in

2< ε.

Thus, we conclude f i (x) ∈ Bε (y). Hence, f is minimal.

5.5.2 Application of the criterion

The conditions on the convergence of the sequence (fn)n∈N and proximity d0

(f i, f in

2n forall i = 0, ..., qn+1 − 1 are fulfilled by Remark 5.4.7 and Lemma 5.4.8 respectively.Let x ∈ Tm and jk ∈ N satisfying

⌈qn4n6

⌉≤ jk ≤ qn−

⌈qn4n6

⌉−1 be arbitrary. We have to show that

the orbitφn Dψn,γn Riαn+1

H−1n (x)

i=0,...,qn+1−1

meets the cube∏mk=1

[jkqn, jk+1qn

]. For this

The ergodic invariant measures 93

purpose, we note that there is i ∈ 0, ..., qn+1 − 1 with Riαn+1H−1

n (x) ∈ D−1ψn,γn

(Sj1,j2,...,jm)by Remark 5.3.1. Then we compute

φn Dψn,γn

(D−1ψn,γn

(Sj1,j2,...,jm))

=[j1qn

4n8 · qn,j1qn

n2 · qn− 1

4n8 · qn

]×[j2qn,j2 + 1qn

m∏i=3

4n6 · qn,ji + 1qn

− 14n6 · qn

]and we can apply Lemma 5.5.2 to prove the minimality of f .

5.6 The ergodic invariant measures

5.6.1 Exactly d ergodic invariant measures

5.6.1.1 Construction of the measures

As announced we will construct the ergodic invariant measures with the aid of the normalizedrestrictions µt of the Lebesgue measure on the sets

Nt := S1 ×Nt × Tm−2, at which Nt :=

[t−1∑k=1

t∑k=1

]⊆ R

i.e. µt (A) =µ(A∩Nt)µ(Nt) for any measurable set A ⊆ Tm. Since each set Nt is Rβ-invariant for any

β ∈ S1, we have (Rβ)∗ µt = µt. With these we define the measures ξnt := (Hn)∗ µt and can provetheir fn-invariance:

(fn)∗ ξnt = (fn)∗ ((Hn)∗ µt) = (fn Hn)∗ µt =

(Hn Rαn+1

)∗ µt = (Hn)∗

(Rαn+1

)∗ µt = ξnt .

Here we used the relation f∗g∗µ = (f g)∗ µ for maps f, g. This holds because we have for anymeasurable set A:

f∗g∗µ (A) = g∗µ(f−1 (A)

(g−1

(f−1 (A)

))= µ

((f g)−1 (A)

)= (f g)∗ µ (A) .

In the next step we want to estimate µ(Hn

)4Hn−1

)). For this purpose, we have to

examine which parts of the set Nt are not mapped back to Nt under hn = gn φn Dψn,γn .Since Nt is gn-invariant, the measure difference is composed of the part, where the conjugationmap φn Dψn,γn is constructed to prove minimality (i.e. on the qn sets

[kqn, kqn + 1

n2·qn

]×Tm−1

for k = 0, ..., qn − 1), and the part that is not mapped back to Nt under Dψn,γn . The last one iscaused by the translation about at most 1

n4 in the r1-coordinate produced by Dψn,γn . Altogether,we obtain:

)4Hn−1

))= µ

)≤ qn ·

1n2 · qn

+2n4≤ 2n2. (5.1)

Now we can use the same approach as in [Win01], chapter 7:By equation 5.1 the sequence

)n∈N

is a Cauchy sequence in the metric on the associated

measure algebra. Since this space is complete (e.g. [Pet83], Proposition 1.4.3.), there exists alimit Bt := limn→∞Hn

)in the measure algebra. For this limit we have µ (Bt) = µ

94 The ergodic invariant measures

because Hn is measure-preserving. The sets Bt and Bs are measurably disjoint due to thedisjointness of the sets Nt and Ns. Moreover, we have weak convergence of the measures (ξnt )n∈Nto a measure ξt, where ξt (A) = µ(A∩Bt)

µ(Bt)for any measurable set A ⊆ Tm. For this absolutely

continuous measure ξt we conclude limn→∞ (fn)∗ ξnt (A) = f∗ξt (A) due to the triangel inequality

µ(H−1n

(f−1n A

)∩ Nt

(f−1n A ∩Hn

))≤ µ

(f−1n A ∩Bt

)≤ µ

(f−1n A4f−1A

(f−1A ∩Bt

)(where the first summand converges to 0 as n→∞ because of fn → f). So we obtain

ξt = limn→∞

ξnt = limn→∞

(fn)∗ ξnt = f∗ξt

using the shown fn-invariance of the measure ξnt . Thus, the measures ξt are f -invariant.Furthermore, these measures ξt are linearly independent because the sets B1, ..., Bd are measur-ably disjoint as noted before. Since any non-ergodic invariant measure can be written as a linearcombination of ergodic measures ([Wa75], Theorem 5.15), there cannot be less than d ergodicmeasures.

5.6.1.2 Trapping property

In the next subsection 5.6.2 we show that f is weak mixing with respect to the measures ξt, inparticular these are ergodic measures. We have to establish that there are no further ones. Forthis purpose, we will prove a result on the Birkhoff sums (see Lemma 5.6.2) and have to gaincontrol over almost everything of every Rt-orbit. In this connection the following sets are useful:In case of 0 ≤ l ≤ qn − 1, 1 ≤ k ≤ n2 − 1, j(1)

i ∈ Z,⌈qn4n6

⌉≤ j(1)

⌉− 1 for i = 1, ...,m

as well as j(u)i ∈ Z, 0 ≤ j(u)

i ≤ qn − 1 for i = 2, ...,m and u = 2, 3 we introduce

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

n2 · qn+

n2 · q2n

n2 · qn+j

(1)1 + 1n2 · q2

t−1∑s=1

ls + lt ·

t−1∑s=1

ls + lt ·

(3)2 + 1q3n

m∏i=3

(3)i + 1q3n

Note that there are qn ·(n2 − 1

)·(qn − 2 ·

⌈qn4n6

⌉)m · q(m−1)·2n such sets ∆t

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,...,j

on Nt. We denote the family of these sets by T tn as well as the union of these sets by T tn. Then

µ(Nt \ T tn

)= lt −

(n2 − 1

)· qn ·

(qn − 2 ·

⌈ qn4n6

⌉)m· q(m−1)·2n · lt

n2 · q2+3·(m−1)n

1− n2 − 1n2

1− 2 · 13n6

)m)· lt ≤

+2m3n6

)· lt,

i.e. µt(Nt \ T tn

)≤ 1

n2 + 2m3n6 ≤ 2

We observe that gn = g[nqσn] on ∆t

l,k,j(1)1 ,...,j

and diam(gn

l,k,j(1)1 ,...,j

(1)m ,j

(2)m ,j

Since ‖DHn−1‖ < ln(qn)n we conclude that diam

(Hn−1 gn

l,k,j(1)1 ,...,j

qnfor n suffi-

ciently large. By the requirements on the number qn in Lemma 5.4.6 we obtain

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,...,j

))≤ Lip (ρi) ·

for every x, y ∈ ∆t

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

and the function ρi ∈ Ξ in case of

i = 1, ..., n. Averaging over all y ∈ ∆t

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

we obtain:∣∣∣∣∣∣∣∣ρi (Hn−1 gn (x))− 1

(Hn−1 gn

l,k,j(1)1 ,...,j

)) ∫Hn−1gn

l,k,j(1)1 ,...,j

) ρi dξnt∣∣∣∣∣∣∣∣ <

(Stl,k,j

(1)1 , ~j2,..., ~jm

)defined in section

5.3.2 is mapped under φn Dψn,γn onto

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−1)·(3·k+

k·(k−1)2 ))

n2 · q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)n

− t(1)2

n2 · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+1n

− ...− t(3+k)2

n2 · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+3+kn

− t(1)3

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+3+k+1n

− ...− t(3+k)3

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+2·(3+k)n

− ...

− t(3+k)m + 1

n2 · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)n

+t((m−1)·(3·k+

k·(k−1)2 )+1)

n2 · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)+1n

+ ...+t

((m−1)·

(3·(n2−1)+

(n2·(n2−1)

2 −k)))

1n6 · γn

n2 · qn+ ...+

((m−1)·

(3·(n2−1)+

(n2·(n2−1)

2 −k)))

1 + 1γn

− 1n6 · γn

[t−1∑s=1

ls + lt ·

qn+ ...+

j(3+k)2

4 · n6 · q3+kn

t−1∑s=1

ls + lt ·

qn+ ...+

j(3+k)2 + 1q3+kn

− 14 · n6 · q3+k

×m∏i=3

qn+ ...+

j(3+k)i

4 · n6 · q3+kn

qn+ ...+

j(3+k)i + 1q3+kn

− 14 · n6 · q3+k

⌈qn4n6

⌉≤ t(1)

i ≤ qn−⌈qn4n6

⌉−1 for i = 2, ...,m as well

as t(l)i ∈ Z, 0 ≤ t(l)i ≤ qn−1, for l = 2, ..., 3+k and i = 2, ...,m as well as t(j)1 ∈ Z, 0 ≤ t(j)1 ≤ qn−1,

for j = 1, ..., (m− 1) ·(

3 · (n2 − 1) + n2·(n2−1)2 − k

)apart from t

((m−1)·(3·k+k·(k−1)

⌉≤ t((m−1)·(3·k+

k·(k−1)2 )+1)

1 ≤ qn −⌈qn4n6

⌉− 1.

(D−1ψn,γn

(Stl,k,j

(1)1 , ~j2,..., ~jm

))is contained in ∆t

l,k,j(1)1 ,j

(1)2 ,...,j

(1)m ,j

(2)m ,j

The same is true for the other allowed values of j(4)i , ..., j

(3+k)i . Thus, there are qk·(m−1)

n trap-ping regions that are mapped into ∆t

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

under φn Dψn,γn .

Hence, we can estimate the number of i ∈ 0, ..., qn+1 − 1 such that φn Dψn,γn Riαn+1(x) is

contained in ∆t

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

by$nt (x)·qk·(m−1)

n ·qn+1 ·1− 8m

n2·q(m−1)·(3+k)n ·q2

for arbitrary x ∈ Tm using Remark 5.3.2.

Lemma 5.6.1. Let ρi ∈ Ξ and i = 1, ..., n. Then for every y ∈ Tm we have

infξn∈Θn

∣∣∣∣∣ 1qn+1

qn+1−1∑k=0

ρi(fkny

)−∫ρi dξ

∣∣∣∣∣ < 17m+ 3n2

· ‖ρi‖0 +1n2,

where Θn is the simplex generated by ξn1 , ..., ξnd .

Proof. Let x ∈ Tm be arbitrary. We introduce the measure ξnx :=∑dt=1$

nt (x) · ξnt ∈ Θn.

The set of numbers k ∈ 0, 1, ..., qn+1 − 1 such that the iterates Rkαn+1(x) are not contained in

one of the trapping regions of the second kind is denoted by Ia. Referred to Remark 5.3.3 there areat most 9m+1

n2 ·qn+1 numbers in Ia. We obtain∣∣∣∑k∈Ia ρi

(Hn Rkαn+1

(x))∣∣∣ ≤ ‖ρi‖0 · 9m+1

n2 ·qn+1.

Moreover, we denote the set of k ∈ 0, 1, ..., qn+1 − 1 such that the iterate φnDψn,γn Rkαn+1(x)

is contained in the corresponding trapping region ∆ ∈ T tn by I∆. By the above considerations

there are at least $nt (x) · qk·(m−1)

n · qn+1 ·1− 8m

n2·q(m−1)·(3+k)n ·q2

= $nt (x) · qn+1 ·

(1− 8m

)·µt (∆) and

at most $nt (x) · qn+1 · µt (∆) many numbers in I∆ for an arbitrary ∆ ∈ T tn. Thus, we obtain for

an arbitrary ∆ ∈ T tn using equation 5.2:∣∣∣∣∣∣ 1qn+1

∑j∈I∆

(x))−∫Hn−1gn(∆)

ρi d ($nt (x) ξnt )

∣∣∣∣∣∣≤ ($n

t (x)µt) (∆)n2

+8mn2·∫Hn−1gn(∆)

|ρi| d ($nt (x) ξnt ) ≤ ($n

t (x)µt) (∆) ·(

+8mn2· ‖ρi‖0

Altogether, we conclude∣∣∣∣∣ 1qn+1

qn+1−1∑k=0

(Hn Rkαn+1

x)−∫ρi dξ

∣∣∣∣∣=

∣∣∣∣∣ 1qn+1

qn+1−1∑k=0

(Hn Rkαn+1

−d∑t=1

∑∆∈T tn

∫Hn−1gn(∆)

ρi d ($nt (x) ξnt ) +

∫Hn−1gn(Nt\T tn)

ρi d ($nt (x) ξnt )

∣∣∣∣∣

∣∣∣∣∣∣d∑t=1

∑∆∈T tn

∑j∈I∆

(x))−∫Hn−1gn(∆)

ρid ($nt (x) ξnt )

∣∣∣∣∣∣+

· ‖ρi‖0 ·9m+ 1n2

· qn+1 + ‖ρi‖0 ·d∑t=1

($nt (x)µt)

(Nt \ T tn

)≤ 1n2

+8mn2· ‖ρi‖0 + ‖ρi‖0 ·

9m+ 1n2

+2 · ‖ρi‖0n2

=17m+ 3n2

· ‖ρi‖0 +1n2.

With x = H−1n (y) we obtain the claim.

We point out that the measure ξnx used in the above proof was dependent on the point x, butindependent of the function ρ ∈ Ξ.

Lemma 5.6.2. For every ρ ∈ Ξ and y ∈ Tm we have

infξn∈Θn

∣∣∣∣∣ 1qn+1

qn+1−1∑k=0

ρ(fk (y)

)−∫ρ dξn

∣∣∣∣∣→ 0 as n→∞,

where Θn is the simplex generated by ξn1 , ..., ξnd .

Proof. By Lemma 5.4.8 we have

d(qn+1)0 (f, fn) := max

i=0,1,...,qn+1−1d0

(f i, f in

) n→∞→ 0.

Then for every ρ ∈ Ξ we have∣∣ρ (f i (x)

)− ρ

(f in (x)

)∣∣ n→∞→ 0 uniformly for i = 0, 1, ..., qn+1 − 1,because every continuous function on the compact space Tm is uniformly continuous. Thus,we get:

∥∥∥ 1qn+1

∑qn+1−1i=0 ρ

(f i (x)

)− 1

∑qn+1−1i=0 ρ

(f in (x)

)∥∥∥0

n→∞→ 0. Applying the previousLemma 5.6.1 we obtain the claim.

Since the family Ξ is dense in C (Tm,R), the convergence holds for every continuous functionby an approximation argument.Now we can prove that the measures ξ1, ..., ξd are the only possible ergodic ones: Assume thatthere is another ergodic invariant probability measure ξ. By the Birkhoff Ergodic Theorem wehave for every ρ ∈ C (Tm,R)

limn→∞

n−1∑k=0

ρ(fk (x)

Tmρ dξ for ξ-a.e. x ∈ Tm.

With the aid of Lemma 5.6.2 we obtain for every ρ ∈ C (Tm,R) and x in a set of ξ-full measure:∫Tm

ρ dξ = limn→∞

n−1∑k=0

ρ(fk (x)

)= limn→∞

qn+1−1∑k=0

ρ(fk (x)

)= limn→∞

ρ dξn,

where ξn is in the simplex generated by ξn1 , ..., ξnd . As noted this measure does not depend onthe function ρ. Thus, we have for every ρ ∈ C (Tm,R): limn→∞

∫Tm ρ dξ

n =∫

Tm ρ dξ. Since thesimplex generated by ξ1, ..., ξd is weakly closed, this implies that ξ is in this simplex. We recallthat ergodic measures are the extreme points in the set of invariant Borel probability measures(see [Wa75], Theorem 5.15.). Then ξ has to be one of the measures ξ1, ..., ξd and we obtain acontradiction.

5.6.2 Weak mixing property w.r.t. ξtWe will prove the weak mixing property for every measure ξt by an approach similar to chapter4 applied on every set Bt respectively Nt.

5.6.2.1 (γ, ε)-distribution

The proof of the weak mixing property will base upon the notion of a (γ, ε)-distribution fromDefinition 4.5.1.

Remark 5.6.3. In the following we will often write the second property of a (γ, ε)-distributionon Jt ⊆ Nt × Tm−2 in the form of∣∣∣µt (I ∩ Φ−1

(S1 × J

))· µt (Jt)− µt

(I)· µt

(J)∣∣∣ ≤ ε · µt (I) · µt (J) ,

where µt is the normalized restriction of the Lebesgue measure on Nt × Tm−2.

In the next step we define the sequence of natural numbers (mn)n∈N analogously to section3.4:

∣∣∣∣m · pn+1

qn+1− 1n · qn

∣∣∣∣ ≤ 260 · (n+ 1)6

m ≤ qn+1 : m ∈ N, inf

∣∣∣∣m · qn · pn+1

qn+1− 1n

∣∣∣∣ ≤ 260 · (n+ 1)6 · qnqn+1

∣∣∣m · qn·pn+1qn+1

− 1n + k

∣∣∣ ≤ 260·(n+1)6·qnqn+1

Proof. In Lemma 5.4.6 we constructed the sequence αn = pnqn

in such a way that qn = 260n6 · qnand pn = 260n6 · pn with pn, qn relatively prime. Then the proof follows along the lines of Lemma3.4.3.

an =(mn ·

qn+1− 1n · qn

By the above construction of mn it holds: |an| ≤ 260·(n+1)6

qn+1. In Lemma 5.4.6 we posed the

condition qn+1 ≥ 260 · (n+ 1)6 · 320 · n13 · |||ψn|||1 · γ2n · 1

lmin. Thus we get:

|an| ≤1

320 · n13 · |||ψn|||1 · γ2n

· lmin.

Lemma 5.6.6. We consider Jt :=[∑t−1

s=1 ls + ltn6 ,∑ts=1 ls −

1n6 , 1− 1

]m−2 ⊂ Nt×Tm−2

and the diffeomorphism Φn := φn Dψn,γn Rmnαn+1D−1

ψn,γn φ−1

n with the conjugating maps φndefined in section 5.3.5. Then the elements In,t of the partition ηtn in Nt are

n·qmn, 1n

)-distributed

on Jt under Φn and Φn(In,t

)⊂ Nt for all t ∈ 1, ..., d and n ≥ 3

√4 · (m− 1).

Proof. The proof is analogous to the proof of Lemma 4.5.5.

Once again we have the next property concerning the conjugating map gn:

Lemma 5.6.7. Let t ∈ 1, ..., d. For every In ∈ ηtn we have: gn(

Φn(In

))= g[nqσn]

(Φn(In

Proof. Analogous to Lemma 4.5.7.

Then we can deduce a statement similar to Lemma 4.6.1:

Lemma 5.6.8. Let t ∈ 1, ..., d, n ≥ 3√

4 · (m− 1) and I ∈ ηtn. We consider the diffeomorphismΦn = φnDψn,γnRmnαn+1

D−1ψn,γn

φ−1n and Jt :=

[∑t−1s=1 ls + lt

n6 ,∑ts=1 ls −

1n6 , 1− 1

]m−2.Then for every m-dimensional cube S of side length q−σn lying in S1 × Jt we get∣∣∣µt (I ∩ Φ−1

n g−1n (S)

)· µt (J)− µt

(I)· µt (S)

∣∣∣ ≤ 21n· µt

(I)· µt (S) . (5.3)

Proof. Since gn Φn(I)⊂ Nt by Lemma 5.6.6 and S ⊂ S1 × Jt ⊂ Nt we observe µ

µ(I ∩ Nt

), µ(I ∩ Φ−1

n g−1n (S)

(I ∩ Φ−1

n g−1n (S) ∩ Nt

)and µ (S) = µ

(S ∩ Nt

). Then

the proof is analogous to the one of Lemma 4.6.1, where the occurring measures can be replacedby the normalized restriction on Nt and Nt×Tm−2 respectively. In the deduction of this criterionfor weak mixing a result like Lemma 5.6.7 is used in the proof.

5.6.2.2 Approximation steps

At first we introduce the maps Ψn,n+k := hn...hn+kRmnαn+k+1h−1

n+k...h−1n and the parameter

κn := d0

(Ψn,n+1, hn Rmnαn+1

h−1n

)+∞∑l=1

d0 (Ψn,n+l+1,Ψn,n+l) .

With the aid of Lemma 3.6.7 we calculate for any l ≥ 0:

(hn ... hn+l+1 Rmnαn+l+2

h−1n+l+1 ... h

−1n , hn ... hn+l Rmnαn+l+1

h−1n+l ... h

(hn ... hn+l+1 Rmnαn+l+2

h−1n+l+1 ... h

−1n , hn ... hn+l+1 Rmnαn+l+1

h−1n+l+1 ... h

)≤|||Hn+l+1|||1 ·mn · |αn+l+2 − αn+l+1|≤|||Hn+l+1|||1 · qn+1 · 2 · |α− αn+l+1|

<|||Hn+l+1|||1 · qn+1 · 2 ·1

2 · kn+l+1 · Ckn+l+1 · |||Hn+l+1|||kn+l+1+1kn+l+1+1

kn+l+1 · Ckn+l+1 · qn+l+1,

at which we used Lemma 5.4.6 in the next-to-last step. So we conclude κn < q−σn → 0 as n→∞.Moreover, for any k ∈ N we note

(Ψn,n+k, hn Rmnαn+1

h−1n

)≤ d0

(Ψn,n+1, hn Rmnαn+1

h−1n

)+k−1∑l=1

d0 (Ψn,n+l+1,Ψn,n+l) < κn.

Next we define the partial partitions νtn for t ∈ 1, ..., d: Consider a disjoint covering of Jt =S1×

[∑t−1s=1 ls + lt

n6 ,∑ts=1 ls −

1n6 , 1− 1

]m−2 ⊂ Nt by cubes An,i with edge length q−σ2

Then νtn consists of all the sets of the form An,i = Hn−1

). Since ‖DHn−1‖0 <

ln(qn)n by

Lemma 5.4.6, we have diam(An,i

)≤ ‖DHn−1‖0 ·

√m · q−

n → 0 as n→∞.

Moreover, we consider the partial partitions ηtn :=

Γn = Hn−1 gn(In

): In ∈ ηtn

for every

t ∈ 1, ..., d. As noted before diam(In

)≤√mq5n

(see Remark 5.3.4) and gn|In = g[nqσn]|In (see

Lemma 5.3.8). So we obtain diam(Γn) ≤ ‖DHn−1‖0 ·∥∥Dg[nqσn]

∥∥0·√mq5n→ 0 as n→∞.

Hence, the sequences (νtn)n∈N as well as (ηtn)n∈N converge to the decomposition into points of

Bt = limn→∞Hn−1

)as n→∞ in each case.

In order to prove the weak mixing property of the measure ξt we want to show

limn→∞

ξt(B ∩ f−mn (A)

)= ξt (B) · ξt (A)

for every measurable sets A,B ⊆ Bt (see [Skl67]). For this purpose, let two measurable setsA,B ⊆ Bt and ε > 0 be given.

• Since ηtn → ε and νtn → ε there is M1 ∈ N such that for every n ≥M1 we have approxima-tions An =

⋃i An,i of A and Bn =

⋃j Γn,j of B such that ξt (A4An) < ε · ξt (A) · ξt (B) as

well as ξt (B4Bn) < ε · ξt (A) · ξt (B). Then we have

)= ξt (fmn (B) ∩A)

≤ ξt(Bn ∩ f−mn (An)

)+ ξt (A4An) + ξt (B4Bn)

≤ ξt(Bn ∩ f−mn (An)

)+ 2 · ε · ξt (A) · ξt (B) .

• We recall Bt = limn→∞Hn−1

)in the measure algebra. Thus, there is M2 ∈ N such

that µ(Hn−1

)< ε · ξt (A) · ξt (B) · µ (Bt) for n ≥ M2. This yields for every

measurable set C ⊆ Tm in case of n ≥M2:

ξt (C) =µ (C ∩Bt)µ (Bt)

≤µ(C ∩Hn−1

))+ µ

(Hn−1

)µ (Bt)

<µ(H−1n−1 (C) ∩ Nt

)µ(Nt

) + ε · ξt (A) · ξt (B) = ξn−1t (C) + ε · ξt (A) · ξt (B) .

In the same way we obtain ξt (C) > ξn−1t (C)− ε · ξt (A) · ξt (B).

• Thus, we get for n ≥ max M1,M2:

)− ξt (A) · ξt (B)

≤ξt(Bn ∩ f−mn (An)

)+ 2 · ε · ξt (A) · ξt (B)− (ξt (An)− ξt (A4An)) · (ξt (Bn)− ξt (B4Bn))

<ξt(Bn ∩ f−mn (An)

)− ξt (An) · ξt (Bn) + 4 · ε · ξt (A) · ξt (B)

<ξn−1t

(Bn ∩ f−mn (An)

)− ξn−1

t (An) · ξn−1t (Bn) + 7 · ε · ξt (A) · ξt (B) .

• In the next step we consider an expression∣∣ξn−1t

(Γn ∩ f−mn

(An))− ξn−1

t (Γn) · ξn−1t

(An)∣∣,

at which An = Hn−1

)with a cube An of edge length a = q

−σ2n contained in Jt. By

Lemma 5.4.8 the sequence (fmnk )k∈N converges to fmn as k →∞ in the Diff0-topology in

particular. Thus f−mnk Hn−1 → f−mn Hn−1 as k → ∞ as well. Since An is a cube,there is K ∈ N such that for every k ≥ K:

ξn−1t

(f−mnn+k

(An)4f−mn

= ξn−1t

(f−mnn+k Hn−1

)4f−mn Hn−1

))≤ ε · ξn−1

(An)· ξn−1t (Γn) .

Hereby, we get

ξn−1t

(Γn ∩ f−mn

(An))≤ ξn−1

(Γn ∩ f−mnn+k

+ ε · ξn−1t

(An)· ξn−1t (Γn) .

• So we have to examine∣∣ξn−1t

(Γn ∩ f−mnn+k

(An))− ξn−1

t (Γn) · ξn−1t

(An)∣∣.

For this we chooseM3 ∈ N such that(

1 + 4q−σ2

)m< 1+ε for all n ≥M3 and approximate

An by sets Q1 =⋃i∈I1Sn,i, Q2 =

⋃i∈I2Sn,i, at which each Sn,i is a cube of sidelength

q−σn contained in Jt, such that Q1 ⊂ An ⊂ Q2, µt(An4Qi

)< ε · µt

)as well as

dist(∂An, ∂Qi

)≥ κn. Such approximations exist because q−σn > κn as seen above and

hence Q2 can be chosen in a way that it is contained in a cube of sidelength a+ 4q−σn withthe same center as An. This yields

µ(An4Q2

)= µ (Q2)− µ

)≤(a+ 4q−σn

)m − am = am ·((

1 +4q−σna

)m− 1)

= am ·

4q−σnq−σ2n

)m− 1

)· ε

by the requirement on the edge length a of the cube An in the definition of the partialpartition νtn. Thus, we obtain the aimed estimate µt

(An4Q2

)< ε ·µt

). Analogously

the relation µt(An4Q1

)< ε · µt

)is deduced.

Since κn > d0

(Ψn,n+k, hn Rmnαn+1

h−1n

)we have the following relations:

hn Rmnαn+1 h−1

n (y) ∈ Q1 =⇒ Ψn,n+k (y) ∈ An,

Ψn,n+k (y) ∈ An =⇒ hn Rmnαn+1 h−1

n (y) ∈ Q2.

Thus, we obtain:

fmnn (x) ∈ Hn−1 (Q1) =: Q1 =⇒ fmnn+k (x) ∈ Hn−1

)= An,

fmnn+k (x) ∈ Hn−1

)= An =⇒ fmnn (x) ∈ Hn−1 (Q2) =: Q2.

This yields

ξn−1t

(Γn ∩ f−mnn

))≤ ξn−1

(Γn ∩ f−mnn+k

(An))≤ ξn−1

(Γn ∩ f−mnn

Hereby, we have

ξn−1t

(Γn ∩ f−mnn+k

(An))− ξn−1

t (Γn) · ξn−1t

≤ξn−1t

(Γn ∩ f−mnn

))− ξn−1

t (Γn) · ξn−1t

)+ ε · ξn−1

t (Γn) · ξn−1t

as well as

ξn−1t

(Γn ∩ f−mnn+k

(An))− ξn−1

t (Γn) · ξn−1t

≥ξn−1t

(Γn ∩ f−mnn

))− ξn−1

t (Γn) · ξn−1t

)− ε · ξn−1

t (Γn) · ξn−1t

• Since each Qi is a disjoint union of sets Hn−1 (Sn), we consider∣∣ξn−1t

(Γn ∩ f−mnn (Hn−1 (Sn))

)− ξn−1

t (Γn) · ξn−1t (Hn−1 (Sn))

∣∣=∣∣∣µt (In ∩ φn Dψn,γn R−mnαn+1

D−1ψn,γn

φ−1n g−1

n (Sn))− µt

)· µt (Sn)

∣∣∣With the aid of Lemma 5.6.8 we obtain:∣∣∣µt (In ∩ φn Dψn,γn R−mnαn+1

D−1ψn,γn

φ−1n g−1

n (Sn))− µt

)· µt (Sn)

∣∣∣≤ 1µt (Jt)

∣∣∣µt (In ∩ φn Dψn,γn R−mnαn+1D−1

ψn,γn φ−1

n g−1n (Sn)

)µt (Jt)− µt

)µt (Sn)

∣∣∣+

1− µt (Jt)µt (Jt)

· µt(In

)· µt (Sn)

≤40 + 2 ·mn

· µt(In

)· µt (Sn)

using µt (Jt) ≥(1− 1

)m−1 ≥ 1 − m−1n ≥ 1

2 and 1−µt(Jt)µt(Jt)

≤ 2·(m−1)n in the last step. By

choosing n sufficiently large this yields:∣∣∣µt (In ∩ φn Dψn,γn R−mnαn+1D−1

ψn,γn φ−1

n g−1n (Sn)

)− µt

)· µt (Sn)

∣∣∣<ε · ξn−1

t (Γn) · ξn−1t (Hn−1Sn) .

Altogether, we conclude for n and k sufficiently large

)− ξt (A) · ξt (B)

<ξn−1t

(Bn ∩ f−mn (An)

)− ξn−1

t (An) · ξn−1t (Bn) + 7 · ε · ξt (A) · ξt (B)

=∑i,j

[ξn−1t

(Γn,j ∩ f−mn

))− ξn−1

)· ξn−1t (Γn,j)

]+ 7 · ε · ξt (A) · ξt (B)

≤∑i,j

[ξn−1t

(Γn,j ∩ f−mnn+k

))− ξn−1

)· ξn−1t (Γn,j) + ε · ξn−1

]+ 7 · ε · ξt (A) · ξt (B)

≤∑i,j

[ξn−1t

(Γn,j ∩ f−mnn (Hn−1Sn,i,l)

)− ξn−1

t (Hn−1Sn,i,l) · ξn−1t (Γn,j)

]+ 2 · ε · ξn−1

]+ 7 · ε · ξt (A) · ξt (B)

<∑i,j

[ε · ξn−1

t (Γn,j) · ξn−1t (Hn−1Sn,i,l)

]+ 2ε · ξn−1

]+ 7ε · ξt (A) · ξt (B)

≤11 · ε · ξt (A) · ξt (B) .

The case of countable many ergodic invariant measures 103

In the same way we obtain an estimate from below:

)− ξt (A) · ξt (B) > −11 · ε · ξt (A) · ξt (B) .

Since ε can be chosen arbitrarily small, the aimed weak mixing property holds true.

Let ω0 denote the standard Riemannian metric on M = S1 × [0, 1]m−1.

Lemma 5.7.1. Let Γn = D−1ψn,γn

(In)∈ ζn. Then hn|Γn is an isometry with respect to ω0.

Proof. By the same calculations as in the proof of Lemma 4.9.1.

Then the aimed f -invariant measurable Riemannian metric ω∞ is constructed in the sameway as in section 3.7 and thus Theorem C is proven.

5.8 The case of countable many ergodic invariant measures

Let r(n) := 1 + (m+ 2)2 · n2·(n2+5)

2 and for n ≥ 2 let t∗n ∈ N be the smallest natural number tsuch that l∗n := 1

t∗n·(t∗n+1) satisfies 1t∗n·(t∗n+1) ≤

2·√m·n3·q5·r(n−1)+1

. In case of n = 1 the number l∗1has to satisfy l∗1 <

12·√m·L1·q1

, at which L1 is the Lipschitz constant of the function ρ1 ∈ Ξ.

This time the measures are constructed with the aid of the sets Nt :=[

tt+1 ,

t+1t+2

)⊆ R

/Z for

t ∈ N, N0 :=[l∗1,

)and N−t :=

[l∗t+1, l

)for t ∈ N. With these we consider Nt = S1×Nt×Tm−2

and the normalized restriction µt of the Lebesgue measure on Nt. Then we will prove TheoremD by a modification of our above constructions.Note that l∗n is the r1-height of the set Nt∗n−1. This height is chosen such that the diame-ter of images of sets with this height under Hn−1 gn can converge to 0 (see equation 5.4).With it we define Nt∗n :=

⋃t≥t∗n−1Nt =

[t∗n−1t∗n

and Nt∗n :=⋃t≥t∗n−1 Nt. Moreover, we put

εn := 1

4·n11·q5·r(n−1)+1n−1

. In Remark 5.8.2 we will explain this choice of εn.

5.8.1 Modifications of the explicit constructions

In step n we use the sets N (1)−n := [0, l∗n) and N (1)

−n := S1×N (1)−n ×Tm−2. Moreover, lt denotes the

r1-length of the set Nt for every t ∈ Z and σn ∈ (0, 1) is a parameter that will be determined insubsection 5.8.2.

5.8.1.1 Trapping map

We introduce the map ψn : [0, 1]→ R as a C∞-function satisfying:

[12 , 1].

104 The case of countable many ergodic invariant measures

• ψn is equal to k · 4εn on[kn2 + 1

n6 ,k+1n2 − 1

]for 0 ≤ k ≤

⌋− 1 and ψn is equal to

k · 4εn on[n2−k−1n2 + 1

n6 ,n2−kn2 − 1

]for 0 ≤ k ≤

⌋− 1. On

[⌊n22

⌋n2 ,

n2−⌊n22

]it is put

to(⌊

⌋− 1)· 4εn.

Thus, ψn is less than n2

2 · 4εn <1

2·n9·q5·r(n−1)+1n−1

=: εn2 , which will be used in order to guarantee

that trapping regions of the second kind stay in Nt under the conjugation map hn.Then we define the trapping map Dψn,γn as in section 5.3.1.

5.8.1.2 The conjugation map hn

This time we set

φ(j)λ := φλ, εn15 ,1,j

and φ(t,lt,j)λ,µ := φt,lt,1,j,λ, εn15 ,µ,

εn7 ,

Hereby, we define the conjugation map φn on the fundamental sector[0, 1

]× [0, 1]× Tm−2:

• On[0, 1

n2·qn

]× Tm−1:

φ(m)n2·qn ... φ

n2·qm−2n φ(2)

n2·qm−1n

• On[

kn2·qn ,

k+1n2·qn

]×Nt × Tm−2 in case of t ∈ Z, −n+ 1 ≤ t ≤ t∗n − 2 as well as k ∈ N and

1 ≤ k ≤ n2 − 1:

φ(t,lt,m)

n2·q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)+(3+k)·(m−2)n ,q3+k

... φ(t,lt,2)

n2·q2+3·(m−1)+...+(3+k−1)·(m−1)n ,q3+k

=φ(t,lt,m)

n2·q2+(m−1)·(3k+ k·(k−1)

2 )+(3+k)·(m−2)n ,q3+k

... φ(t,lt,2)

n2·q2+(m−1)·(3k+ k·(k−1)

2 )n ,q3+k

• On[

kn2·qn ,

k+1n2·qn

]×N (1)

−n × Tm−2 in case of k ∈ N and 1 ≤ k ≤ n2 − 1 with:

φ(−n,l∗n,m)

n2·q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)+(3+k)·(m−2)n ,q3+k

... φ(−n,l∗n,2)

n2·q2+3·(m−1)+...+(3+k−1)·(m−1)n ,q3+k

=φ(−n,l∗n,m)

n2·q2+(m−1)·(3k+ k·(k−1)

2 )+(3+k)·(m−2)n ,q3+k

... φ(−n,l∗n,2)

n2·q2+(m−1)·(3k+ k·(k−1)

2 )n ,q3+k

• On[

kn2·qn ,

k+1n2·qn

]× Nt∗n ×Tm−2 in case of k ∈ N, 1 ≤ k ≤ n2− 1 with l∗n := 1− t∗n−1

t∗n= 1

t∗n:

φ(t∗n,l

∗n,m)

n2·q2+3·(m−2)+4·(m−2)+...+(3+k−1)·(m−2)+(3+k)·(m−3)n ,q3+k

... φ(t∗n,l∗n,3)

n2·q2+3·(m−2)+...+(3+k−1)·(m−2)n ,q3+k

=φ(t∗n,l∗n,m)

n2·q2+(m−2)·(3k+ k·(k−1)

2 )+(3+k)·(m−3)n ,q3+k

... φ(t∗n,l∗n,3)

n2·q2+(m−2)·(3k+ k·(k−1)

2 )n ,q3+k

Since the map φn coincides with the identity on a neighbourhood of the different sections, we canpiece it together smoothly in this way and obtain a diffeomorphism on

]×Tm−1. Moreover,

we extend it to a diffeomorphism on Tm using the rule φn R uqn

= R uqn φn, where u ∈ Z.

Alike we construct the smooth measure-preserving diffeomorphism gn on the fundamentalsector

]×Tm−1 initially with a parameter σn ∈ (0, 1) that will be determined subsequently:

• On[0, 1

n2·qn

]× Tm−1: gn = g[n·qσnn ].

• On[

kn2·qn ,

k+1n2·qn

]×Nt×Tm−2 in case of k ∈ Z, 1 ≤ k ≤ n2−1, and t ∈ Z, −n+1 ≤ t < t∗n−1:

gn = gt,n2·q

2+(m−1)·(3·(k+1)+ (k+1)·k2 )

n ,[n·qσnn ],lt·εn

2 , εn8

We note that the condition a·b·δε = a·b

4 ∈ Z is satisfied, because qn is a multiple of 4 due tosubsection 5.8.2.

• On[

kn2·qn ,

k+1n2·qn

]× Nt × Tm−2 in case of k ∈ Z, 1 ≤ k ≤ n2 − 1, and t ∈ Z, t ≥ t∗n−1:

gn = g[n·qσnn ].

• On[

kn2·qn ,

k+1n2·qn

]×N (1)

−n × Tm−2 in case of k ∈ Z, 1 ≤ k ≤ n2 − 1: gn = g[n·qσnn ].

Since gn coincides with the map g[n·qσnn ] in a neighbourhood of the boundary of the differentsections, this yields a smooth map and we can extend it to a smooth measure-preserving diffeo-morphism on Tm using the description gn R l

qn= R l

qn gn for l ∈ Z.

This time the conjugation map hn is defined to be equal to

• gn φn Dψn,γn on sections[lqn, lqn

+ 1n2·qn

]× Tm−1

• gn D−1ψn,γn

φn Dψn,γn on sections[lqn

+ kn2·qn ,

+ k+1n2·qn

]×Tm−1 for k = 1, ..., n2 − 1.

Note that it is still a smooth map because hn coincides with g[nqσnn ] in a neighbourhood of theboundary of the sections.

5.8.1.3 Trapping regions

Once again there are different kinds of trapping regions.In order to prove minimality we consider on

[lqn, lqn

+ 1n2·qn

]×Tm−1 for l ∈ Z the following sets

Sl,j2,...,jm :[l

jmn2 · q2

+jm−1

n2 · q3n

+ ...+j2

n2 · qmn,l

jmn2 · q2

+jm−1

n2 · q3n

+ ...+j2 + 1n2 · qmn

m∏i=2

[εn, 1− εn] .

Then the set of trapping regions of this first kind consists of all sets D−1ψn,γn

(Sl,j2,...,jm), whereall ji ∈ Z satisfy dεnqne ≤ ji ≤ qn − dεnqne − 1.By our constructions every orbit

Rkαn+1

H−1n (x)

k=0,...,qn+1−1

meets every trapping region of

the first kind (compare with Remark 5.3.1).For t ∈ −n+ 1, ...− 1, 0, 1, ..., t∗n − 2 we modify the sets St

l,k,j(1)1 , ~j2,..., ~jm

Stl,k,j

(1)1 , ~j2,..., ~jm

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−1)·(3·k+

k·(k−1)2 ))

n2 · q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)n

n2q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+1n

+ ...+j

(3+k)2

n2 · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+3+kn

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+3+k+1n

+ ...+j

(3+k)3

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+2·(3+k)n

(3+k)m

n2q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)n

+t((m−1)·(3·k+

k·(k−1)2 )+1)

n2 · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)+1n

+ ...+t

((m−1)·

(3·(n2−1)+

n2·(n2−1)2 −k

1n2 · γn

n6 · γn,

n2 · qn+ ...+

((m−1)·

(3·(n2−1)+

n2·(n2−1)2 −k

))1 + 1

γn− 1n2 · γn

− 1n6 · γn

[l∗n +

t−1∑s=−n+1

ls + lt ·

(t(1)2

qn+ ...+

t(3+k)2

l∗n +t−1∑

s=−n+1

ls + lt ·

(t(1)2

qn+ ...+

t(3+k)2 + 1q3+kn

− εn

×m∏i=3

[t(1)i

qn+ ...+

t(3+k)i

,t(1)i

qn+ ...+

t(3+k)i + 1q3+kn

− εn

where the union is taken over all t(1)i ∈ Z, dεnqne ≤ t

(1)i ≤ qn − dεnqne − 1 for i = 2, ...,m

as well as t(l)i ∈ Z, 0 ≤ t(l)i ≤ qn − 1, for l = 2, ..., 3 + k and i = 2, ...,m as well as t(j)1 ∈ Z,⌈

εnl∗nqn

⌉≤ t(j)1 ≤ qn −

⌈εnl∗nqn

⌉− 1, for j = 1, ..., (m− 1) ·

(3 · (n2 − 1) + n2·(n2−1)

2 − k).

Then for t ∈ −n+ 1, ...,−1, 0, 1, ..., t∗n − 2 the set of trapping regions of the second kind on

Nt consists of all sets D−1ψn,γn

(Stl,k,j

(1)1 , ~j2,..., ~jm

), where l, k ∈ Z, 1 ≤ k ≤ n2 − 1, as well as all

j(u)i ∈ Z satisfy

⌈εnl∗nqn

⌉≤ j(u)

i ≤ qn −⌈εnl∗nqn

⌉− 1.

On N (1)−n we consider the sets S−n

l,k,j(1)1 , ~j2,..., ~jm

S−nl,k,j

(1)1 , ~j2,..., ~jm

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−1)·(3·k+

k·(k−1)2 ))

n2 · q2+3·(m−1)+4·(m−1)+...+(3+k−1)·(m−1)n

n2 · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+1n

+ ...+j

(3+k)2

n2 · q2+3(m−1)+4(m−1)+...+(3+k−1)(m−1)+3+kn

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+3+k+1n

+ ...+j

(3+k)3

n2 · q2+3·(m−1)+...+(3+k−1)·(m−1)+2·(3+k)n

(3+k)m

n2 · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)n

+t((m−1)·(3·k+

k·(k−1)2 )+1)

n2 · q2+3(m−1)+...+(3+k−1)(m−1)+(m−1)·(3+k)+1n

+ ...+t

((m−1)·

(3·(n2−1)+

n2·(n2−1)2 −k

1n2 · γn

n6 · γn,

n2 · qn+ ...+

((m−1)·

(3·(n2−1)+

n2·(n2−1)2 −k

))1 + 1

γn− 1n2 · γn

− 1n6 · γn

[l∗n ·

(t(1)2

qn+ ...+

t(3+k)2

), l∗n ·

(t(1)2

qn+ ...+

t(3+k)2 + 1q3+kn

− εn

×m∏i=3

[t(1)i

qn+ ...+

t(3+k)i

,t(1)i

qn+ ...+

t(3+k)i + 1q3+kn

− εn

(1)i ≤ qn − dεnqne − 1 for i = 2, ...,m

εnl∗nqn

⌉≤ t(j)1 ≤ qn −

⌈εnl∗nqn

⌉− 1, for j = 1, ..., (m− 1) ·

(3 · (n2 − 1) + n2·(n2−1)

2 − k).

Then the set of trapping regions of the second kind on N (1)−n consists of allD−1

ψn,γn

(S−nl,k,j

(1)1 , ~j2,..., ~jm

where l, k ∈ Z, 1 ≤ k ≤ n2 − 1 as well as all j(u)i ∈ Z satisfy

⌈εnl∗nqn

⌉≤ j(u)

⌉− 1.

Remark 5.8.1. Note that we have chosen that trapping regions D−1ψn,γn

(Stl,k,j

(1)1 , ~j2,..., ~jm

isfying D−1ψn,γn

φn(Stl,k,j

(1)1 , ~j2,..., ~jm

)⊂ Nt for t ∈ −n,−n+ 1, ...,−1, 0, 1, ..., t∗n − 2. Here we

used the bound(

1 + 1q3n

+ ...+ 1q3+n2−1

)· ψn < εn.

On Nt∗n we consider trapping regions of the form D−1ψn,γn

(Ut∗n

l,k,j(1)1 , ~j3,..., ~jm

), at which

Ut∗n

l,k,j(1)1 , ~j3,..., ~jm

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−2)·(3·k+

k·(k−1)2 ))

n2 · q2+3·(m−2)+4·(m−2)+...+(3+k−1)·(m−2)n

n2 · q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+1n

+ ...+j

(3+k)3

n2q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+3+kn

n2 · q2+3·(m−2)+...+(3+k−1)·(m−2)+3+k+1n

+ ...+j

(3+k)4

n2 · q2+3·(m−2)+...+(3+k−1)·(m−2)+2·(3+k)n

(3+k)m

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+(m−2)·(3+k)n

+t((m−2)·(3·k+

k·(k−1)2 )+1)

n2q2+3(m−2)+...+(3+k−1)(m−2)+(m−2)·(3+k)+1n

+ ...+t

((m−1)·

(3·n2+

n2·(n2−1)2

)−(m−2)·(3+k)

1n2 · γn

n6 · γn,

n2 · qn+ ...+

((m−1)·

(3·n2+

n2·(n2−1)2

)−(m−2)·(3+k)

)1 + 1

γn− 1n2 · γn

− 1n6 · γn

×[t∗n − 1t∗n

+ εn, 1− εn]×

m∏i=3

[t(1)i

qn+ ...+

t(3+k)i

,t(1)i

qn+ ...+

t(3+k)i + 1q3+kn

− εn

(1)i ≤ qn − dεnqne − 1 for i = 3, ...,m

εnl∗nqn

⌉≤ t(j)1 ≤ qn −

⌈εnl∗nqn

⌉− 1 for j = 1, ..., (m− 1) ·

(3 · n2 +

n2·(n2−1)2

)− (m− 2) · (3 + k).

Then the set of trapping regions of the second kind on Nt∗n consists of all D−1ψn,γn

(Ut∗n

l,k,j(1)1 ,..., ~jm

where l, k ∈ Z, 1 ≤ k ≤ n2 − 1 and all j(u)i ∈ Z satisfy

⌈εnl∗nqn

⌉≤ j(u)

⌉− 1.

By the requirements on the numbers t(u)i and j

(u)i underlying 1

γn-sections on the θ-axis, that

are part of trapping regions belonging to N(1)−n and Nt in case of t ∈ −n+ 1, ...t∗n − 2, are

also part of trapping regions belonging to Nt∗n and vice versa. Then the number of iteratesRiαn+1

(x)i=0,...,qn+1−1

captured by one of the t∗n + n− 1 blocks covering such a 1γn

-section is

at least(n2 − 10 · (m− 1)

)·(⌊qn+1 ·

1− 2n4

n2·γn

⌋− 2)≥(n2 − 10 · (m− 1)

)· qn+1 ·

1− 3n4

n2·γn for everyx ∈ Tm.

Since there are qn ·(n2 − 1

)·(qn − 2 ·

⌈εnl∗nqn

⌉)(m−1)·(

3n2+n2·(n2−1)

such 1γn

-sections, thenumber of iterates trapped by the regions of the second kind is at least

qn ·(n2 − 1

)·(qn − 2 ·

⌈εnl∗nqn

⌉)(m−1)·(

3·n2+n2·(n2−1)

·(n2 − 10 · (m− 1)

)· qn+1 ·

1− 3n4

n2 · γn

1− 1n2

1− 3 · 4 ·√m

)m·(n4+3·n2)·(

1− 10 · (m− 1)n2

1− 3n4

)· qn+1

1− 1n2

1− 12m2

1− 10 · (m− 1)n2

1− 3n4

)· qn+1

1− 20 ·m2

)· qn+1

for n sufficiently large. Here we used

εnl∗n≤ 1

n9 · q5·r(n−1)+1n

· 4 ·√m · n3 · q5·r(n−1)+1

n =4 ·√m

On the contrary, at most 20·m2

n2 · qn+1 iterates are not captured by these regions.

Remark 5.8.2. Since we need this portion to converge to zero as n → ∞, this explains ourchoice of εn.

Remark 5.8.3. In order of a handsome description of the subsequent sets we introduce anothernotation: By Dψn,∗ : [0, 1]m → Rm we denote the transformation sending (θ, r1, r2, ..., rm−1) to

(θ, r1, r2 −

+ ...+1

q3+n2−1n

)· ψn (θ) , ..., rm−1 −

+ ...+1

q3+n2−1n

)· ψn (θ)

Using the maps Cγn (θ, r1, ...rm−1) = (γn · θ, r1, ..., rm−1) we construct the map

Dψn,γn,∗ := C−1γn Dψn,∗ Cγn :

]× Tm−1 →

]× Tm−1.

Inside D−1ψn,γn

(Ut∗n

l,k,j(1)1 , ~j3,..., ~jm

)we consider for t ≥ t∗n the sets D−1

ψn,γn,∗

(V tl,k,j

(1)1 , ~j3,..., ~jm

Nt, at which

V tl,k,j

(1)1 , ~j3,..., ~jm

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−2)·(3·k+

k·(k−1)2 ))

n2 · q2+3·(m−2)+4·(m−2)+...+(3+k−1)·(m−2)n

n2q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+1n

+ ...+j

(3+k)3

n2 · q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+3+kn

n2 · q2+3·(m−2)+...+(3+k−1)·(m−2)+3+k+1n

+ ...+j

(3+k)4

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+2·(3+k)n

(3+k)m

n2q2+3(m−2)+...+(3+k−1)(m−2)+(m−2)·(3+k)n

+t((m−2)·(3·k+

k·(k−1)2 )+1)

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+(m−2)·(3+k)+1n

+ ...+t

((m−1)·

(3·n2+

n2·(n2−1)2

)−(m−2)·(3+k)

1n2 · γn

n6 · γn,

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−2)·(3·k+

k·(k−1)2 ))

n2 · q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)n

n2q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+1n

+ ...+j

(3+k)3

n2 · q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+3+kn

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+3+k+1n

+ ...+j

(3+k)4

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+2·(3+k)n

(3+k)m

n2q2+3(m−2)+...+(3+k−1)(m−2)+(m−2)·(3+k)n

+t((m−2)·(3·k+

k·(k−1)2 )+1)

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+(m−2)·(3+k)+1n

+ ...+t

((m−1)·

(3·n2+

n2·(n2−1)2

)−(m−2)·(3+k)

)1 + 1

γn− 1n2 · γn

− 1n6 · γn

t+ 1,t+ 1t+ 2

m∏i=3

[t(1)i

qn+ ...+

t(3+k)i

,t(1)i

qn+ ...+

t(3+k)i + 1q3+kn

− εn

(note that these are part of D−1ψn,γn

(Ut∗n

l,k,j(1)1 , ~j3,..., ~jm

), because the trapping map causes a trans-

lation of at least 4 · εn in the r1-coordinate on these regions)

and for t < −n the sets D−1ψn,γn,∗

(V tl,k,j

(1)1 , ~j3,..., ~jm

)⊂ Nt, at which

V tl,k,j

(1)1 , ~j3,..., ~jm

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−2)·(3·k+

k·(k−1)2 ))

n2 · q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)n

n2q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+1n

+ ...+j

(3+k)3

n2 · q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+3+kn

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+3+k+1n

+ ...+j

(3+k)4

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+2·(3+k)n

(3+k)m

n2q2+3(m−2)+...+(3+k−1)(m−2)+(m−2)·(3+k)n

+t((m−2)·(3·k+

k·(k−1)2 )+1)

n2 · q2+3(m−2)+...+(3+k−1)(m−2)+(m−2)·(3+k)+1n

+ ...+t

((m−1)·

(3·n2+

n2·(n2−1)2

)−(m−2)·(3+k)

1n2 · γn

n6 · γn,

n2 · qn+ ...+

((m−1)·

(3·n2+

n2·(n2−1)2

)−(m−2)·(3+k)

)1 + 1

γn− 1n2 · γn

− 1n6 · γn

×[l∗−t+1, l

∗−t)×

m∏i=3

[t(1)i

qn+ ...+

t(3+k)i

,t(1)i

qn+ ...+

t(3+k)i + 1q3+kn

− εn

Note that the last-mentioned sets are part of D−1ψn,γn

(Ut∗n

l,k,j(1)1 , ~j3,..., ~jm

), because l∗n+1 < εn and

the trapping map causes a translation of at least 4 · εn in the r1-coordinate on these regions.Furthermore, we introduce the sets V t

∗n−1

l,k,j(1)1 , ~j3,..., ~jm

and V −nl,k,j

(1)1 , ~j3,..., ~jm

such that

D−1ψn,γn

(Vt∗n−1

l,k,j(1)1 , ~j3,..., ~jm

)= D−1

ψn,γn

(Ut∗n

l,k,j(1)1 , ~j3,..., ~jm

)∩(S1 ×Nt∗n−1 × Tm−2

D−1ψn,γn

(V −nl,k,j

(1)1 , ~j3,..., ~jm

)= D−1

ψn,γn

(Ut∗n

l,k,j(1)1 , ~j3,..., ~jm

)∩(S1 ×N−n × Tm−2

For arbitrary x ∈ Tm $nt (x) denotes the portion of the orbit

Riαn+1

(x)i=0,...,qn+1−1

trapped

by all the regions D−1ψn,γn

(Stl,k,j

(1)1 , ~j2,..., ~jm

)and $n

t (x) is the portion of the orbit trapped by all

the regions D−1ψn,γn

(V tl,k,j

(1)1 , ~j3,..., ~jm

5.8.1.4 Partial partitions

Partial partitions ηtn

For every t ∈−n+ 1, ..., n2 − 1

the partial partition ηtn will be constructed on the fundamental

sector[0, 1

]×Nt × Tm−2 initially. For this we divide the fundamental sector in sections:

• On[

kn2·qn ,

k+1n2·qn

]×Nt×Tm−2 in case of k, t ∈ N, −n+ 1 ≤ t ≤ n2− 1 and 1 ≤ k ≤ n2− 2

the partial partition ηtn consists of all multidimensional intervals of the following form:

n2 · qn+

n2 · q2n

+ ...+j(1+(m−1)·(3(k+1)+

k(k+1)2 ))

n2 · q2+(m−1)·(3·(k+1)+k·(k+1)

lt · n2q2+(m−1)·(3·(k+1)+

k·(k+1)2 )

n2 · qn+

n2 · q2n

+ ...+j(1+(m−1)·(3(k+1)+

k(k+1)2 ))

n2 · q2+(m−1)·(3(k+1)+k·(k+1)

− εn

lt · n2q2+(m−1)·(3·(k+1)+

k·(k+1)2 )

[l∗n +

t−1∑s=−n+1

ls + lt ·

qn+ ...+

j(3+k+1)2

q3+k+1n

+εn · lt

8 · q3+k+1n

l∗n +t−1∑

s=−n+1

ls + lt ·

qn+ ...+

j(3+k+1)2 + 1q3+k+1n

− εn · lt8 · q3+k+1

×m∏i=3

qn+ ...+

j(3+k+1)i

q3+k+1n

+εn · lt

8 · q3+k+1n

qn+ ...+

j(3+k+1)i + 1q3+k+1n

− εn · lt8 · q3+k+1

where j(l)1 ∈ Z,

⌉≤ j(l)

1 ≤ qn−⌈qn

⌉−1 for l = 1, ..., 1+(m−1)·

(3 (k + 1) + k·(k+1)

j(1)2 ∈ Z and

⌈εnltqn

⌉≤ j(1)

2 ≤ qn−⌈εnltqn

⌉−1, j(1)

i ∈ Z and⌈qn

⌉≤ j(1)

i ≤ qn−⌈qn

⌉−1

for i = 3, ...,m as well as j(l)i ∈ Z and

⌉≤ j(l)

i ≤ qn −⌈qn

⌉− 1 for i = 2, ...,m and

l = 2, ..., 3 + k + 1.

• On[0, 1

n2·qn

]×Nt×Tm−2 as well as

[n2−1n2·qn ,

]×Nt×Tm−2 there are no elements of the

partial partition ηtn.

]×Nt×Tm−2 is extended to

a partial partition of Tm.

Partial partition ζn

]× Tm−1 initially and therefore divide this sector into sections:

In case of k, t ∈ N, −n+ 1 ≤ t ≤ t∗n−1 − 1 and 1 ≤ k ≤ n2 − 1 the partial partition ζn consists ofall sets D−1

ψn,γn

n2 · qn+

n2 · q2n

+ ...+j

(1+(m−1)·

(3·n2+

n2·(n2−1)2

n2 · γn+

1n6 · γn

n2 · qn+

n2 · q2n

+ ...+j

(1+(m−1)·

(3·n2+

n2·(n2−1)2

s+ 1n2 · γn

− 1n6 · γn

[l∗n +

t−1∑s=−n+1

ls + lt ·

qn+ ...+

j(3+k+1)2

q3+k+1n

+ ...+j(2+(m−1)·(3·(k+1)+

k·(k+1)2 ))

q2+(m−1)·(3·(k+1)+

k·(k+1)2 )

j(3+(m−1)·(3·(k+1)+

k·(k+1)2 ))

2 · εn

2 · n2 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσnn ]

+2 · ε2

n · lt

n2 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσnn ]

l∗n +t−1∑

s=−n+1

ls + lt ·

qn+ ...+

j(3+k+1)2

q3+k+1n

+ ...+j(2+(m−1)·(3·(k+1)+

k·(k+1)2 ))

q2+(m−1)·(3·(k+1)+

k·(k+1)2 )

(j(3+(m−1)·(3·(k+1)+

k·(k+1)2 ))

2 + 1)· εn

2n2 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσnn ]

− 2 · ε2n · lt

n2 · q2+(m−1)·(3·(k+1)+k·(k+1)

2 )n · [nqσnn ]

×m∏i=3

qn+ ...+

j(3+k)i

+4εnq3+kn

qn+ ...+

j(3+k)i + 1q3+kn

− 4εnq3+kn

where s ∈ Z and 0 ≤ s ≤ n2 − 1, j(l)1 ∈ Z and d4εn · qne ≤ j

(l)1 ≤ qn − d4εn · qne − 1 for

l = 1, ..., 1+(m− 1)·(

3n2 + n2·(n2−1)2

)except for l = 2+3(m−1)+...+(3+k−1)(m−1), for which

we demand⌈(

⌉≤ j

(l)1 ≤ qn −

⌈(εnlt

⌉− 1; j(

3+(m−1)·(3(k+1)+k(k+1)

2 ))2 ∈ Z,

8n2 · [nqσnn ] ≤ j(3+(m−1)·(3(k+1)+

k(k+1)2 ))

2 ≤ n2 ·⌊

⌋· [nqσnn ] − 8n2 · [nqσnn ] − 1; j(l)

2 ∈ Z and

d4εn · qne ≤ j(l)2 ≤ qn − d4εn · qne − 1 for l = 1, ..., 2 + (m− 1) ·

(3 · (k + 1) + k·(k+1)

)as well as

j(l)i ∈ Z and d4εn · qne ≤ j(l)

i ≤ qn − d4εn · qne − 1 for i = 3, ...,m and l = 1, .., 3 + k.

Lemma 5.8.4. The conjugation map hn acts as an isometry on the elements of the partitionζn.

Proof. As in Lemma 5.7.1 φn Dψn,γn acts as an isometry on any element D−1ψn,γn

(In)∈ ζn

and φn(In)lies in the “good area” of the map gn. But the prior application of D−1

ψn,γncauses

a translation of(

1 + 1q3n

+ ...+ 1

q3+n2−1n

)· u · 4εn with some u ≤ n2

2 in the ~r-coordinates, in

particular in the r1-coordinate. At first we observe that D−1ψn,γn

φn(In)is still contained

in the same definition section of gn by our choice of j(2+3(m−1)+...+(3+k−1)(m−1))1 . Thus, we

compare the caused translation with an εb·a = lt·εn

2·n2·q2+(m−1)·(3·(k+1)+ k·(k+1)

2 )n ·[nqσnn ]

-domain of the

map gn = ga,b,ε,δ on the r1-axis. In case of 2 + (m− 1) ·(

3 · (k + 1) + k·(k+1)2

)≥ 3 + n2 − 1 the

shifting is a multiple of such a domain and then D−1ψn,γn

φn(In)is still contained in the “good

area” of gn. In the other case we write 1 + 1q3n

+ ...+ 1

q3+n2−1n

q2+(m−1)·(3·(k+1)+ k·(k+1)

+R with

l ∈ Z and some rest term R < 2

q2+(m−1)·(3·(k+1)+ k·(k+1)

2 )+1n

. Since l

q2+(m−1)·(3·(k+1)+ k·(k+1)

· u · 4εn

is a multiple of εb·a , we consider 2

q2+(m−1)·(3·(k+1)+ k·(k+1)

2 )+1n

· u · 4εn. We have

n2 · 16 · u · 1l2t· [nqσnn ] · 1

≤ n2 · 16 · n2

1l∗n−1

· n · qσnn · 4 · n11 · q5·r(n−1)+1n−1

≤ n2 · 16 · n2

4 ·√m · (n− 1)3 · q5·r(n−2)+1

· n · q4·r(n−1)n · 4 · n11 · q5·r(n−1)+1

≤ 512 · n22 ·m · q9·r(n−1)+3n−1 < qn

by our assumptions on the numbers qn and σn in the next section. So this deviation is boundedby

q2+(m−1)·(3·(k+1)+

k·(k+1)2 )+1

· u · 4εn <2

q2+(m−1)·(3·(k+1)+

k·(k+1)2 )

· l2t · εnn2 · 16 · u · [nqσnn ]

· u · 4εn

= εn · lt ·εn·lt

[nqσnn ] · n2 · q2+(m−1)·(3·(k+1)+k·(k+1)

Then D−1ψn,γn

φn(In)is still contained in the “good area” of gn.

Thus, hn acts as an isometry on the elements of the partition ζn.

Since the elements of the partition ζn cover a set of measure at least 1 − 3m+1n2 in case of

n ≥ 4, we can apply the same approach as in section 3.7 to prove the existence of a measurablef -invariant Riemannian metric.

5.8.2 Norm estimates and convergence of (fn)n∈N

By the same approach as in section 5.4.1 we obtain estimates on the norms of the conjugationmaps:

<∞.We obtain

|||Hn|||kn+1 ≤ Cn · qkn+1n · γ(m+2)·(kn+1)

n ≤ Cn · q(kn+1)·(m+2)2·

n2·(n2+5)2

where the constant Cn is independent of qn.

Proof. From section 5.4.1 we recall the results |||Dψn,γn |||k ≤ C · γkn, |||φn|||k ≤ C · γ(m−1)·kn ,

|||gn|||k ≤ C · qkn · γkn and apply the formula of Faà di Bruno: |||φn Dψn,γn |||k ≤ C · γm·kn ,|||D−1

ψn,γnφn Dψn,γn |||k ≤ C ·γ

(m+1)·kn and finally |||hn|||k ≤ C ·qkn ·γ

(m+2)·kn . Here the constant

depends on l1, ..., lt∗n−2, l∗n, l∗n amongst others. In particular, it is independent of qn.

Then we can prove convergence of (fn)n∈N with the aid of Lemma 5.4.6 under the additionalassumptions qn > maxi=1,...,n+1 Lip (ρi), qn ≥ Cn, 24

εn= 96 · n11 · q5·r(n−1)+1

n−1 divides the number

qn, qn > 512 · n22 ·m · q9·r(n−1)+3n−1 , qn ≥ 24

εn· 56ε2n−1

· |||ψn−1|||1 · 1l∗n−2· (n − 1) · γ2

n−1 and a minormodification of the condition

|α− αn| <|α− αn−1|

2n+1 · kn · Ckn · q1+(kn+1)2·(m+2)2·n2·(n2+5)n

Remark 5.8.6. In particular, |||Hn|||1 ≤ q1+(m+2)2·

n2·(n2+5)2

n motivates our definition of the

number r(n) = 1 + (m+ 2)2 · n2·(n2+5)

In the construction of a diffeomorphism with countable many ergodic invariant measuresspecial attention is paid to the number σn ∈ (0, 1): We choose it in such a way that qσnn =q

4·r(n−1)n−1 .

Remark 5.8.7. We define the partial partition ηtn :=Hn−1 gn D−1

ψn,γn

): In ∈ ηtn

After the application of D−1ψn,γn

on In ∈ ηtn the diameter is at most√m ·(

)≤ 2 ·

√m · εn.

Unfortunately, on this set gn = g[nqσnn ] is not necessarily true, but the set is strictly containedin such a cube of sidelength 2εn that is a union of domains of ga,b,ε. Then we obtain for thediameter of such a partition element:

diam(Hn−1 gn

))≤ ‖DHn−1‖0 ·

∥∥Dg[nqσnn ]

∥∥0· 2 ·√m · εn

≤ qr(n−1)n−1 · n · q4·r(n−1)

n−1 · 2 ·√m

n9 · q5·r(n−1)+1n−1

as n→∞. Thus, this sequence of partial partitions converges to the decomposition into points.

5.8.3 Proof of minimality

We compute for any trapping region D−1ψn,γn

(Sj1,j2,...,jm):

φn Dψn,γn

(D−1ψn,γn

(Sj1,j2,...,jm))

=[j1qn

n2 · qn,j1qn

n2 · qn− εnn2 · qn

]×[j2qn,j2 + 1qn

m∏i=3

+εnqn,ji + 1qn

− εnqn

After the subsequent application of D−1ψn,γn

the diameter is at most√m ·(

)≤ 2 ·

√m · εn.

As before on this set gn = g[nqσnn ] is not necessarily true, but it is strictly contained in such acube of sidelength 2εn that is a union of domains of ga,b,ε. Thus, by our choice of σn the diameterof Hn−1 gn D−1

ψn,γn φn Dψn,γn

(D−1ψn,γn

(Sl,j2,...,jm))is less than

‖DHn−1‖0 ·∥∥Dg[nqσnn ]

∥∥0· 2 ·√m · εn ≤ qr(n−1)

n−1 · n · q4·r(n−1)n−1 · 2

√m · 1

n9 · q5·r(n−1)+1n−1

=2 ·√m

n8 · qn−1,

which converges to 0 as n → ∞. Hence, we can apply a modified criterion 5.5.2 for minimalityto conclude that f is minimal.

5.8.4 The ergodic invariant measures

5.8.4.1 Constructions of the measures ξt

At first we observe that after each application of the trapping maps Dψn,γn and D−1ψn,γn

re-spectively a set of measure at most εn is mapped from Nt into another set Ns in case of−n ≤ t ≤ t∗n − 1. Moreover, the part used to prove minimality is not mapped back into Ntunder the conjugation map hn. Thus, we obtain

)4Hn−1

))≤ 2·qn·

1n2 · qn

·µ(Nt

)+4·εn ≤

2n2·µ(Nt

16 ·√m

n6·l∗n ≤

3n2·µ(Nt

)for −n ≤ t ≤ t∗n − 1.In case of t ≥ t∗n and t < −n we have to examine this measure difference more carefully: On each

Nt we take the mentioned family of sets D−1ψn,γn,∗

(V tl,k,j

(1)1 , ~j3,..., ~jm

)with the above restrictions.

These are mapped back into the respective Nt under hn and have a total measure of at least(1− 20·m2

)· µ(Nt

). Consequently µ

)4Hn−1

))≤ 2 · 20·m2

n2 · µ(Nt

This yields the convergence ofHn

)n∈N

in the measure algebra and we can continue as insection 5.6.1.1.

5.8.4.2 Trapping property

By the same approach as in section 5.6.1.2 we consider the sets ∆t

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,...,j

(1)m ,j

(2)m ,j

with 0 ≤ l ≤ qn − 1, 1 ≤ k ≤ n2 − 1 and j(u)i ∈ Z satisfying

⌈εnl∗nqn

⌉≤ j

(u)i ≤ qn −

⌈εnl∗nqn

⌉− 1

on Nt in case of −n + 1 ≤ t ≤ t∗n − 2. In particular, D−1ψn,γn

l,k,j(1)1 ,...,j

)is still contained

in Nt. As above we denote the union of these sets by T tn, estimate µt(Nt \ T tn

n2 and notethat there are at least

$nt (x)·

(1− 20 ·m2

n2 · q2+3·(m−1)n

·qn+1 = $nt (x)·

(1− 20 ·m2

)·qn+1·µt

l,k,j(1)1 ,j

(1)2 ,...,j

)iterates φnDψn,γn R

iαn+1

(x) contained in ∆t

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,...,j

(1)m ,j

(2)m ,j

for arbitrary x ∈ Tm.For the diameter of such a set we obtain by the same arguments as in Remark 5.8.7:

diam(Hn−1 gn D−1

ψn,γn

l,k,j(1)1 ,...,j

))≤ ‖DHn−1‖0 · [nq

σnn ] · 2 ·

√m · εn

≤ qr(n−1)n−1 · n · q4·r(n−1)

n−1 · 2 ·√m

n9 · q5·r(n−1)+1n−1

n2 · qn−1.

Hereby, we conclude for every x, y ∈ ∆t

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,...,j

(1)m ,j

(2)m ,j

and i ∈ 1, ..., n:

(∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,...,j

))≤ qn−1 ·

1n2 · qn−1

We point out that the number l∗1 was chosen in such a way that

diam(g1 D−1

ψ1,γ1

(∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

(3)2 ,...,j

))≤ q1 ·

√m · (ε1 + l∗1) ≤ q1 ·

√m · 2 · l∗1 <

112 · L1

In case of t ≥ t∗n we use the sets D−1ψn,γn,∗

(Etl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

)for 1 ≤ k ≤ n2−1, j(u)

i ∈ Z

satisfying⌈εnl∗nqn

⌉≤ j(u)

⌉− 1, at which

Etl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

n2 · qn+

n2 · q2n

n2 · qn+j

(1)1 + 1n2 · q2

t+ 1,t+ 1t+ 2

m∏i=3

(3)i + 1q3n

Once again we denote the union of these sets by Etn and compute

µ(Etn

)= qn ·

(n2 − 1

)·(qn − 2 ·

⌈εnl∗nqn

⌉)1+3·(m−2) 1

n2 · q2+3·(m−2)n

1− 1n2

1− 36m2

)· lt ≥

(1− 2

)· µ(Nt

i.e. we get µt(Nt \ Etn

)≤ 2

n2 . Now we examine the sets D−1ψn,γn,∗

(V tl,k,j

(1)1 , ~j3,..., ~jm

). Under

D−1ψn,γn

φn Dψn,γn these are mapped onto the image of

n2 · qn+

n2 · q2n

+t(1)1

n2 · q3n

+ ...+t((m−2)·(3·k+

k·(k−1)2 ))

n2 · q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)n

− t(1)3

n2q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+1n

− ...− t(3+k)3

n2 · q2+3(m−2)+4(m−2)+...+(3+k−1)(m−2)+3+k

− t(3+k)m + 1

n2 · q2+3(m−2)+4(m−2)+...+(3+k)(m−2)+

t((m−2)·(3·k+

k·(k−1)2 )+1)

n2 · q2+3(m−2)+4(m−2)+...+(3+k)(m−2)+1n

((m−1)·

(3·n2+

n2·(n2−1)2

)−(m−2)·(3+k)

1n2 · γn

n6 · γn,

n2 · qn+ ...+

((m−1)·

(3·n2+

n2·(n2−1)2

)−(m−2)·(3+k)

)1 + 1

γn− 1n2 · γn

− 1n6 · γn

t+ 1,t+ 1t+ 2

m∏i=3

qn+ ...+

j(3+k)i

qn+ ...+

j(3+k)i + 1q3+kn

− εn

under D−1ψn,γn,∗

. In particular, they are still contained in Nt.

Moreover, in this case at least(qn − 2 ·

⌈εnl∗n· qn⌉)(m−2)·k

sets of the formD−1ψn,γn,∗

(V tl,k,j

(1)1 , ~j3,..., ~jm

are mapped into one such D−1ψn,γn,∗

(Etl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

)under D−1

ψn,γnφnDψn,γn .

So we estimate the number of i ∈ 0, ..., qn+1 − 1 such that D−1ψn,γn

φn Dψn,γn Riαn+1

(x) is

contained in it by $nt (x) ·

(1− 20·m2

)· qn+1 · µt

(D−1ψn,γn,∗

(Etl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

))for arbitrary x ∈ Tm.

Note that diam(D−1ψn,γn,∗

(Etl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

))≤√m · (εn + lt). Then we can

estimate the diameter

ψn,γn,∗

(Etl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

))≤‖DHn−1‖0 · [nq

σnn ] ·

√m · (εn + lt)

≤qr(n−1)n−1 · n · q4·r(n−1)

n−1 ·√m ·

n9 · q5·r(n−1)+1n−1

2 ·√m · n3 · q5·r(n−1)+1

)≤ 1n2 · qn−1

and we deduce |ρi (Hn−1 gn (x))− ρi (Hn−1 gn (y))| < 1n2 for x, y ∈ D−1

ψn,γn,∗

(Etl,k,j

(1)1 ,j

(1)3 ,...,j

)and i = 1, ..., n.

We have to examine where the rest of the trapping regions D−1ψn,γn

(Ut∗n

l,k,j(1)1 , ~j3,..., ~jm

)is mapped

into. Therefore, we introduce

Et∗n−1

l,k,j(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

n2 · qn+

n2 · q2n

n2 · qn+j

(1)1 + 1n2 · q2

]×[t∗n − 1t∗n

,t∗n

t∗n + 1

m∏i=3

(3)i + 1q3n

as well as for t ≤ −n

Etl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

(1)m ,j

(2)m ,j

n2 · qn+

n2 · q2n

n2 · qn+j

(1)1 + 1n2 · q2

]×[l∗−t+1, l

∗−t)×

m∏i=3

(3)i + 1q3n

in case of 1 ≤ k ≤ n2−1, j(u)i ∈ Z satisfying

⌈εnl∗nqn

⌉≤ j(u)

i ≤ qn−⌈εnl∗nqn

⌉−1. Then we consider

the sets D−1ψn,γn,∗

(Et∗n−1

l,k,j(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

)in Nt∗n−1 and D−1

ψn,γn,∗

(Etl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

)in Nt, which capture the setsD−1

ψn,γn

(Vt∗n−1

l,k,j(1)1 , ~j3,..., ~jm

)andD−1

ψn,γn

(V tl,k,j

(1)1 , ~j3,..., ~jm

)respectively

under D−1ψn,γn

φn Dψn,γn . The same computations concerning the diameter and the captured

iterates as above apply. On N−n the sets D−1ψn,γn,∗

(E−nl,k,j

(1)1 ,j

(1)3 ,j

(2)3 ,j

(3)3 ,...,j

)capture the sets

D−1ψn,γn

(S−nl,k,j

(1)1 , ~j2,..., ~jm

)(for all possible values of ~j2) as well as D−1

ψn,γn

(V −nl,k,j

(1)1 , ~j3,..., ~jm

Now we have assembled all the ingredients for a proof of a result like Lemma 5.6.1 on the simplexξtt∈Z. Then the proof follows along the lines of section 5.6.1.2. In conclusion we get that themeasures ξt for t ∈ N are the only absolutely continuous f -invariant ergodic measures.

5.8.4.3 Weak mixing property w.r.t. ξt

In the proof of the weak mixing property of the measures ξt we use the partial partitions ηtn,which converge to the decomposition into points by Remark 5.8.7 as well as sets of the formHn−1

)and Hn−1 (Sn), at which An and Sn are cubes of sidelenght q−

σn2 resp. q−σn . The

diameter of these sets is bounded by

‖DHn−1‖0 ·√m · q−

σn2 ≤ qr(n−1)

n−1 ·√m · q−2·r(n−1)

n−1 =√m · q−r(n−1)

Hence, their diameters converge to 0 as n → ∞ and we are able to execute the above approx-imation steps. Once again we compute that a partitition element In ∈ ηtn is mapped underΦn := φn Dψn,γn R

mnαn+1

D−1ψn,γn

φ−1n in the interior of one step-by-step domain of the map

gn = ga,b,ε and the θ-length γ of this image satisfies γ ≥ 1−2εa . Furthermore, we deduce a result

on (γ, ε)-distribution of In ∈ ηtn under D−1ψn,γn

Φn on Jt :=[∑t−1

s=1 ls + 2 · εn,∑ts=1 ls − 2 · εn

[2 · εn, 1− 2 · εn]m−2 as in Lemma 5.6.6 for every t ∈−n+ 1, ..., n2 − 1

in step n. Hereby, we

can prove a statement similar to Lemma 5.6.8 for the diffeomorphism D−1ψn,γn

Φn instead of Φn(such a criterion is exhibited in section 6.5.2). This suffices to prove the weak mixing propertyfor any ξt because we can choose n sufficiently large.

Thus, Theorem D is deduced.

Chapter 6

Natural measures ofdiffeomorphisms with arbitraryLiouvillean rotation number

6.1 Introduction

By the well-known Brouwer fixed-point theorem every continuous function on the disc D2 has afixed point. Indeed, Bourgin proved with the aid of the Brouwer translation theorem that forevery area-preserving orientation-preserving homeomorphism of the disc there is a fixed pointinside the disc ([Bo68]). Hence, any area- and orientation-preserving diffeomorphism of the dischas at least three ergodic invariant measures: The Dirac-measure δ at a fixed point in the interiorof the disc, a measure supported at the boundary and any ergodic component of the area. In[FK04], §3, Fayad and Katok constructed diffeomorphisms with this minimal number of ergodicinvariant measures. In fact, they proved that the set of such diffeomorphisms is a residual subsetin the closure A′

(D2)in the C∞-topology of the conjugates of rotations with conjugacies fixing

every point of the boundary and the fixed points of the action by rotations (the boundary pointsand the fixed points of the action are called singularities).As noted in [FK04] the pictures of rotations and conjugacies are essentially identical on the discD2 and the annulus S1× [0, 1]: We have polar coordinates (θ, r) and the rotations of the standardcircle action R = Rtt∈S1 are given by Rt (θ, r) = (θ + t, r). In this connection the origin ofthe disc, which is a fixed point of the circle action, corresponds to the boundary S1 × 0 in thecase of the annulus (so considering the ergodic invariant measures the δ-measure at the fixedpoint of the circle action in the disc-case corresponds to the Lebesgue measure on the boundarycomponent S1 ×0). Since all the conjugation maps of our constructions will coincide with theidentity near r = 0 and r = 1, the differences between the disc and the annulus are insignificant.For the sake of convenience we will present our constructions in case of the annulus S1 × [0, 1].In both cases the Lebesgue measure µ on the manifold, the δ-measures at the fixed points ofthe rotations and the Lebesgue measures on the boundary components are called the naturalmeasures.We will extend the result of [FK04] by constructing diffeomorphisms with the minimal numberof ergodic invariant measures in the restricted space

A′α (M) := H Rα H−1 : H ∈ Diff∞ (M,µ) , H = id on the singularitiesC∞

122 Preliminaries

for every Liouvillean number α ∈ S1. In addition, our constructed diffeomorphisms are weakmixing with respect to the area and preserve a measurable Riemannian metric.So this result is in line with the previous chapters, where constructions of diffeomorphisms withergodic properties, that preserve a measurable Riemannian metric, are exhibited. At this juncturein chapters 4 and 5 the number of ergodic invariant measures for diffeomorphisms on the torusTm of dimension m ≥ 2 is examined. By Theorem B the set of weak mixing and strictly ergodicdiffeomorphisms is a dense Gδ-set in Aα (Tm) = h Rα h−1 : h ∈ Diff∞ (Tm, µ)

for everyLiouvillean number α. However, other numbers of ergodic invariant measures are possible as well:According to Theorem C for any d ∈ N the set of minimal diffeomorphisms preserving exactly dergodic measures and a measurable Riemannian metric is dense in Aα (Tm). The second resultis connected to [Win01], where for any d ∈ N A. Windsor constructed minimal diffeomorphismswith d ergodic invariant measures in A (M) := h St h−1 : h ∈ Diff∞ (M,ν) , t ∈ S1

onany compact and connected smooth boundaryless manifold M of dimension m ≥ 2 admitting afree C∞-action S = Stt∈S1 preserving a smooth volume ν.In this chapter we consider the manifolds D2 and S1× [0, 1] with boundary. Indeed, we will prove:

Theorem E. Let M be the disc D2 or the annulus S1× [0, 1] and R = Rtt∈S1 be the respectivestandard action by rotations. Then for every Liouvillean number α ∈ S1 the set of smoothdiffeomorphisms f ∈ A′α (M) that have exactly three ergodic invariant measures, namely thenatural measures onM , are weak mixing with respect to the Lebesgue measure onM and preservea measurable Riemannian metric is a dense subset of A′α (M) in the C∞-topology.

In subsection 6.2.2 we will conclude as well

Corollary B. The set of smooth diffeomorphisms f ∈ A′α (M) that have exactly three ergodicinvariant measures, namely the natural measures on M , and are weak mixing with respect to theLebesgue measure on M is a residual set (i.e. it contains a dense Gδ-set) in the C∞-topology inA′α (M) for every Liouvillean number α ∈ S1.

6.2 Preliminaries

6.2.1 Definitions and notations

In addition to the definitions presented in section 2.2 we introduce the subsequent notations:

Definition 6.2.1. 1. For a continuous function F : [0, 1]× [−1, 2]→ R

‖F‖0,ext := maxz∈[0,1]×[−1,2]

|F (z)| .

2. Let f ∈ Diffk(S1 × [−1, 2]

)with coordinate functions fi be given. Then we consider fi as

a function [0, 1]× [−1, 2]→ R and define

‖Df‖0,ext := maxi,j∈1,2

‖Djfi‖0,ext

|||f |||k,ext := max‖D~afi‖0,ext ,

∥∥D~a (f−1i

)∥∥0,ext : i = 1, 2, ~a with 0 ≤ |~a| ≤ k

Preliminaries 123

6.2.2 First steps of the proofTheorem E will follow from the subsequent Proposition:

Proposition 6.2.2. For every Liouvillean number α there is a sequence (αn)n∈N of rationalnumbers αn = pn

measure-preserving diffeomorphisms satisfying hn R 1qn

= R 1qn hn as well as hn = id in a

neighbourhood of the boundary, such that the diffeomorphisms fn = Hn Rαn+1 H−1n with

Hn = h1 h2 ...hn converge in the Diff∞ (M)-topology and the diffeomorphism f = limn→∞ fnhas exactly three ergodic invariant measures (namely the Lebesgue measure µ on M = S1× [0, 1],the Lebesgue measures δ0 and δ1 on the boundary components S1×0 and S1×1 respectively),is weak mixing with respect to µ, admits an invariant measurable Riemannian metric and satisfiesf ∈ A′α (M).Furthermore, for every ε > 0 the parameters in the construction can be chosen in such a waythat d∞ (f,Rα) < ε.

By Proposition 6.2.2 and the same arguments as in section 3.2.1 we deduce Theorem E.

Moreover, we can show that the set of weak mixing diffeomorphisms is generic in A′α (M)(i.e. it is a dense Gδ-set) using Proposition 6.2.2 and the same technique as in Remark 3.2.2.Next let Ξ be a countable dense subset of C (M,R). For ρ ∈ Ξ and ε > 0 we consider the set

S (ρ, ε) :=

f ∈ A′α (M) : ∃m ∈ N : inf

ξ∈Θ

∣∣∣∣∣ 1m

m−1∑i=0

ρ(f i (x)

)−∫M

ρ dξ

∣∣∣∣∣ < ε for every x ∈M

at which Θ is the simplex generated by the measures µ, δ0 and δ1. Obviously such a set S (ρ, ε) isopen. It is also a dense subset of A′α (M) because every constructed diffeomorphism f ∈ A′α (M)is an element of S (ρ, ε) due to Lemma 6.5.3 and the set of constructed diffeomorphisms is denseas seen above. By the same reasoning as at the end of section 6.5.1⋂

⋂k∈N

which as a countable intersection of open and dense sets is a dense Gδ-set, is contained in theset of diffeomorphisms f ∈ A′α (M) with the natural measures as the only ergodic invariantmeasures. Since the intersection of dense Gδ-sets is a dense Gδ-set, Corollary B is proven.

6.2.3 Sketch of the proofIn our setting the conjugation map hn is made up of three maps introduced in section 6.3:hn = gn D−1

ψn,γn φn Dψn,γn , which coincides with the identity in a neighbourhood of the

boundary.At this juncture the trapping map Dψn,γn is used to gain control of almost everything of everyorbit

Hn Rkαn+1

(x)k=0,...,qn+1−1

with the aid of the trapping regions. This allows us to prove

a convergence result on Birkhoff sums (see Lemma 6.5.3), which in turn enables us to excludethe existence of further ergodic invariant measures besides the natural measures.The conjugation map φn is used to map the trapping regions (which have nearly full lengthin the r-coordinate) on sets of small diameter and contrariwise to map elements of a partialpartition ηn on stripes with r-length almost 1. The second property is used in the proof of theweak mixing property which is based on the notion of a (γ, ε)-distribution. In this proof we also

need a map introducing shear in the θ-coordinate. The map gn has to play this role. Since theconjugation maps have to act as an isometry on large parts of the manifold in order to constructan f -invariant measurable Riemannian metric, a careful design of each conjugation map is re-quired. The application of D−1

ψn,γnis necessary to make hn to a diffeomorphism onto S1 × [0, 1]

(see Remark 6.3.11).Then we will construct the f -invariant measurable Riemannian metric by the same approach asin section 3.7: The conjugation maps are constructed in such a way that they act as isometrieson elements of a partial partition ζn with respect to the standard metric ω0. Since these par-tial partitions converge to the decomposition into points, we can prove the convergence of theRiemannian metrics ωn :=

(H−1n

)∗ω0 to an f -invariant measurable Riemannian metric.

Let r(n) := r(n) = 8 ·n ·(n+5) and we put εn := 1

4·n11·q5·r(n−1)+1n−1

. In Remark 6.5.1 we will explain

this choice of εn. Moreover, σn ∈ (0, 1) is a parameter that will be determined in Remark 6.4.10.Furthermore, we fix an arbitrary countable set Ξ = ρ1, ρ2, ... of Lipschitz continuous functionsρi : S1 × [0, 1]→ R that is dense in C

(S1 × [0, 1] ,R

). This set Ξ will be used in section 6.5.1 to

prove that the natural measures are the only ergodic invariant ones.

6.3.1 The trapping mapTo exclude the existence of further ergodic measures we have to gain control over a large pro-portion of the orbit

Hn Riαn+1

(x)i=0,1,...,qn+1−1

for every x ∈ S1 × [0, 1] with the aid of a

similar trapping map as in subsection 4.4.1. For this purpose, we use for every n ∈ N a smoothmap ψn : [0, 1]→ R satisfying

[12 , 1].

• ψn is equal to k · 4εn on[kn2 + 1

n4 ,k+1n2 − 1

]for 0 ≤ k ≤

⌋− 1 and ψn is equal to

k · 4εn on[n2−k−1n2 + 1

n4 ,n2−kn2 − 1

]for 0 ≤ k ≤

⌋− 1. On

[⌊n22

⌋n2 ,

n2−⌊n22

]it is put

to(⌊

⌋− 1)· 4εn.

With it we define the map Dψn : [0, 1]× R→ R2 by:

(θ, r) 7→(θ, r +

+ ...+1

q3+n−1n

)· ψn (θ)

Using the maps Cγn (θ, r) = (γn · θ, r) we construct the map

Dψn,γn := C−1γn Dψn Cγn :

]× R→

]× R.

Since this map coincides with the identity in a neighbourhood of the boundary of the sector onthe θ-axis, we can extend it to a smooth map Dψn,γn : S1 × R → S1 × R using the descriptionDψn,γn R l

γn= R l

γnDψn,γn for any l ∈ Z. In our construction we use

γn = n · q2+3+4+...+(3+n−1)n = n · q2+3·n+

n·(n−1)2

Remark 6.3.1. The trapping mapDψn,γn causes a r-translation by at most 2·(⌊

⌋− 1)·4εn ≤

4n2 · εn.

Remark 6.3.2. We have Dψn,γn

(S1 × [0, 1]

)⊂ S1 × [−1, 2]. This motivates our definition of

‖·‖0,ext and is used in the norm estimates in section 6.4.1 implicitly.

6.3.2 Trapping regions

We introduce three kinds of trapping regions:In the interior of S1 × [0, 1] and for l ∈ Z as well as k = 0, ..., n− 1 we consider the sets

Sintl,k,j

(1)1 , ~j2

n · qn+

n · q2n

+t(1)1

n · q3n

+ ...+t(3·k+

k·(k−1)2 )

n · q2+3+4+...+(3+k−1)n

n · q2+3+4+...+(3+k−1)+1n

(3+k)2

nq2+3+4+...+(3+k−1)+3+kn

+t(3·k+

k·(k−1)2 +1)

nq2+3+...+(3+k−1)+(3+k)+1n

+ ...+t(3·(n−1)+

n·(n−1)2 −k)

1n4 · γn

n · qn+ ...+

t(3·(n−1)+

n·(n−1)2 −k)

1 + 1γn

− 1n4 · γn

[t(1)2

qn+ ...+

t(3+k)2

,t(1)2

qn+ ...+

t(3+k)2 + 1q3+kn

− εn

where the union is taken over t(1)2 ∈ Z,

⌈(4n2 + 1

)εn · qn

⌉≤ t

(1)2 ≤ qn −

⌈(4n2 + 1

)εn · qn

⌉− 1

and t(j)1 ∈ Z, 0 ≤ t(j)1 ≤ qn − 1, for j = 1, ..., 3 · (n− 1) + n·(n−1)

2 − k apart from t(3·k+

k·(k−1)2 +1)

satisfying dεn · qne ≤ t(3·k+

k·(k−1)2 +1)

1 ≤ qn − dεn · qne − 1 and t(l)2 ∈ Z, 0 ≤ t

(l)2 ≤ qn − 1, for

l = 2, ..., 3 + k.

Then the set of trapping regions of the first kind consists of all sets D−1ψn,γn

(Sintl,k,j

(1)1 , ~j2

), where

all j(1)i ∈ Z satisfy

⌈18n2εn · qn

⌉≤ j

(1)i ≤ qn −

⌈18n2εn · qn

⌉− 1 for i = 1, 2 and j

(s)2 ∈ Z,

0 ≤ j(s)2 ≤ qn − 1 for s = 2, ..., 3 + k.

In the neighbourhood of the boundary S1 × 0 we introduce the trapping regions of the

second kind S0

l,k,j(1)1 , ~j2

:= D−1ψn,γn

l,k,j(1)1 , ~j2

)∩(S1 × [0, 1]

), at which

l,k,j(1)1 , ~j2

n · qn+

n · q2n

+t(1)1

n · q3n

+ ...+t(3·k+

k·(k−1)2 )

n · q2+3+4+...+(3+k−1)n

n · q2+3+4+...+(3+k−1)+1n

(3+k)2

nq2+3+4+...+(3+k−1)+3+kn

+t(3·k+

k·(k−1)2 +1)

nq2+3+...+(3+k−1)+(3+k)+1n

+ ...+t(3·(n−1)+

n·(n−1)2 −k)

1n4 · γn

n · qn+ ...+

t(3·(n−1)+

n·(n−1)2 −k)

1 + 1γn

− 1n4 · γn

]×[0, 4 · n2 · εn

where the union is taken over all t(j)1 ∈ Z, 0 ≤ t(j)1 ≤ qn− 1, for j = 1, ..., 3 · (n− 1) + n·(n−1)2 − k

apart from t(3·k+

k·(k−1)2 +1)

1 satisfying dεn · qne ≤ t(3·k+

k·(k−1)2 +1)

1 ≤ qn − dεn · qne − 1.Then the set of trapping regions of the second kind consists of all sets S0

l,k,j(1)1 , ~j2

, where all j(1)i ∈ Z

satisfy⌈18n2εn · qn

⌉≤ j

(1)i ≤ qn −

⌈18n2εn · qn

⌉− 1 for i = 1, 2 and j(s)

2 ∈ Z, 0 ≤ j(s)2 ≤ qn − 1,

for s = 2, ..., 3 + k.

In the neighbourhood of the boundary S1×1 we introduce the trapping regions of the third

kind S1

l,k,j(1)1 , ~j2

:= D−1ψn,γn

l,k,j(1)1 , ~j2

)∩(S1 × [0, 1]

), at which

l,k,j(1)1 , ~j2

n · qn+

n · q2n

+t(1)1

n · q3n

+ ...+t(3·k+

k·(k−1)2 )

n · q2+3+4+...+(3+k−1)n

n · q2+3+4+...+(3+k−1)+1n

(3+k)2

nq2+3+4+...+(3+k−1)+3+kn

+t(3·k+

k·(k−1)2 +1)

nq2+3+...+(3+k−1)+(3+k)+1n

+ ...+t(3·(n−1)+

n·(n−1)2 −k)

1n4 · γn

n · qn+ ...+

t(3·(n−1)+

n·(n−1)2 −k)

1 + 1γn

− 1n4 · γn

]×[1− 4 · n2 · εn, 1

where the union is taken over all t(j)1 ∈ Z, 0 ≤ t(j)1 ≤ qn− 1, for j = 1, ..., 3 · (n− 1) + n·(n−1)2 − k

apart from t(3·k+

k·(k−1)2 +1)

1 satisfying dεn · qne ≤ t(3·k+

k·(k−1)2 +1)

1 ≤ qn − dεn · qne − 1.Then the set of trapping regions of the third kind consists of all sets S1

l,k,j(1)1 , ~j2

, where all j(1)i ∈ Z

satisfy⌈18n2εn · qn

⌉≤ j

(1)i ≤ qn −

⌈18n2εn · qn

⌉− 1 for i = 1, 2 and j(s)

2 ∈ Z, 0 ≤ j(s)2 ≤ qn − 1,

for s = 2, ..., 3 + k.

Remark 6.3.3. By the requirements on the numbers t(u)1 and j(u)

i underlying 1γn

-sections on theθ-axis that are part of trapping regions belonging to one kind are also part of trapping regionsbelonging to the other kinds.Let x = (θ, r) ∈ S1 × [0, 1] be arbitrary. By the construction of the map Dψn there are at

most four sections[kn2 + 1

n4 ,k+1n2 − 1

]on the domain [0, 1] such that r does not belong to ei-

ther D−1ψn

([kn2 + 1

n4 ,k+1n2 − 1

]×[0, 4n2 · εn

]), D−1

([kn2 + 1

n4 ,k+1n2 − 1

]×[1− 4n2 · εn, 1

D−1ψn

([kn2 + 1

n4 ,k+1n2 − 1

4n2 + 1)· εn, 1−

(4n2 + 1

)· εn]).

We have to bear the gaps of our trapping region in the r-coordinate in mind. Therefore, wenote that

(1 + 1

+ ...+ 1

q3+k−1n

)· 4εn is a multiple of 1

by our assumptions in Lemma 6.4.6

and this deviation translates by full 1

-blocks in the r-coordinate. Hence, there are at most

four further sections[

kn2γn

+ 1n4γn

, k+1n2γn

− 1n4γn

]on[0, 1

]such that r does not belong to either

D−1ψn,γn

n2γn+ 1

n4γn, k+1n2γn

− 1n4γn

]×[0, 4n2 · εn

])or D−1

ψn,γn

n2γn+ 1

n4γn, k+1n2γn

− 1n4γn

]×[1− 4n2 · εn, 1

])orD−1

ψn,γn

n2γn+ 1

n4γn, k+1n2γn

− 1n4γn

]×[t(1)2qn

+ ...+ t(3+k)2

+ εnq3+kn

,t(1)2qn

+ ...+ t(3+k)2 +1

− εnq3+kn

For l = 0, ..., qn − 1, k = 0, 1, ..., n − 1 a trapping region on[lqn

+ kn·qn ,

+ k+1n·qn

]× [0, 1]

consists of at least (1− 3 · εn) · q3n+n·(n−1)

2 −(3+k)n many 1

γn-sections. We fix l, k, j(1)

1 , ~j2. Sincei · αn+1i=0,...,qn+1−1 is equidistributed on S1, the number of iterates i, such that the orbitRiαn+1

(x)i=0,...,qn+1−1

is captured by one of the 3 trapping regions D−1ψn,γn

(Stl,k,j

(1)1 , ~j2

S1 × [0, 1]), t ∈ int, 0, 1, is at least

(1− 3 · εn) · q3n+n·(n−1)

2 −(3+k)n ·

(n2 − 8

)·⌊qn+1 ·

1− 2n2

n2 · γn

Depending on the point x ∈ S1 × [0, 1] there is a portion $nt (x) of these iterates spent in

trapping regions of the specific kind, t ∈ int, 0, 1. This portion does not depend on the indicesl, k, j

(1)1 , ~j2. Then the number of iterates i, such that the orbit

Riαn+1

(x)i=0,...,qn+1−1

an arbitrary trapping region D−1ψn,γn

(Stl,k,j

(1)1 , ~j2

)∩(S1 × [0, 1]

), is not less than

n·(n−1)2 −(3+k)

n ·$nt (x) ·

(n2 − 8

)· qn+1 ·

1− 4n2

n2 · γn

≥$nt (x) · qn+1 ·

(n2 − 8

1− 4n2

n3 · q2+3+kn

≥$nt (x) · qn+1 ·

(1− 12

)· 1n · q5+k

iterates. Moreover, for every t ∈ int, 0, 1 there are(qn − 2 ·

⌈18n2εn · qn

⌉)2 · q2+kn trapping

regions of the specific kind on[lqn

+ kn2·qn ,

+ k+1n2·qn

]×Nt × Tm−2 for l = 0, ..., qn − 1 as well

as k = 0, ..., n− 1 and so not less than(qn − 2 ·

⌈18n2εn · qn

⌉)2 · q2+kn · qn+1 ·

(1− 12

)· 1n · q2+3+k

≥qn+1 ·(

1− 1n6

1− 12n2

)· 1n · qn

≥ qn+1 ·(

1− 14n2

)· 1n · qn

iterates are trapped here. Altogether, at least qn+1 ·(1− 14

)iterates are captured.

Remark 6.3.4. On the contrary, at most 14n2 · qn+1 iterates are not captured by the trapping

regions.

In this subsection we define the two announced sequences of partial partitions (ηn)n∈N and(ζn)n∈N of M = S1 × [0, 1].

]× [0, 1]. For this purpose, we

divide the fundamental sector in n sections:

• On[

kn·qn ,

k+1n·qn

]× [0, 1] in case of k ∈ N and 0 ≤ k ≤ n− 2 the partial partition ηn consists

of all multidimensional intervals of the following form:

n · qn+

n · q2n

+ ...+j(1+3·(k+1)+

k·(k+1)2 )

n · q2+3·(k+1)+k·(k+1)

+18n2 · εn

n · q2+3·(k+1)+k·(k+1)

n · qn+

n · q2n

+ ...+j(1+3·(k+1)+

k·(k+1)2 )

n · q2+3·(k+1)+k·(k+1)

− 18n2 · εn

n · q2+3·(k+1)+k·(k+1)

qn+ ...+

j(3+k+1)2

q3+k+1n

4 · q3+k+1n

qn+ ...+

j(3+k+1)2 + 1q3+k+1n

− εn

4 · q3+k+1n

where j(l)2 ∈ Z,

⌈18n2 · εn · qn

⌉≤ j(l)

2 ≤ qn −⌈18n2 · εn · qn

⌉− 1 for l = 1, ..., 3 + k + 1 and

j(l)1 ∈ Z,

⌈18n2 · εn · qn

⌉≤ j(l)

1 ≤ qn−⌈18n2 · εn · qn

⌉−1 for l = 1, ..., 1+3·(k + 1)+ k·(k+1)

• On[n−1n·qn ,

]× [0, 1] there are no elements of the partial partition ηn.

]× [0, 1] is extended to a

partial partition of S1 × [0, 1].

]× [0, 1] initially and therefore divide this sector into n sections:

kn·qn ,

k+1n·qn

]× [0, 1] in case of k ∈ N and 0 ≤ k ≤ n− 1 the partial partition ζn consists of all

sets Γn = D−1ψn,γn

n · qn+

n · q2n

+ ...+j(1+3·n+

n·(n−1)2 )

n2 · γn+

1n4 · γn

n · qn+

n · q2n

+ ...+j(1+3·n+

n·(n−1)2 )

s+ 1n2 · γn

− 1n4 · γn

qn+ ...+

j(3+k+1)2

q3+k+1n

+ ...+j(2+3·(k+1)+

k·(k+1)2 )

q2+3·(k+1)+

k·(k+1)2

j(3+3·(k+1)+

k·(k+1)2 )

2 · 16n2 · εn

n · q2+3·(k+1)+k·(k+1)

2n · [nqσnn ]

+1600n4 · ε2

n · q2+3·(k+1)+k·(k+1)

2n · [nqσnn ]

qn+ ...+

(j(3+3·(k+1)+

k·(k+1)2 )

2 + 1)· 16n2 · εn

n · q2+3·(k+1)+k·(k+1)

2n · [nqσnn ]

− 1600n4 · ε2n

n · q2+3·(k+1)+k·(k+1)

2n · [nqσnn ]

⌈100n2 · εn · qn

⌉≤ j(l)

1 ≤ qn−⌈100n2 · εn · qn

⌉−1 for l = 1, ..., 1+3·n+ n·(n−1)

(l)2 ∈ Z and

⌈100n2 · εn · qn

⌉≤ j(l)

2 ≤ qn−⌈100n2 · εn · qn

⌉−1 for l = 1, ..., 2+3·(k + 1)+ k·(k+1)

j(3+3(k+1)+

k(k+1)2 )

2 ∈ Z, [nqσnn ] · n ≤ j(3+3(k+1)+

k(k+1)2 )

2 ≤ [nqσnn ]16n·εn − [nqσnn ] · n− 1 as well as s ∈ N

and 0 ≤ s ≤ n2 − 1.

Remark 6.3.6. For every n the partial partition ζn consists of disjoint sets, covers a set ofmeasure at least 1− 3

n2 in case of n ≥ 3 and the sequence (ζn)n∈N converges to the decompositioninto points.

Remark 6.3.7. Note that Dψn,γn acts as an isometry on all the partition elements Γn ∈ ζn.

]such that 1

ε ∈ Z. Moreover, we consider δ > 0 such that 1δ ∈ Z and

a·b·δε ∈ Z. We denote

]×[0, ε

b·a]by ∆a,b,ε. Using the maps

Da,b,ε : R2 → R2, (θ, r) 7→(a · θ, b · a

ε· r)

and gε from Lemma 3.3.4 we define the measure-preserving map ga,b,ε : ∆a,b,ε → gb (∆a,b,ε) byga,b,ε = D−1

a,b,ε gε Da,b,ε. Using the fact that a·b·δε ∈ Z we extend it to a smooth diffeomorphism

ga,b,ε,δ :[0, 1

]× [δ, 1− δ]→ gb

([0, 1

]× [δ, 1− δ]

ga,b,ε,δ

(θ, r + l · ε

b · a

)=(l · εa, l · ε

b · a

)+ ga,b,ε (θ, r)

for r ∈[0, ε

b·a]and l ∈ Z satisfying δ

ε · b · a ≤ l ≤1−δε · b · a− 1.

With the choice δ = 12n2 · εn we construct the smooth measure-preserving diffeomorphism gn

on the fundamental sector[0, 1

]×[12n2 · εn, 1− 12n2 · εn

]initially and for this divide it into

n sections:On

n·qn ,k+1n·qn

]×[12n2 · εn, 1− 12n2 · εn

]in case of k ∈ Z and 0 ≤ k ≤ n− 1:

gn = gn·q

2+3·(k+1)+ (k+1)·k2

n ,[n·qσnn ],16n2·εn,12n2·εn

Since gn coincides with the map g[n·qσnn ] in a neighbourhood of the boundary of the differentsections on the θ-axis, this yields a smooth map and we can extend it to a smooth measure-preserving diffeomorphism on S1 ×

[12n2 · εn, 1− 12n2 · εn

]using the description gn R l

R lqn gn for l ∈ Z.

Moreover, let χn : [0, 1]→ [0, 1] be a smooth function satisfying the subsequent properties:

• χn is equal to 0 on[0, 4n2 · εn

]as well as on

[1− 8n2 · εn, 1

[6n2 · εn, 1− 10n2 · εn

]χn takes the value 1.

• χn is non-decreasing on[4n2 · εn, 6n2 · εn

[1− 10n2 · εn, 1− 8n2 · εn

With it we define gn : S1 ×[0, 12n2 · εn

]→ S1 ×

[0, 12n2 · εn

]and gn : S1 ×

[1− 12n2 · εn, 1

S1 ×[1− 12n2 · εn, 1

gn (θ, r) = (θ + χn (r) · [n · qσnn ] · r, r)

Since all the constructed maps gn coincide with g[nqσnn ] in a neighbourhood of the boundary of therespective domain, we can piece them together smoothly to a diffeomorphism gn : S1 × [0, 1] →S1 × [0, 1].We note that the assumption a·b·δ

ε = a·b·34 ∈ Z is satisfied, because 1

εn= 4 · n11 · q5·r(n−1)+1

divides qn by our construction of the sequence (αn)n∈N in Lemma 6.4.6. Moreover, gn = id inthe neighbourhoods S1 ×

[0, 4n2 · εn

]and S1 ×

[1− 8n2 · εn, 1

]of the boundary components.

Remark 6.3.8. We will call the parts of the domains ∆a,b,ε,δ corresponding to ∆ (4ε) of gε the“good area” of gn.

We modify Lemma 3.3.6:

Lemma 6.3.9. For every j ∈ N and 0 < ε < 14·j there exists a smooth measure-preserving

diffeomorphism ϕε,j on R2, which is the rotation in the plane by π/2 about the point(

)∈ R2

on [(j + 1) · ε, 1− (j + 1) · ε]2 and coincides with the identity outside of [j · ε, 1− j · ε]2.

Proof. First of all we recall the notation ∆ (ε) = [ε, 1− ε]2 from subsection 3.3.2. Let ψε be asmooth diffeomorphism satisfying

ψε (x, y) =

(x, y) on R2 \∆ (j · ε)(

12 + 1

5 ·(x− 1

2 + 15 ·(y − 1

))on ∆ ((j + 1) · ε)

τε (x, y) =

(1− y, x) on(x− 1

)2 +(y − 1

)2 ≤ 150

(x, y) on

(x− 1

)2 +(y − 1

)2 ≥ 116

We define ϕε := ψ−1

ε τε ψε. Then the diffeomorphism ϕε coincides with the rotation on∆ ((j + 1) · ε) and with the identity on R2 \∆ (j · ε). From this we conclude that det (Dϕε) > 0.

Moreover, ϕε is measure-preserving on Uε :=(R2 \∆ (j · ε)

)∪∆ ((j + 1) · ε).

As in the proof of Lemma 3.3.4 we construct a diffeomorphism ϕε, that is measure-preservingon the whole R2 and agrees with ϕε on Uε with the aid of “Moser’s trick”.

Furthermore, for λ ∈ N we define the maps Cλ (x1, x2) = (λ · x1, x2) and Dλ (x1, x2) =(λ · x1, λ · x2). Let µ ∈ N, 1

δ ∈ N and 1δ divides µ. We construct a diffeomorphism ψµ,δ,ε2 in the

following way:

• Consider [0, 1− 2δ]2: Since 1δ divides µ, we can divide [0, 1− 2δ]2 in cubes of sidelength 1

• Under the map Dµ any of these cubes of the form∏2i=1

[liµ ,

li+1µ

]with li ∈ N is mapped

onto∏2i=1 [li, li + 1].

• On [0, 1]2 we will use the diffeomorphism ϕ−1ε2,1

constructed in Lemma 6.3.9. Since this isthe identity outside of ∆ (ε2), we can extend it to a diffeomorphism ϕ−1

ε2,1on R2 using the

instruction ϕ−1ε2,1

(x1 + k1, x2 + k2) = (k1, k2) + ϕ−1ε2,1

(x1, x2), where ki ∈ Z and xi ∈ [0, 1].

• Now we define the smooth measure-preserving diffeomorphism

ψµ,δ,ε2 = D−1µ ϕ−1

ε2,1Dµ : [0, 1− 2δ]2 → [0, 1− 2δ]2 .

• Hereby, we define

ψµ,δ,ε2 (x1, x2) =([ψµ,δ,ε2 (x1 − δ, x2 − δ)

+ δ,[ψµ,δ,ε2 (x1 − δ, x2 − δ)

on [δ, 1− δ]2

(x1, x2) else

This is a smooth map, because ψµ,δ,ε2 is the identity in a neighbourhood of the boundaryby the construction.

Remark 6.3.10. For every set W =∏2i=1

[liµ + ri,

li+1µ − ri

], where li ∈ Z and ri ∈ R satisfies

|ri · µ| ≤ ε2, we have ψµ,δ,ε2 (W ) = W .

Using these maps we build the following smooth measure-preserving diffeomorphism:

φλ,ε,j,µ,δ,ε2 :[0,

]× R→

]× R, φλ,ε,j,µ,δ,ε2 = C−1

λ ψµ,δ,ε2 ϕε,j Cλ.

Afterwards, φλ,ε,j,µ,δ,ε2 is extended to a diffeomorphism on S1 × R by the description

φλ,ε,j,µ,δ,ε2

λ, x2 + k2

λ, k2

)+ φλ,ε,j,µ,δ,ε2 (x1, x2)

for ki ∈ Z.For convenience we will use the following notation: φλ,µ = φλ,εn,4n2,µ,4n2·εn, εn3 . Hereby, we

define the diffeomorphism φn on the fundamental sector: On[

kn·qn ,

k+1n·qn

]× R in case of k ∈ N

and 0 ≤ k ≤ n− 1

φn = φn·q2+3+4+...+(3+k−1)

n ,q3+kn

= φn·q

2+3k+ k·(k−1)2

n ,q3+kn

Now we extend φn to a diffeomorphism on S1 × R using the description φn R 1qn

= R 1qn φn.

132 Convergence of (fn)n∈N in Diff∞(S1 × [0, 1] , µ

)Remark 6.3.11. Since ϕε,j coincides with the identity outside of [j · ε, 1− j · ε]2, we haveφn(Dψn,γn

(S1 × [0, 1]

))= Dψn,γn

(S1 × [0, 1]

). Hence

D−1ψn,γn

φn Dψn,γn : S1 × [0, 1]→ S1 × [0, 1] .

6.4 Convergence of (fn)n∈N in Diff∞(S1 × [0, 1] , µ

)By the same methods as in section 3.6 we show that the sequence of constructed measure-preserving smooth diffeomorphisms fn = Hn Rαn+1 H−1

n converges.

We estimate the norms of the occurrent maps.

|||Dψn,γn |||k,ext ≤ C · γkn,

Proof. By construction of the map Dψn,γn = C−1γn Dψn Cγn we have

Dψn,γn (θ, r) = (θ, r + dn · ψn (γn · θ))

as well asD−1ψn,γn

(θ, r) = (θ, r − dn · ψn (γn · θ))

using the abbreviation dn := 1 + 1q3n

+ ...+ 1q3+n−1n

Since dn ≤ 2 we obtain: |||Dψn,γn |||k,ext ≤ C · dn · γkn ≤ C · qk·(2+3·n+

n·(n−1)2 )

Next we can prove an estimate on the norms of the map φn:

|||φn|||k,ext ≤ C · γkn,

where C is a constant depending on k and n, but is independent of qn.

Proof. First of all we consider the map φλ,µ := φλ,ε,j,µ,δ,ε2 = C−1λ ψµ,δ,ε2 ϕε,j Cλ introduced

in subsection 6.3.5:

φλ,µ (x1, x2) =(

1λ· [ψµ ϕε,j ]1 (λ · x1, x2) , [ψµ ϕε,j ]2 (λ · x1, x2)

Let k ∈ N. We compute for a multiindex ~a with 0 ≤ |~a| ≤ k:∥∥∥D~a [φλ,µ]1

∥∥∥0,ext

≤ λk−1 · |||ψµ ϕε,j |||k,ext and∥∥∥D~a [φλ,µ]

∥∥∥0,ext

≤ λk · |||ψµ ϕε,j |||k,ext.

Therefore, we examine the map ψµ. For any multiindex ~a with 0 ≤ |~a| ≤ k and u ∈ 1, 2we obtain:

∥∥D~a [ψµ]u∥∥

0,ext ≤ µk−1 · |||ϕε2 |||k,ext = Ck,ε2 · µk−1 and in the same way we get∥∥∥D~a [ψ−1µ

∥∥∥0,ext

≤ Ck,ε2 · µk−1. Hence: |||ψµ|||k,ext ≤ C · µk−1.

Convergence of (fn)n∈N in Diff∞(S1 × [0, 1] , µ

In the next step we use the formula of Faà di Bruno mentioned in remark 3.6.3. With it wecompute for any multiindex ~ν with |~ν| = k in the same way as in the proof of Lemma 3.6.4:

∥∥∥D~ν [(ψµ ϕε,j)−1]u

∥∥∥0,ext

≤∑

~λ∈N20, 1≤|~λ|≤k

∥∥∥D~λ [ϕ−1ε,j

∥∥∥0,ext·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏i=1

|||ψ−1µ ||||~ki||~li|,ext

~ki! ·(~li!)|~ki|

Using |||ψ−1µ ||||~ki||~li|,ext

≤ C · µ|~ki|·|~li|:∏si=1 |||ψ−1

µ ||||~ki||~li|,ext

≤ C · µ∑si=1|~li|·|~ki| ≤ C · µk, where

C is independent of µ. Finally∥∥∥D~ν [(ψµ ϕε,j)−1

∥∥∥0,ext

≤ C · µk. Analogously we compute∥∥D~ν [ψµ ϕε,j ]u∥∥

0,ext ≤ C ·|||ψµ|||k,ext ≤ C ·µk−1. Altogether, we obtain |||ψµϕε,j |||k,ext ≤ C ·µk.

Hereby, we estimate∥∥∥D~a [φλ,µ]

∥∥∥0,ext

≤ C ·λk ·µk and analogously∥∥∥D~a [φ−1

∥∥∥0,ext

≤ C ·λk ·µk.

In conclusion this yields |||φλ,µ|||k,ext ≤ C · µk · λk.In the setting of our explicit construction of the map φn in section 6.3.5 we have ε = εn, ε2 = εn

λmax = n · q2+3·(n−1)+(n−1)·(n−2)

2n and µmax = q3+n−1

n . Thus:

|||φn|||k,ext ≤ C (k, n) ·(n · q2+3·(n−1)+

(n−1)·(n−2)2

)k·(q3+n−1n

)k≤ C (k, n) · γkn,

where C (k, n) is a constant independent of qn.

Combining the last results with the aid of the formula of Faà di Bruno yields

|||D−1ψn,γn

φn Dψn,γn |||k ≤ C · γ3·kn ,

In the next step we consider the map hn = gn D−1ψn,γn

φn Dψn,γn , where gn is constructedin section 6.3.4:

|||hn|||k ≤ C · qkn · γ4·kn ,

Proof. We label φn := D−1ψn,γn

φn Dψn,γn . Outside of S1 × [δ, 1− δ]m−1 we have:

hn (x1, x2) = gn φn (x1, x2)

=([φn (x1, x2)

+ χn (x2) · [n · qσnn ] ·[φn (x1, x2)

]2,[φn (x1, x2)

h−1n (x1, x2) = φ−1

n g−1n (x1, x2)

=([φ−1n (x1 − χn (x2) · [n · qσnn ] · x2, x2)

]1,[φn (x1 − χn (x2) · [n · qσnn ] · x2, x2)

134 Convergence of (fn)n∈N in Diff∞(S1 × [0, 1] , µ

)Since σn < 1 we can estimate:

|||hn|||k ≤ 2 · Cn,k · [n · qσnn ]k · |||φn|||k ≤ C · qσn·kn · γ3·kn ≤ C · qkn · γ3·k

with a constant C independent of qn.

In the other case we have

gn φn (x1, x2) =([ga,b,ε

([φn]1,[φn]2

(x1, x2) ,[ga,b,ε

([φn]1,[φn]2

(x1, x2)).

Then we use the formula of Faà di Bruno as in the proof of Lemma 3.6.5 and obtain the claim:|||gn φn|||k ≤ C(n, k) · qkn · γ4·k

Finally, we are able to prove an estimate on the norms of the map Hn:

|||Hn|||k ≤ C · qk·4·n·(n+5)n ,

where C is a constant depending solely on k, n and Hn−1. Since Hn−1 is independent of qn inparticular, the same is true for C.

Proof. By Lemma 6.4.4 and γn = n · q2+3n+(n−1)·n

2n = n · q2+

n·(n+5)2

n we have

|||hn|||k ≤ C · qk·(1+4·2+4·n·(n+5)

2 )n ≤ C · qk·4·n·(n+5)

Then we can verify the claim by the same arguments as in Lemma 3.6.6.

6.4.2 Proof of convergenceWe show that we can satisfy the conditions from Lemma 3.6.8 in our constructions:

qnof rational numbers with 1

εndivides qn and qn > maxi=1,...,n+1 Li (where Li

denotes the Lipschitz constant of ρi ∈ Ξ), such that our conjugation maps Hn constructed insection 6.3 fulfil the following conditions:

|α− αn| <1

2 · kn · Ckn · |||Hn|||kn+1kn+1

|α− αn| <1

2n+1 · qn · |||Hn|||1

Proof. In Lemma 6.4.5 we deduced the estimate |||Hn|||kn+1 ≤ Cn · q(kn+1)·4·n·(n+5)n , where the

constant Cn was independent of qn. Thus, we can choose qn ≥ Cn for every n ∈ N. Hence, weobtain: |||Hn|||kn+1 ≤ q8·n·(n+5)·(kn+1)

Besides qn ≥ Cn we set the conditions qn ≥ 1εn· 12εn−1·|||ψn−1|||1·(n− 1)2·q2·(2+3·(n−1)+

(n−1)·(n−2)2 )

n−1 ,

qn > n13 · q9·r(n−1)+1n−1 and qn > maxi=1,...,n+1 Li. Since α is a Liouvillean number, we find a

The invariant measures 135

sequence of rational numbers αn = pnqn, pn, qn relatively prime, converging to α under the above

restrictions (formulated for qn) satisfying:

|α− αn| =∣∣∣∣α− pn

∣∣∣∣ < |α− αn−1| · ε1+8·n·(n+5)·(kn+1)2

2n+1 · kn · Ckn · q1+8·n·(n+5)·(kn+1)2

Put qn := qnεn

and pn := pnεn

. Then we obtain:

|α− αn| <|α− αn−1|

2n+1 · kn · Ckn · q1+8·n·(n+5)·(kn+1)2

Thus, we have |α− αn|n→∞→ 0 monotonically. Because of |||Hn|||kn+1

kn+1 ≤ q8·n·(n+5)·(kn+1)2

n thisyields: |α− αn| < 1

. Thus, the first property of this Lemma is fulfilled.

We can verify the second property by the same arguments as in Lemma 3.6.9.

Remark 6.4.7. Lemma 6.4.6 shows that the conditions of Lemma 3.6.8 are satisfied. There-fore, our sequence of constructed diffeomorphisms fn converges in the Diff∞(M)-topology to adiffeomorphism f ∈ Aα(M).

Remark 6.4.8. In particular |||Hn|||1 ≤ q8·n·(n+5)n motivates our definition of the number r(n) =

8 · n · (n+ 5).

As in Lemma 3.6.11 we can conclude:

(f m, f mn

)≤ 1

Remark 6.4.10. We determine the parameter σn ∈ (0, 1) in such a way that qσnn = q4·r(n−1)n−1 ,

i.e. we have [nqσnn ] = n · q4·r(n−1)n−1 .

6.5 The invariant measures

As above µ is the Lebesgue measure on S1× [0, 1] and δ0 (resp. δ1) denotes the Lebesgue measureon the boundary component S1 × 0 (resp. S1 × 1). We aim for showing that these are theonly ergodic f -invariant measures. For this purpose, we deduce a statement on the Birkhoff sumsfor arbitrary x ∈ S1 × [0, 1] (see Lemma 6.5.3). In order to prove such a statement we have togain control over a large proportion of every Rt-orbit. This is done with the aid of the trappingmaps and regions.Furthermore, λ denotes the Lebesgue measure on S1 and λ the Lebesgue measure on [0, 1].

6.5.1 Trapping property

In case of 0 ≤ l ≤ qn − 1, 0 ≤ k ≤ n− 1, j(1)i ∈ Z,

⌈18n2εn · qn

⌉≤ j

(1)i ≤ qn −

⌈18n2εn · qn

⌉− 1

for i = 1, 2 as well as j(t)2 ∈ Z, 0 ≤ j(t)

2 ≤ qn − 1 for t = 2, 3 we introduce the sets

∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

n · qn+

n · q2n

n · qn+j

(1)1 + 1n · q2

(3)2 + 1q3n

136 The invariant measures

We observe that there are qn · n ·(qn − 2 ·

⌈18n2εn · qn

⌉)2 · q2n such sets ∆

l,k,j(1)1 ,j

(1)2 ,j

(2)2 ,j

denote the union of these sets by T intn and the collection of these sets by T int

n . Then

µ(S1 × [0, 1] \ T int

)= 1−n ·qn ·

(qn − 2 ·

⌈18n2εn · qn

⌉)2 ·q2n ·

1n · q5

≤ 1−(

1− 2 · 14n4

≤ 1n4.

Note that D−1ψn,γn

(∆l,k,j

(1)1 ,...,j

)⊆ S1 ×

[12n2 · εn, 1− 12n2 · εn

]. Unfortunately, gn = g[nqσnn ]

is not necessarily true on D−1ψn,γn

(∆l,k,j

(1)1 ,...,j

), but this set is strictly contained in a cube of

sidelength 1n·q2

n+ 4n2 · εn ≤ 8n2 · εn that is a union of domains of ga,b.ε. Then we obtain

diam(Hn−1 gn

(D−1ψn,γn

(∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

)))≤ ‖DHn−1‖0 · n · q

σnn ·√

2 · 8n2 · εn

≤ qr(n−1)n−1 · q4·r(n−1)

n−1 · 8n3 ·√

4 · n11 · q5·r(n−1)+1n−1

n8 · qn−1

by the construction of the number σn in Remark 6.4.10.By the requirements on the number qn in Lemma 6.4.6 we obtain

∣∣∣ρi (Hn−1 gn D−1ψn,γn

(x))− ρi

(Hn−1 gn D−1

ψn,γn(y))∣∣∣

≤Lip (ρi) · diam(Hn−1 gn

(D−1ψn,γn

(∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

)))≤qn−1 ·

4n8 · qn−1

for every x, y ∈ ∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

and the function ρi ∈ Ξ in case of i = 1, ..., n.

Remark 6.5.1. Since we need this expression to converge to 0 as n → ∞, this explains ourchoice of εn.

Averaging over all y ∈ ∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

we obtain:

∣∣∣∣∣∣ρi(Hn−1 gn D−1

ψn,γn(x))− 1

∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

) ∫Hn−1gn

(∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

) ρidµ∣∣∣∣∣∣ < 4

(Sintl,k,j

(1)1 , ~j2

)defined in section 6.3.2

is mapped under φn Dψn,γn onto

n · qn+

n · q2n

+t(1)1

n · q3n

+ ...+t(3·k+

k·(k−1)2 )

n · q2+3+4+...+(3+k−1)n

− t(1)2

n · q2+3+4+...+(3+k−1)+1n

− ...

− t(3+k)2 + 1

nq2+3+4+...+(3+k−1)+3+kn

+t(3·k+

k·(k−1)2 +1)

nq2+3+...+(3+k−1)+(3+k)+1n

+ ...+t(3·(n−1)+

n·(n−1)2 −k)

1n4 · γn

n · qn+ ...+

t(3·(n−1)+

n·(n−1)2 −k)

1 + 1γn

− 1n4 · γn

qn+ ...+

j(3+k)2

qn+ ...+

j(3+k)2 + 1q3+kn

− εn

where the union is taken over t(1)2 ∈ Z,

⌈(4n2 + 1

)εn · qn

⌉≤ t

(1)2 ≤ qn −

⌈(4n2 + 1

)εn · qn

⌉− 1

and t(j)1 ∈ Z, 0 ≤ t(j)1 ≤ qn − 1, for j = 1, ..., 3 · (n− 1) + n·(n−1)

2 − k apart from t(3·k+

k·(k−1)2 +1)

satisfying dεn · qne ≤ t(3·k+

k·(k−1)2 +1)

1 ≤ qn − dεn · qne − 1 and t(l)2 ∈ Z, 0 ≤ t

(l)2 ≤ qn − 1, for

l = 2, ..., 3 + k.

(D−1ψn,γn

(Sintl,k,j

(1)1 , ~j2

))is contained in ∆

l,k,j(1)1 j

(1)2 ,j

(2)2 ,j

. The same

is true for the other allowed values of j(4)2 , ..., j

(3+k)2 . Thus, there are qkn trapping regions that

are mapped into ∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

under φn Dψn,γn . Hence, we can estimate the numberof i ∈ 0, ..., qn+1 − 1 such that φn Dψn,γn Riαn+1

(x) is contained in ∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

qkn ·$nint(x) · qn+1 ·

1− 12n2

n·q3+kn ·q2

= $nint(x) · qn+1 ·

(1− 12

)· µ(

∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

)from below and

by $nint(x) · qn+1 · µ

(∆l,k,j

(1)1 ,j

(1)2 ,j

(2)2 ,j

)from above for arbitrary x ∈ S1 × [0, 1] using Remark

6.3.3.Let x ∈ S1 × [0, 1] be arbitrary. We denote the set of iterates j ∈ 0, ..., qn+1 − 1 such thatφn Dψn,γn Rjαn+1

(x) is contained in ∆ ∈ T intn by I∆. With the aid of equation 6.1 we obtain:∣∣∣∣∣∣ 1

∑j∈I∆

(Hn−1 gn D−1

ψn,γn φn Dψn,γn Rjαn+1

(x))−$n

int(x) ·∫Hn−1gn(∆)

ρidµ

∣∣∣∣∣∣≤4 · µ (∆)

12n2·∫Hn−1gn(∆)

|ρi| dµ.

Furthermore, we examine the trapping regions in the neighbourhoods of the boundaries. Forl = 0, 1, ..., qn−1, k = 0, 1, ..., n−1 and

⌈18n2εn · qn

⌉≤ j(1)

1 ≤ qn−⌈18n2εn · qn

⌉−1 we introduce

the sets

l,k,j(1)1

n · qn+

n · q2n

n · qn+j

(1)1 + 1n · q2

]×[0, 4 · n2 · εn

l,k,j(1)1

n · qn+

n · q2n

n · qn+j

(1)1 + 1n · q2

]×[1− 8 · n2 · εn, 1

Again, T tn denotes the collection of these sets ∆t

l,k,j(1)1

in case of t = 0, 1 as well as I∆0 and I∆1

respectively label the set of iterates such that Dψn,γn Rjαn+1(x) is contained in ∆t ∈ T tn for

t = 0 and accordingly t = 1.We observe that for t = 0, 1 the map Hn−1 gn acts as the identity on these sets ∆t

l,k,j(1)1

)≤ 16 · n2 · εn. Then we conclude for i = 1, ..., n and x, y ∈ ∆t

l,k,j(1)1

|ρi (Hn−1 gn (x))− ρi (Hn−1 gn (y))| < Lip (ρi) · diam(

l,k,j(1)1

n9 · q5·r(n−1)n−1

In particular, this holds true for y =(

+ kn·qn + j

(1)1n·q2

). We consider such points

n·q2n, t)

for u = 0, ..., nq2n − 1 and calculate for z ∈

n·q2n− 1

2·n·q2n, un·q2

2·n·q2n

∣∣∣∣ρi(( u

n · q2n

))− ρi ((z, t))

∣∣∣∣ < Lip (ρi) ·1

2 · n · q2n

n · qn.

Averaging over all z ∈[

un·q2

n− 1

2·n·q2n, un·q2

2·n·q2n

]yields

∣∣∣∣∣∣ 1n · q2

· ρi(

n · q2n

)−∫[

un·q2n

− 12·n·q2n

, un·q2n

+ 12·n·q2n

] ρi dδt∣∣∣∣∣∣ < 1

n · qn· 1n · q2

Summing over all u = 0, ..., nq2n − 1 gives∣∣∣∣∣∣ 1

n · q2n

·nq2n−1∑u=0

n · q2n

)−∫

S1ρi dδ

∣∣∣∣∣∣ < 1n · qn

The set of u ∈

0, ..., nq2n − 1

such that

n·q2n, t)is contained in one of the blocks ∆t

l,k,j(1)1∈ T tn

is denoted by U tn. Since there are at least qn · n ·(qn − 2 ·

⌈18n2εn · qn

⌉)≥(1− 1

)· n · q2

n suchblocks there are at most

⌈1n4 · n · q2

⌉numbers u ∈

0, ..., nq2

n − 1outside of U tn. Hereby, we get∣∣∣∣∣∣ 1

∑∆t∈T tn

∑j∈I∆t

(Hn Rjαn+1

(x))−$n

t (x) ·∫

S1ρi (θ, t) dδt

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1qn+1

∑∆t∈T tn

∑j∈I∆t

(Hn Rjαn+1

(x))−$n

t (x) · 1n · q2

·nq2n−1∑u=0

n · q2n

)∣∣∣∣∣∣+$n

t (x) ·

∣∣∣∣∣∣ 1n · q2

·nq2n−1∑u=0

n · q2n

)−∫

S1ρi (θ, t) dδt

∣∣∣∣∣∣≤∑u∈Utn

∣∣∣∣∣∣ 1qn+1

∑j∈I∆tu

(Hn Rjαn+1

(x))− $n

t (x)n · q2

· ρi(

n · q2n

)∣∣∣∣∣∣+1n2· ‖ρi‖0 +

1n · qn

Under the mapHn

(qn − 2 ·

⌈18n2εn · qn

⌉)·q2+kn trapping regionsD−1

ψn,γn

(Stl,k,j

(1)1 , ~j2

)are mapped

into one such ∆t

l,k,j(1)1

. Thus, for arbitrary x ∈ S1 × [0, 1] the set ∆t

l,k,j(1)1

captures at least

$nt (x)·qn+1 ·

(1− 14

)· 1n·q2

nand at most $n

t (x)·qn+1 · 1n·q2

niterates D−1

ψn,γnφnDψn,γn Rjαn+1

(x)by Remark 6.3.3. Then we can estimate with the aid of equation 6.2

∣∣∣∣∣∣ 1qn+1

∑j∈I∆tu

(Hn Rjαn+1

(x))− $n

t (x)n · q2

n · q2n

)∣∣∣∣∣∣ < $nt (x)n · q2

· 1n8

+14n2· $

nt (x)n · q2

· ‖ρi‖0 .

In continuation of the above estimate we conclude

∣∣∣∣∣∣ 1qn+1

∑∆t∈T tn

∑j∈I∆t

(Hn Rjαn+1

(x))−$n

t (x) ·∫

S1ρi (θ, t) dδt

∣∣∣∣∣∣≤ 1n8

+14n2· ‖ρi‖0 +

1n2· ‖ρi‖0 +

1n · qn

≤15n2· ‖ρi‖0 +

Using this preparatory work we can prove the following result on the Birkhoff sums:

Lemma 6.5.2. Let ρi ∈ Ξ and i = 1, ..., n. Then for every y ∈M = S1 × [0, 1] we have

infξn∈Θ

∣∣∣∣∣∣ 1qn+1

qn+1−1∑j=0

ρi(f jn (y)

)−∫M

ρi dξn

∣∣∣∣∣∣ < 60n2· ‖ρi‖0 ,

where Θ is the simplex generated byµ, δ0, δ1

Proof. Let x ∈ S1 × [0, 1] be arbitrary. We introduce the measure

ξnx := $nint(x) · µ+$n

0 (x) · δ0 +$n1 (x) · δ1 ∈ Θ.

The set of numbers k ∈ 0, 1, ..., qn+1 − 1, such that the iterates Rkαn+1(x) are not contained

in one of the trapping regions, is denoted by Ia. Referring to Remark 6.3.4 there are at most14n2 · qn+1 numbers in Ia. We obtain

∣∣∣∑j∈Ia ρi

(Hn Rjαn+1

(x))∣∣∣ ≤ ‖ρi‖0 · 14

n2 · qn+1.Hereby, we obtain:

∣∣∣∣∣∣ 1qn+1

qn+1−1∑j=0

(Hn Rjαn+1

(x))−$n

int(x) ·∫M

ρi dµ−$n0 (x) ·

∫S1ρi dδ

0 −$n1 (x) ·

∫S1ρi dδ

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1qn+1

∑∆int∈T int

∑j∈I∆int

(Hn Rjαn+1

(x))−$n

int(x) ·∫M

ρi dµ

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1qn+1

∑∆0∈T 0

∑j∈I∆0

(Hn Rjαn+1

(x))−$n

0 (x) ·∫

S1ρi (θ, 0) dδ0

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1qn+1

∑∆1∈T 1

∑j∈I∆1

(Hn Rjαn+1

(x))−$n

1 (x) ·∫

S1ρi (θ, 1) dδ1

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1qn+1

∑j∈Ia

(Hn Rjαn+1

(x))∣∣∣∣∣∣

≤ 4n8

+12n2· ‖ρi‖0 + µ

(M \ T int

)· ‖ρi‖0 + 2 ·

(15n2· ‖ρi‖0 +

14n2· ‖ρi‖0 ≤

60n2· ‖ρi‖0 .

With x = H−1n (y) we obtain the statement of the Lemma.

We point out that the measure ξnx used in the above proof was dependent on the point x, butindependent of the function ρ ∈ Ξ.

Lemma 6.5.3. For every ρ ∈ Ξ and y ∈ S1 × [0, 1] we have

infξn∈Θ

∣∣∣∣∣ 1qn+1

qn+1−1∑k=0

ρ(fk (y)

)−∫ρ dξn

∣∣∣∣∣→ 0 as n→∞,

where Θ is the simplex generated byµ, δ0, δ1

Proof. By Lemma 6.4.9 we have

d(qn+1)0 (f, fn) := max

i=0,1,...,qn+1−1d0

(f i, f in

) n→∞→ 0.

Then for every ρ ∈ Ξ we have∣∣ρ (f i (x)

)− ρ

(f in (x)

)∣∣ n→∞→ 0 uniformly for i = 0, 1, ..., qn+1 − 1,because every continuous function on the compact space S1×[0, 1] is uniformly continuous. Thus,we get:

∥∥∥ 1qn+1

∑qn+1−1i=0 ρ

(f i (x)

)− 1

∑qn+1−1i=0 ρ

(f in (x)

)∥∥∥0

n→∞→ 0. Applying the previousLemma 6.5.2 we obtain the claim.

Since the family Ξ is dense in C(S1 × [0, 1] ,R

), the convergence holds for every continuous

function by an approximation argument.Now we can prove that the measures µ, δ0, δ1 are the only possible ergodic ones: Assume thatthere is another ergodic invariant probability measure ξ. By the Birkhoff Ergodic Theorem wehave for every ρ ∈ C

(S1 × [0, 1] ,R

limn→∞

n−1∑k=0

ρ(fk (x)

S1×[0,1]

ρ dξ for ξ-a.e. x ∈ S1 × [0, 1] .

With the aid of Lemma 6.5.3 we obtain for every ρ ∈ C(S1 × [0, 1] ,R

)and x in a set of ξ-full

measure:∫S1×[0,1]

ρ dξ = limn→∞

n−1∑k=0

ρ(fk (x)

)= limn→∞

qn+1−1∑k=0

ρ(fk (x)

)= limn→∞

∫S1×[0,1]

ρ dξn,

where ξn is in the simplex generated byµ, δ0, δ1

. As noted this measure does not depend on the

function ρ. Thus, we have for every ρ ∈ C(S1 × [0, 1] ,R

): limn→∞

∫S1×[0,1]

ρ dξn =∫

S1×[0,1]ρ dξ.

Since the simplex generated byµ, δ0, δ1

is weakly closed, this implies that ξ is in this simplex.

We recall that ergodic measures are the extreme points in the set of invariant Borel probabilitymeasures (see [Wa75], Theorem 5.15.). Then ξ has to be one of the measures

µ, δ0, δ1

and we

obtain a contradiction.

6.5.2 Weak mixing with respect to Lebesgue measure on S1 × [0, 1]

The proof of the weak mixing-property will be based on the notion of a (γ, ε)-distribution:

Definition 6.5.4. Let Φ : S1 × [0, 1]→ S1 × [0, 1] be a diffeomorphism and J be an interval in[0, 1]. We say that an element I of a partial partition is (γ, ε)-distributed on J under Φ, if thefollowing properties are satisfied:

• [c, c+ γ′]× J ⊆ Φ(I)⊆ [c, c+ γ]× [0, 1] for some c ∈ S1 and 0 < γ′ ≤ γ ≤ γ.

• For every interval J ⊆ J it holds:∣∣∣∣∣∣µ(I ∩ Φ−1

(S1 × J

))µ(I) −

λ (J)

∣∣∣∣∣∣ ≤ ε ·λ(J)

λ (J).

In the next step we define the sequence of natural numbers (mn)n∈N:

∣∣∣∣m · pn+1

qn+1− 1n · qn

∣∣∣∣ ≤ 1εn+1 · qn+1

m ≤ qn+1 : m ∈ N, inf

∣∣∣∣m · qn · pn+1

qn+1− 1n

∣∣∣∣ ≤ qnεn+1 · qn+1

∣∣∣m · qn·pn+1qn+1

− 1n + k

∣∣∣ ≤ qnεn+1·qn+1

non-empty for every n ∈ N, i.e. mn exists.

Proof. In Lemma 6.4.6 we constructed the sequence αn = pnqn

in such a way that qn = 1εn· qn

and pn = 1εn· pn with pn, qn relatively prime. Therefore, the set

j · qn·pn+1

qn+1: j = 1, 2, ..., qn+1

contains εn+1·qn+1

gcd(qn,qn+1) different equally distributed points on S1. Hence, there are at least εn+1·qn+1qn

different such points and so for every x ∈ S1 there is a j ∈ 1, ..., qn+1, such that

infk∈Z

∣∣∣∣x− j · qn · pn+1

qn+1+ k

∣∣∣∣ ≤ qnεn+1 · qn+1

In particular, this is true for x = 1n .

an =(mn ·

qn+1− 1n · qn

By the above construction of mn it holds: |an| ≤ 1εn+1·qn+1

. In the proof of Lemma 6.4.6 we setthe condition qn+1 ≥ 1

εn+1· 12 · 1

εn· |||ψn|||1 · γ2

n. Thus, we get:

|an| ≤εn

12 · |||ψn|||1 · γ2n

Lemma 6.5.7. We consider the interval J :=[25n2 · εn, 1− 25n2 · εn

]as well as the diffeomor-

phism Φn := D−1ψn,γn

φn Dψn,γn Rmnαn+1D−1

ψn,γnφ−1

n with the conjugating map φn defined in

section 6.3.5. Then the elements of the partition ηn are(

1n·q2

)-distributed on J under Φn.

Proof. Analogus to Lemma 4.5.5.

In order to prove the weak mixing property we modify the criterion 4.6.2: Indeed, in our casewe will use the subsequent sequence of partial partitions and we will need that it converges tothe decomposition into points.

Lemma 6.5.8. With the aid of the partial partitions (ηn)n∈N constructed in section 6.3.3.1 wedefine the partial partitions

Γn = Hn−1 gn D−1ψn,γn

): In ∈ ηn

Then we get νn → ε.

Proof. Since the trapping map D−1ψn,γn

causes a r-translation by at most 4n2 · εn, we have

D−1ψn,γn

)⊆ S1 ×

[12n2 · εn, 1− 12n2 · εn

]due to the choice of j(1)

After the application of D−1ψn,γn

on In ∈ ηn the diameter is at most√

+ 4n2 · εn)≤

2 ·√

2 · 4n2εn. Unfortunately, on this set gn = g[nqσnn ] is not necessarily true, but it is strictlycontained in such a cube of sidelength 2 ·

√2 · 4n2εn that is a union of domains of ga,b,ε. Then

we obtain for the diameter of such a partition element:

ψn,γn

))≤ ‖DHn−1‖0 · [nq

σnn ] · 2 ·

√2 · 4n2 · εn

≤ qr(n−1)n−1 · n · q4·r(n−1)

n−1 · 2 ·√

n9 · q5·r(n−1)+1n−1

as n→∞. Thus, this sequence of partial partitions converges to the decomposition into points.

Lemma 6.5.9. Let n ≥ 5. For the number mn as above we consider

Φn = D−1ψn,γn

φn Dψn,γn Rmnαn+1D−1

ψn,γn φ−1

and J :=[25n2 · εn, 1− 25n2 · εn

Then for every cube S of side length q−σnn lying in S1 × J we get∣∣∣µ(I ∩ Φ−1n g−1

n (S))· λ (J)− µ

(I)· µ (S)

∣∣∣ ≤ 21n· µ(I)· µ (S) . (6.3)

Proof. According to Lemma 6.5.7 Φn(

1n·q2

)-distributes the partition element In ∈ ηn on

J , in particular Φn(In

)⊆ [c, c+ γ] × [0, 1] for some c ∈ S1 and some γ ≤ 1

n·q2n. Furthermore,

one computes that φn Dψn,γn Rmnαn+1 D−1

ψn,γn φ−1

)is contained in the interior of the

step-by-step domains of the map gn and µ(

Φn(I))≥ 1

a ·(1− 2

)· λ (J) as well as on its

boundary gn = g[nqσnn ] holds. Particularly it follows γ ≥ 1−2εa in case of gn = ga,b,ε.

Then the proof follows along the lines of Lemma 3.5.5.

Now we are able to prove the aimed weak mixing property:

Proposition 6.5.10. Let fn = Hn Rαn+1 H−1n and the sequence (mn)n∈N be constructed

as above. Suppose additionally that d0 (fmn , fmnn ) < 12n for every n ∈ N and that the limit

f = limn→∞ fn exists.Then f is weak mixing.

Proof. To apply Lemma 3.5.2 we consider the partial partitions νn := Hn−1 gn D−1ψn,γn

(ηn).As proven in Lemma 6.5.8 these partial partitions satisfy νn → ε. We have to establish equation3.2. For it let ε > 0 and a cube A ⊆ S1 × (0, 1) be given. There exists N ∈ N such thatA ⊆ S1×

[25n2 · εn, 1− 25n2 · εn

]for every n ≥ N . Because of Lemma 6.5.7 we obtain for every

In ∈ ηn: Φn(In

)⊇ [c, c+ γ] ×

[25n2 · εn, 1− 25n2 · εn

]for some γ ≤ 1

n·q2n. Furthermore, we

note fmnn = Hn Rmnαn+1H−1

n = Hn−1 gn Φn Dψn,γn g−1n H−1

n−1.Let Sn be a cube of sidelength q−σnn contained in S1 ×

[25n2 · εn, 1− 25n2 · εn

]= S1 × J . We

look at Cn := Hn−1 (Sn), Γn ∈ νn, and compute (since gn and Hn−1 are measure-preserving):∣∣µ (Γn ∩ f−mnn (Cn))− µ (Γn) · µ (Cn)

∣∣ =∣∣∣µ(In ∩ Φ−1

n g−1n (Sn)

)− µ

)· µ (Sn)

∣∣∣≤ 1λ (J)

·∣∣∣µ(In ∩ Φ−1

n g−1n (Sn)

)· λ (J)− µ

)· µ (Sn)

∣∣∣+1− λ (J)λ (J)

· µ(In

)· µ (Sn) .

Since λ (J) ≥ 12 and 1−λ(J)

λ(J) ≤ 2 · (1− λ (J)) ≤ 2n we continue by applying Lemma 6.5.9:∣∣µ (Γn ∩ f−mnn (Cn)

)− µ (Γn) · µ (Cn)

∣∣ ≤ 2 · 21n· µ(In

)· µ (Sn) +

2n· µ(In

)· µ (Sn)

=44n· µ(In

)· µ (Sn) .

Moreover, it holds diam(Cn) ≤ ‖DHn−1‖0 · diam (Sn) ≤ qr(n−1)n−1 ·

qσnn= q

r(n−1)n−1 ·

q4·r(n−1)n−1

, i.e.

diam(Cn) → 0 as n → ∞. Thus, we can approximate A by a countable disjoint union ofsets Cn = Hn−1 (Sn) with Sn ⊆ S1 ×

[25n2 · εn, 1− 25n2 · εn

]a cube of sidelength q−σnn in

given precision, when n is chosen big enough. Consequently for n sufficiently large there aresets A1 =

⋃i∈Σ1

nCin and A2 =

⋃i∈Σ2

nCin with countable sets Σ1

n and Σ2n of indices satisfying

A1 ⊆ A ⊆ A2 as well as |µ(A)− µ(Ai)| ≤ ε3 · µ(A) for i = 1, 2.

Additionally we choose n such that 44n < ε

3 holds. It follows∣∣µ (Γn ∩ f−mnn (A))− µ (Γn) · µ (A)

∣∣ ≤ ε · µ (Γn) · µ (A)

by the same calculations as in the proof of Proposition 3.5.6.

By Lemma 6.4.9 the requirement of the proximity between f and fn is fulfilled. Hence, f isweak mixing.

144 Construction of the f -invariant measurable Riemannian metric

Let ω0 denote the standard Riemannian metric on M = S1× [0, 1]. The following Lemma showsthat the conjugation map hn = gn D−1

ψn,γnφn Dψn,γn constructed in section 6.3 is an isometry

with respect to ω0 on the elements of the partial partition ζn.

Lemma 6.6.1. Let D−1ψn,γn

)∈ ζn. Then hn|D−1

ψn,γn(In) is an isometry with respect to ω0.

Proof. As noted in Remark 6.3.7 Dψn,γn acts as an isometry on any element D−1ψn,γn

(In)∈ ζn.

Next we observe that φn is an isometry on such an element In by the choices of ε1 and ε2 in theconstruction of the conjugation map φn as well as the positioning of the elements In. Here the“inner rotation map” is important.Moreover, we compute that φn

(In)lies in the “good area” of the map gn. But the prior appli-

cation of D−1ψn,γn

causes a translation of(

1 + 1q3n

+ ...+ 1q3+n−1n

)· u · 4εn with some u ≤ n2

the r-coordinate. At first we observe that D−1ψn,γn

φn(In)is still contained in the same defini-

tion section of gn by our choice of j(2+3+...+(3+k−1))1 . Thus, we compare the caused translation

with an εb·a = 16n2·εn

n·q2+3·(k+1)+ k·(k+1)

2n ·[nqσnn ]

-domain of the map gn = ga,b,ε,δ on the r-axis. In case of

2 + 3 · (k + 1) + k·(k+1)2 ≥ 3 + n− 1 there is a number l ∈ Z such that

l · 16n2 · εn

n · q2+3·(k+1)+k·(k+1)

2n · [nqσnn ]

=u · 4εnq3+n−1

because 1εn−1

= 4 · (n− 1)11 · q5·r(n−2)+1n−2 divides qσnn = q

4·r(n−1)n−1 . So the shifting is a multiple

of such a domain and then D−1ψn,γn

φn(In)is still contained in the “good area” of gn. In the

other case we write 1 + 1q3n

+ ... + 1q3+n−1n

q2+3·(k+1)+ k·(k+1)

+ R with l ∈ Z and some rest

term R < 2

q2+3·(k+1)+ k·(k+1)

. Since l

q2+3·(k+1)+ k·(k+1)

· u · 4εn is a multiple of εb·a we consider

q2+3·(k+1)+ k·(k+1)

· u · 4εn. We have

n · u · [nqσnn ] · 12n2 · εn

≤ n · n2

2· n · qσnn · 2 · n9 · q5·r(n−1)+1

=n13 · q4·r(n−1)n · q5·r(n−1)+1

n−1 = n13 · q9·r(n−1)+1n−1 < qn

by our assumptions on the numbers qn and σn in section 6.4.2. So this deviation is bounded by

q2+3·(k+1)+

k·(k+1)2 +1

· u · 4εn <2

q2+3·(k+1)+

k·(k+1)2

· 2n2 · εnn · u · [nqσnn ]

· u · 4εn

= εn ·16n2 · εn

[nqσnn ] · n · q2+3·(k+1)+k·(k+1)

Then D−1ψn,γn

φn(In)is still contained in the “good area” of gn.

Thus, hn acts as an isometry on the elements of the partition ζn.

Then the aimed f -invariant measurable Riemannian metric ω∞ is constructed as in section3.7. Hence, Proposition 6.2.2 is proven.

Chapter 7

Smooth diffeomorphisms withhomogeneous spectrum anddisjointness of convolutions

7.1 Introduction

Let M be a smooth compact connected manifold of dimension m ≥ 2 admitting a smooth non-trivial circle action S = Stt∈S1 preserving a smooth volume ν. In this setting we consider the

closure of conjugates A = h St h−1 : h ∈ Diff∞ (M,ν) , t ∈ S1C∞

and more precisely forα ∈ S1 the restricted spaces Aα = h Sα h−1 : h ∈ Diff∞ (M,ν)

. In [FK04], Problem7.11., the following question is posed:Given a circle action S and the corresponding space A, is there a diffeomorphism f ∈ A withany of the following properties:

1. a good approximation of type (h, h+ 1);

2. a maximal spectral type disjoint with its convolutions;

3. a homogeneous spectrum of multiplicity two for the Cartesian square f × f?

This question takes up problems in the category of measure-preserving transformations on aLebesgue space (X,µ). For instance, there is extensive research on the construction of transfor-mations possessing specific essential valuesMUT of the spectral multiplicities. Here the propertyof admitting an approximation of type (h, h+ 1) is often used to find an upper bound forMUT

and there exist two standard points of view: to consider the spectrum of T (and in particularMUT ) either on L2 (X,µ) or on the orthogonal complement L2

0 (X,µ) of the constant functions.In [KL95] it was proved that all possible subsets of N ∪ ∞ can be realized as MUT for someergodic transformation T in the first case (since 1 is always an eigenvalue because of the constantfunctions, “possible” means any subset of N∪∞ with 1 as an element). In the second case theCartesian powers of a generic transformation provide a good opportunity for the construction ofexamples with the infimum of essential spectral multiplicities larger than 1. Although it seemsvery unlikely that these Cartesian powers have finite maximal spectral multiplicity, this is thegeneric case: Independently Ageev and Ryzhikov proved a celebrated result, that for a genericautomorphism T the Cartesian square T × T has homogeneous spectrum of multiplicity 2 (see

146 Preliminaries

[Ag99] resp. [Ry99a]). Ageev was even able to show that for the n-th power Tn = T × ...× T ofa generic transformation T it holds M (Tn) = n, n · (n− 1) , ..., n! (cf. [Ag99], Theorem 2). Healso proved for every n ∈ N the existence of an ergodic transformation with homogeneous spec-trum of multiplicity n in the orthogonal complement of the constant functions ([Ag05], Theorem1) solving Rokhlin’s problem on homogeneous spectrum in ergodic theory.The second question is linked to a conjecture of Kolmogorov respectively Rokhlin and Fomin(after verifying that the property held for all dynamical systems known at that time) that everyergodic transformation possesses the so-called group property, i.e. the maximal spectral type σis symmetric and dominates its square σ ∗ σ. This conjecture was proven to be false. Indeed,in [St66] A.M. Stepin gave the first example of a dynamical system without the group property.Later he showed that for a generic transformation all convolutions σk0 , k ∈ N, of the maximalspectral type σ0 on L2

0 (X,µ) are mutually singular (see [St87]).In the smooth category there are only few results in this direction: In [St87], §4, Stepin con-structed diffeomorphisms, for which the convolutions of the maximal spectral type on L2

0 (M,µ)are mutually singular, on manifolds M as above using a smooth variant of the method of ap-proximation by periodic transformations.Extending this result of Stepin and answering the beforehand cited question affirmatively weprove the following Theorem:

Theorem F. Let M be a smooth compact connected manifold of dimension m ≥ 2 admitting asmooth non-trivial circle action S = Stt∈S1 preserving a smooth volume ν and α a Liouvilleannumber. Then the set of smooth diffeomorphisms, that have a maximal spectral type disjointwith its convolutions, a homogeneous spectrum of multiplicity 2 for f × f and admit a goodapproximation of type (h, h+ 1), is residual (i.e. it contains a dense Gδ-set) in Aα in theDiff∞ (M)-topology.

7.2 Preliminaries

7.2.1 First steps of the proof

With the aid of section 2.4 we can concentrate on the case of(S1 × [0, 1]m−1

,R, µ): Let f be

a diffeomorphism on S1 × [0, 1]m−1 with the aimed properties and f defined as in section 2.4.We observed that f and f are metrically isomorphic. Thus, f admits a good approximation oftype (h, h+ 1) because the speed and the type of a periodic approximation are invariant underisomorphisms. Moreover, f and f are unitarily equivalent (see Remark 7.4.1). Since the spectraltypes and multiplicities are spectral invariants, we conclude that f has the aimed properties.Hence, it is sufficient to prove Theorem F in case of

(S1 × [0, 1]m−1

,R, µ). In this setting we

will show the subsequent statement:

qnconverging monotonically to α and a sequence of measure-preserving smooth

diffeomorphisms φn, that coincide with the identity in a neighbourhood of the boundary and satisfyφnR 1

qn= R 1

qnφn such that the diffeomorphisms fn = HnRαn+1H−1

n , where Hn = Hn−1φn,converge in the Diff∞-topology to a limit f = limn→∞ fn, which satisfies f ∈ Aα and admits agood linked approximation of type (h, h+ 1) as well as a good cyclic approximation.Furthermore, for every ε > 0 the parameters in the construction can be chosen in such a waythat d∞ (f,Rα) < ε.

In section 7.9 we will deduce Theorem F from this Proposition.

Periodic approximation in Ergodic Theory 147

7.2.2 Sketch of the proof

By the previous section it is enough to construct a diffeomorphism f with the aimed propertiesin case of M = S1 × [0, 1]m−1. Using results on periodic approximation as well as spectraltheory of dynamical systems stated in the successive two sections we are able to reduce thistask to the construction of a diffeomorphism admitting a good linked approximation of type(h, h+ 1) and a good cyclic approximation (see section 7.9). For this purpose, we define theconjugation maps very explicitly in section 7.5. Here the map φn is constructed in such away that the image of the likewise explicitly defined bases c(n)

0,1 and c(n)0,2 under H−1

n consistsof several stripes of nearly full volume in [0, 1]m−1. By the posterior choice of the parameters~an these stripes are positioned in order to guarantee that a great portion of Rmn H−1

(c(n)0,1

)(resp. Rmn+1 H−1

(c(n)0,2

)) is mapped back into H−1

(c(n)0,1

)(resp. H−1

(c(n)0,2

)) for a prescribed

height mn (resp. mn + 1) of the particular tower. Afterwards, we prove that the sequencefn = Hn Rαn+1 H−1

n composed of these conjugation maps converges in the C∞-topology toa measure-preserving smooth diffeomorphism f ∈ Aα under some conditions on the sequence(αn)n∈N of rational numbers (cf. Lemma 7.6.4). In the adjacent two sections we show thatthis constructed limit f admits the required types of approximation with the respective speed ofapproximation.

7.3 Periodic approximation in Ergodic Theory

In this section we give a short introduction to the concept of periodic approximation in ErgodicTheory. A more comprehensive presentation can be found in [Ka03].Let (X,µ) be a Lebesgue space. A tower t of height h(t) = h is an ordered sequence of disjointmeasurable sets t = c1, ..., ch of X having equal measure, which is denoted by m (t). The setsci are called the levels of the tower, especially c1 is the base. Associated with a tower there is acyclic permutation σ sending c1 to c2, c2 to c3,... and ch to c1. Using the notion of a tower wecan give the next definition:

Definition 7.3.1. A periodic process is a collection of disjoint towers covering the space Xtogether with an equivalence relation among these towers identifying their bases.

There are two partial partitions associated with a periodic process: The partition ξ into allsets of all towers and the partition η consisting of the union of bases of towers in each equivalenceclass and their images under the iterates of σ, where when we go beyond the height of a certaintower in the class we drop this tower and continue until the highest tower in the equivalenceclass has been exhausted. Obviously we have η ≤ ξ.A sequence (ξn, ηn, σn) of periodic processes is called exhaustive if ηn → ε. Such an exhaustivesequence of periodic processes is consistent if for every measurable subset A ⊆ X the sequenceσn (A) converges to a set B, i.e. µ (σn (A)4B)→ 0 as n→∞. Moreover, we will call a sequenceof towers t(n) from the periodic process (ξn, ηn, σn) substantial if there exists r > 0 such thath(t(n))·m(t(n))> r for every n ∈ N.

Definition 7.3.2. Let T : (X,µ)→ (X,µ) be a measure-preserving transformation. An exhaus-tive sequence of periodic processes (ξn, ηn, σn) forms a periodic approximation of T if

d (ξn, T, σn) =∑c∈ξn

µ (T (c)4σn (c))→ 0 as n→∞.

148 Periodic approximation in Ergodic Theory

Given a sequence g (n) of positive numbers we will say that the transformation T admits aperiodic approximation with speed g (n) if for a certain subsequence (nk)k∈N there exists anexhaustive sequence of periodic processes (ξk, ηk, σk) such that d (ξk, T, σk) < g (nk).

In order to define the type of the periodic approximation we need the notion of equivalencefor sequences of periodic processes:

Definition 7.3.3. Two sequences of periodic processes Pn = (ξn, ηn, σn) and P ′n = (ξ′n, η′n, σ

are called equivalent if for every n ∈ N there is a bijective correspondence θn between subsetsSn and S′n of the sets of towers of Pn resp. P ′n such that

• For t ∈ Sn: h (θn (t)) = h (t).

•∑t∈Sn h (t)m (t)→ 1 as n→∞.

•∑t∈Sn h (t) · |m (t)−m (θn (t))| → 0 as n→∞.

• If two towers from Sn are equivalent in Pn then their images under θn are equivalent in P ′n.

There are various types of approximation. We introduce the most important ones:

Definition 7.3.4. 1. A cyclic process is a periodic process which consists of a single tower ofheight h. An approximation by an exhaustive sequence of cyclic processes is called a cyclicapproximation. In particular, we will refer to a cyclic approximation with speed o

a good cyclic approximation.

2. An approximation generated by periodic processes equivalent to periodic processes consist-ing of two substantial towers whose heights differ by one is said to be of type (h, h+ 1).Equivalently the heights of the two towers t1 and t2 with base B1 resp. B2 are equal toh and h + 1 and for some r > 0 we have µ (B1) > r

h as well as µ (B2) > rh+1 . We will

call the approximation of type (h, h+ 1) with speed o(

)good and with speed o

h·(h+1)

)excellent.

3. An approximation of type (h, h+ 1) will be called linked approximation of type (h, h+ 1) ifthe two towers involved in the approximation are equivalent. This insures that the partitionη generated by the union of the bases of the two towers and the iterates of this set is fine.

Remark 7.3.5. As noted in [Ry06] a good linked approximation of type (h, h+ 1) implies theconvergence

Uk·(h+1)T −→w r · UkT + (1− r) · Id

in the weak operator topology for every k ∈ N and some r ∈ (0, 1), where UT is the Koopman-operator of T (see section 7.4.1).

From the different types of approximations various ergodic properties can be derived. Forexample in [KS67], Corollary 2.1., the subsequent Lemma is proven.

Lemma 7.3.6. Let T : (X,µ)→ (X,µ) be a measure-preserving transformation. If T admits agood cyclic approximation, then T is ergodic.

In [KS70] Katok and Stepin proved the genericity of automorphisms having a continuousspectrum in the set of measure-preserving homeomorphisms (recall that a transformation has acontinuous spectrum, i.e. the corresponding operator UT in the space L2 (M,µ) has no eigen-functions other than constants, if and only if it is weak mixing). For this purpose, they deducedthe following result ([KS70], Theorem 5.1.):

Spectral theory of dynamical systems 149

Lemma 7.3.7. Let T : (X,µ)→ (X,µ) be a measure preserving transformation. If T is ergodicand admits a good approximation of type (h, h+ 1), then T has continuous spectrum.

Moreover, the theory of periodic approximation can be used to prove genericity of constructedproperties. The applied statement can be summarized as follows (cf. [Ka03], Theorem 2.1.):

Lemma 7.3.8. Given a type τ and a speed g (n), the set of all measure-preserving transforma-tions of a Lebesgue space which admit a periodic approximation of type τ with speed g (n) is aresidual set in the weak topology.

7.4 Spectral theory of dynamical systems

Besides the concept of periodic approximation we will need further mathematical tools. We referto [Na98] and [Go99] for more details.

7.4.1 Spectral typesLet (X,µ) be a Lebesgue space and T : (X,µ) → (X,µ) be an automorphism. Then we definethe induced Koopman-operator UT : L2 (X,µ)→ L2 (X,µ) by UT f = f T . Since

〈UT f, UT g〉 =∫X

f T · g T dµ =∫X

f · g dµ = 〈f, g〉 for every f, g ∈ L2 (X;µ)

and U−1T = UT−1 this is an unitary operator on the Hilbert space L2 (X,µ).

Remark 7.4.1. If two measure-preserving dynamical systems (X1, µ1, T1) and (X2, µ2, T2) aremetrically isomorphic, their isomorphism h : X1 → X2 induces an isomorphism of Hilbert spacesVh : L2 (X2, µ2) → L2 (X1, µ1) by (Vhf) = f h. Then we have UT1 = Vh UT2 V −1

h and thisrelation is called unitary equivalence of operators. Hence, any invariant of unitary equivalencedefines an invariant of isomorphisms. Such invariants are said to be spectral invariants or spectralproperties.Moreover, we note that 1 is always an eigenvalue of UT because of the constant functions. Sowhen we discuss the spectral properties of UT we mean its spectral properties restricted to theorthogonal complement of the constants. Hence, we consider the properties of UT in the spaceL2

0 (X,µ) of all L2-functions with zero integral.

One of the important spectral invariants are the so-called spectral measures: Let f ∈ L20 (X,µ)

and Z (f) := span UnT f : n ∈ ZL2

0(X,µ). Using Bochner’s theorem one can prove the existence

of a finite Borel measure σf defined on the unit circle S1 in the complex plane satisfying

〈UnT f, f〉 =∫

S1zn dσf (z) for every n ∈ Z.

Then σf is called the spectral measure of f with respect to UT .Moreover, by the Hahn-Hellinger Theorem, there is a sequence of functions fn ∈ L2

0 (X,µ), n ∈ N,for which

L20 (X,µ) = ⊕n∈NZ (fn) and σf1 σf2 ...

These measures are unique in the sense that for any other family of functions gn ∈ L20 (X,µ),

n ∈ N, for which L20 (X,µ) = ⊕n∈NZ (gn) and σg1 σg2 ... we have σfn ∼ σgn for every

n ∈ N.

Definition 7.4.2. The spectral type of σf1 is called the maximal spectral type σ of UT .

150 Spectral theory of dynamical systems

7.4.2 Spectral multiplicitiesBesides the maximal spectral type an important characterization of UT is the multiplicityfunction MUT : S1 → N ∪ ∞, which is σf1 - almost everywhere defined by

MUT (z) =∞∑i=1

χAi (z) , where Ai =z ∈ S1 :

dσfidσf1

(z) > 0.

Here dσfidσf1

is the Radon-Nikodym derivative of σfi with respect to σf1 .Using this multiplicity function we establish the setMUT of essential spectral multiplicities,which is the essential range of MUT with respect to σf1 . Then we define the maximal spectralmultiplicity mUT as the essential supremum (with respect to σf1) ofMUT .

Definition 7.4.3. UT is said to have homogeneous spectrum of multiplicity m ifMUT = m.In particular, UT has simple spectrum ifMUT = 1. Otherwise UT has a non-simple spectrum.

Another interpretation of the multiplicity function can be given by the the subsequent for-mulation on the canonical form of an unitary operator (see [CFS82], Appendix 2):For any unitary operator U on a separable complex Hilbert space H and any m ∈ N ∪ ∞ wecan find a Borel set Am of the circle S1 and a sequence of vectors hm,k ∈ H, k = 1, ..,m (resp.k = 1, 2, ... in case of m =∞), such that

•⋃m∈N Am = S1 and Am1 ∩Am2 = ∅ for m1 6= m2.

• ⊕m∈N ⊕mk=1 Z (hm,k) = H, Z (hm,k)⊥Z (hm1,k1) for (m, k) 6= (m1, k1).

• σ(m) := σhm,k = σhm,l for 1 ≤ k, l ≤ m and σ(m)(S1 \Am

The multiplicity function is defined on S1 by the relation m (λ) = m for λ ∈ Am. Note that themeasures σ(m) are not the spectral measures, but the spectral type of the measure σ(m) is calleda spectral type of multiplicity m.

In connection with the previous chapter 7.3 we state the following result ([KS67], Theorem3.1.):

Lemma 7.4.4. Let T be an automorphism of a Lebesgue space. If T admits a cyclic approxi-mation of speed θ

h , where θ <12 , then the spectrum of UT is simple.

For automorphisms with simple spectrum we have the subsequent theorem of Ryzhikov([Ry99b], Theorem 2.1.):

Lemma 7.4.5. Let (X,µ) be a Lebesgue space with µ (X) = 1 and T : (X,µ) → (X,µ) be anautomorphism with simple spectrum. Suppose that the weak convergence

UknT −→w (a · UT + (1− a) · Id)

holds for some a ∈ (0, 1) and some strictly increasing sequence (kn)n∈N of natural numbers.Then the Cartesian square T × T has a homogeneous spectrum of multiplicity 2.

7.4.3 Disjointness of convolutionsIn this section we study the convolutions of the maximal spectral type σ. Therefore, we statethe definition of a convolution of measures:

Construction of the conjugation maps 151

Definition 7.4.6. Let G be a topological group and µ, ν finite Borel measures on G. Then theirconvolution µ ∗ ν is defined by

(µ ∗ ν) (A) =∫ ∫

1A (x · y) dµ(x) dν(y)

for each measurable set A of G.

If all the convolutions σk = σ ∗ ... ∗ σ for k ∈ N are pairwise mutually singular, one speaksabout disjointness of convolutions. To guarantee this pairwise singularity of convolutions of themaximal spectral type of a measure-preserving transformation the following property is useful:

Definition 7.4.7. An automorphism T of a Lebesgue space (X,µ) is said to be κ-weak mixing,κ ∈ [0, 1], if there exists a strictly increasing sequence (kn)n∈N of natural numbers such that theweak convergence

UknT −→w (κ · Pc + (1− κ) · Id)

holds, where Pc is the projection on the subspace of constants.

Remark 7.4.8. By [St87], Proposition 3.1., we can characterise this property in geometriclanguage: A transformation T is κ-weak mixing if and only if there is an increasing sequence(kn)n∈N of natural numbers such that for all measurable sets A and B

limn→∞

µ(A ∩ T knB

)= κ · µ (A) · µ (B) + (1− κ) · µ (A ∩B) .

We recognize that 0-weak mixing corresponds to rigidity and 1-weak mixing to the usual notionof weak mixing.

As announced this property has connections with certain properties of the maximal spectraltype (see [St87], Theorem 1):

Lemma 7.4.9. If the transformation T is κ-weak mixing for some 0 < κ < 1 and σ is themaximal spectral type for UT |L2

0(X,µ), then σ and all its convolutions σk = σ ∗ ... ∗ σ are pairwisemutually singular.

7.5 Construction of the conjugation maps

We fix an arbitrary Liouvillean number α. In order to construct the conjugation map φn we willneed two types of maps introduced in the subsequent subsections.

7.5.1 The map φ(i)λ,ε

We recall Lemma 3.3.6:

(12 , ...,

)∈ Rm on [ε, 1− ε]m and coincides with the identity outside of

[ε2 , 1−

]m.Furthermore, for λ ∈ N we define the maps Cλ (x1, x2, ..., xm) = (λ · x1, x2, ..., xm). Hereby,

we define a smooth measure-preserving diffeomorphism

φ(i)λ,ε :

]× [0, 1]m−1 →

]× [0, 1]m−1

, φ(i)λ,ε := C−1

λ ϕε,1,i Cλ.

152 Construction of the conjugation maps

Since φ(i)λ,ε coincides with the identity on a neigbourhood of the boundary, we can proceed it using

the description φ(i)λ,ε

(x1 + 1

λ , x2, ..., xm)

1λ , 0, ..., 0

(i)λ,ε (x1, ..., xm) to a diffeomorphism on

S1 × [0, 1]m−1.

7.5.2 The map ψk,q,~a,ε

The map ψk,q,~a,ε is constructed in the spirit of [Be], Lemma 4.3.:Let k, q ∈ Z, ε > 0 and ρ : R→ R be a smooth increasing function that equals 0 for x ≤ −1 and1 for x ≥ 0. Moreover, for every 0 ≤ i ≤ k − 1 we have a(i) ∈ Z with 0 ≤ a(i) ≤ q − 1. This setof parameters is denoted by ~a. With it we define the map ψk,q,~a,ε : [0, 1]→ [0, 1] by

ψk,q,~a,ε (x) =a(0)q

+a(1)− a(0)

q· ρ(x

ε− 1k · ε

)+ ...+

a(k − 1)− a(k − 2)q

· ρ(x

ε− k − 1

k · ε

Note that for every 0 ≤ i ≤ k − 1 we have ψk,q,~a,ε|[ ik , i+1k −ε] = a(i)

q and we can estimate∥∥∥Dlψk,q,~a,ε

∥∥∥0≤ 1

εl·∥∥Dlρ

∥∥0. In our constructions we will have a(0) = 0 as well as a(k − 1) = 0.

Besides this map ψk,q,~a,ε we use a smooth map σε : R → [0, 1] satisfying σε (x) = 0 for x ≤ ε2 ,

σε (x) = 1 for ε ≤ x ≤ 1−ε and σε (x) = 0 for x ≥ 1− ε2 . Then we define the measure-preserving

diffeomorphism ψk,q,~a,ε : S1 × [0, 1]m−1 → S1 × [0, 1]m−1 by

ψk,q,~a,ε (θ, r1, ..., rm−1) =(θ + ψk,q,~a,ε (r1) · σε (r2) · ...σε (rm−1) , r1, ..., rm−1

We emphasize that the maps σε are introduced to guarantee that ψk,q,~a,ε coincides with theidentity in a neigbourhood of the boundary. Moreover, we observe ψk,q,~a,ε R 1

q= R 1

q ψk,q,~a,ε

and |||ψk,q,~a,ε|||l ≤ C (ε, l).

Using the maps from the precedent subsections we construct the conjugation map φn:

φn = φ(m)

4qn−1

φ(m−1)

14qn−1

... φ(2)

qm−1n , 1

4qn−1

ψqn,qn,~an, 14qn−1

4qn−1

14qn−1

where the parameters ~an = (an (0) , ..., an (qn − 1)) with 0 ≤ an(i) ≤ qn − 1 will be determinedlater.

Remark 7.5.2. By “good area” of φn we denote the domain, where all the occuring maps φ(j)λj ,ε

act as the particular rotation and the map ψk,q,~a,ε acts as one of the translations by a(i)q .

In order to guarantee that a strip of our partition element is contained in the “good area”completely we choose εn := 1

2qn−1slightly larger than ε = 1

4qn−1. Hereby, we observe that for an

interval[lqn, l+1qn

]on the θ-axis the length (1− 2 · εn)m · 1

qn≥ (1−m · 2 · εn) · 1

qnis part of the

“good area”.

7.6 Convergence of (fn)n∈N in Diff∞ (M)

7.6.1 Properties of the conjugation maps φn

In order to estimate the norm of the conjugating maps φ(m)

4qn−1

φ(m−1)

14qn−1

... φ(2)

qm−1n , 1

4qn−1

well as(φ

4qn−1

14qn−1

we will need the technical results 3.6.1 and 3.6.2. With these

we can prove the following norm estimates:

Lemma 7.6.1. For every k ∈ N we have

|||φ(m)

4qn−1

... φ(2)

qm−1n , 1

4qn−1

|||k ≤ C1 (m, k, qn−1) · q(m−1)2·kn ,

|||(φ

4qn−1

14qn−1

|||k ≤ C2 (m, k, qn−1) · q4·kn ,

where C1 (m, k, qn−1) as well as C2 (m, k, qn−1) are constants depending on m, k and qn−1, butare independent of qn.

Proof. First of all we consider the map φ(i)λ,ε = C−1

λ ϕε Cλ introduced in subsection 7.5.1:

φ(i)λ,ε (x1, ..., xm) =

[ϕε]1 (λx1, x2, ..., xm) , [ϕε]2 (λx1, x2, ..., xm) , ..., [ϕε]m (λx1, x2, ..., xm)).

Let k ∈ N. We compute for a multiindex ~a with 0 ≤ |~a| ≤ k:∥∥∥D~a [φ(i)

∥∥∥0≤ λk−1 · |||ϕε|||k and

for r ∈ 2, ...,m:∥∥∥D~a [φ(i)

∥∥∥0≤ λk · |||ϕε|||k. Since

(i)λ,ε

is of the same form, we have

|||φ(i)λ,ε|||k ≤ C · λk.

In the next step we consider φ := φ(m)λm,ε

... φ(2)λ2,ε

. Let λmax := max λ2, ..., λm. By the same

induction proof as in Lemma 3.6.4 we show |||φ|||k ≤ C (m, k, ε) ·λ(m−1)·kmax for every k ∈ N, where

C (m, k, ε) is a constant depending on m, k and ε.In the setting of our explicit construction of the map φ(m)

4qn−1

...φ(2)

qm−1n , 1

4qn−1

we have ε = 14qn−1

and λmax = qm−1n . Thus:

|||φ(m)

4qn−1

... φ(2)

qm−1n , 1

4qn−1

|||k ≤ C (m, k, qn−1) ·(qm−1n

)(m−1)·k= C (m, k, qn−1) · q(m−1)2·k

where C (m, k, qn−1) is a constant independent of qn.

In the same spirit we obtain the estimate on(φ

4qn−1

14qn−1

With the aid of the formula of Faà di Bruno presented in Remark 3.6.3 we can prove anestimate on the norms of the map φn:

|||φn|||k ≤ C · q((m−1)2+4)·kn ,

where C is a constant depending on m, k, n and qn−1, but is independent of qn.

Proof. First of all we consider the map φ(1) := ψqn,qn,~an, 14qn−1

φ(1), at which we use the

notation φ(1) :=(φ

4qn−1

14qn−1

. Let k ∈ N. According to subsection 7.5.2 we have

|||ψqn,qn,~an, 14qn−1

|||k ≤ C (m, k, qn−1), at which C (m, k, qn−1) is a constant independent of qn.

Let r ∈ 1, ...,m and ~ν be any multiindex with |~ν| = k. We compute with the aid of the formulaof Faà di Bruno in the same manner as in Lemma 3.6.5:∥∥∥D~ν [φ(1)

∥∥∥0

=∥∥∥∥D~ν [ψqn,qn,~an, 1

4qn−1 φ(1)

∥∥∥∥0

≤∑

∥∥∥∥D~λ [ψqn,qn,~an, 14qn−1

∥∥∥∥0

·k∑s=1

∑ps(~ν,~λ)

~ν! ·s∏j=1

|||φ(1)||||~kj||~lj|

~kj ! ·(~lj !)|~kj|

As seen in Lemma 7.6.1: |||φ(1)||||~kj||~lj| ≤ C · q4·|~lj|·|~kj|

n . By the same calculations as in Lemma

3.6.5 we get∥∥∥D~ν [φ(1)

∥∥∥0≤ C · q4·k

Analogously we compute∥∥∥∥D~ν [(φ(1)

)−1 ψ−1qn,qn,~an,

14qn−1

∥∥∥∥0

≤ C · |||φ(1)|||k ≤ C · q4kn . Alto-

gether, we obtain |||φ(1)|||k ≤ C · q4kn with a constant C independent of qn.

In the next step we denote φ(2) := φ(m)

4qn−1

... φ(2)

qm−1n , 1

4qn−1

and consider φn = φ(2) φ(1). By

the same calculations as above we obtain for any multiindex ~ν with |~ν| = k:∥∥∥D~ν [φ(2) φ(1)]r

∥∥∥0≤ C · |||φ(2)|||k · q4·k

n ≤ C · q(m−1)2·kn · q4·k

where we used Lemma 7.6.1 in the last step and the constants C, C are independent of qn.

Analogously we show the same estimate on(φ(1)

(φ(2)

)−1.

Finally, we conclude:

|||φn|||k ≤ C (m, k, qn−1) · q(m−1)2·kn · q4·k

n = C (m, k, qn−1) · q((m−1)2+4)·kn ,

where C (m, k, qn−1) is a constant independent of qn.

Again using the formula of Faà di Bruno we are able to prove an estimate on the norms ofthe map Hn as in Lemma 3.6.6:

|||Hn|||k ≤ C · q((m−1)2+4)·kn ,

where C is a constant depending solely on m, k, qn−1 and Hn−1. Since Hn−1 is independent ofqn in particular, the same is true for C.

In particular, we see that this norm can be estimated by a power of qn.

7.6.2 Proof of convergenceIn Lemma 3.6.8 we proved convergence of the sequence (fn)n∈N to f ∈ Aα in the Diff∞ (M)-topology under some assumptions on the sequence (αn)n∈N. We show that we can satisfy theconditions from this Lemma in our constructions:

Lemma 7.6.4. Let (kn)n∈N be a strictly incr. sequence of natural numbers with∑∞n=1

qnof rational numbers with

2 · qn−2 · qmn−1 divides qn (A)

(αn)n∈N converges to α monotonically (B)

such that our conjugation maps Hn constructed in section 7.5 fulfil the following conditions:

|α− αn| <1

2 · kn · Ckn · |||Hn|||kn+1kn+1

‖DHn−1‖0 ·16 · qn−2 · qmn−1 ·

In particular, we have qn > n · 16 · qn−2 · qmn−1.

Proof. The sequence of rational numbers αn = pnqn

will be created out of αn = pnqn, at which

pn ≤ pn and qn ≤ qn are relatively prime.

In Lemma 7.6.3 we saw |||Hn|||kn+1 ≤ Cn · q((m−1)2+4)·(kn+1)n , where the constant Cn was in-

dependent of qn. Thus, we can require qn ≥ Cn for every n ∈ N. Hereby, we get the estimate

|||Hn|||kn+1 ≤ q((m−1)2+5)·(kn+1)n . Furthermore, we can demand

qn > 8 · n · Cn−1 · q(m−1)2+5n−1 ·

√m ≥ 8 · n · ‖DHn−1‖0 ·

√m. (7.1)

Since α is a Liouvillean number, we find a sequence of rational numbers αn = pnqn, pn, qn relatively

prime, under the above restrictions satisfying:

|α− αn| =∣∣∣∣α− pn

∣∣∣∣ < |α− αn−1|

2 · kn · Ckn ·(2qn−2qmn−1

)((m−1)2+5)·(kn+1)2

· q((m−1)2+5)·(kn+1)2

Put qn := 2 · qn−2 · qmn−1 · qn and pn := 2 · qn−2 · qmn−1 · pn. Then we obtain:

|α− αn| <|α− αn−1|

2 · kn · Ckn · q((m−1)2+5)·(kn+1)2

Thus, we have |α− αn| → 0 monotonically as n→∞.

Because of |||Hn|||kn+1kn+1 ≤ q

((m−1)2+5)·(kn+1)2

n this yields: |α− αn| < 1

2·kn·Ckn ·|||Hn|||kn+1kn+1

. Thus,

the first property of this Lemma is fulfilled.Equation 7.1 implies the second property, because

qn = 2 · qn−2 · qmn−1 · qn > 16 · qn−2 · qmn−1 · n · ‖DHn−1‖0 ·√m.

156 Proof of (h, h+ 1)-property

Remark 7.6.5. Lemma 7.6.4 shows that the conditions of Lemma 3.6.8 are satisfied. There-fore, our sequence of constructed diffeomorphisms fn converges in the Diff∞(M)-topology to adiffeomorphism f ∈ Aα.

The numbers αn+1 = pn+1qn+1

can be written in the following form:

αn+1 = αn ±γnqn+1

= αn ±γnqn+1

where γn ∈ N and γnqn+1

= |αn+1 − αn| ≤ 2 · |α− αn|. In particular, we have

γnqn+1

≤ 2 · |α− αn| ≤ 2 · 1

2 · kn · Ckn · q((m−1)2+5)·(kn+1)2

≤ 1kn · Ckn · qm+2

. (7.2)

Remark 7.6.6. We point out that the sequence (αn)n∈N is independent of the choices of theparameters an (i). So we can determine these parameters afterwards, which will depend on either“case +” (i.e. αn+1 = αn + γn

qn+1) or “case -” (i.e. αn+1 = αn − γn

qn+1).

In the proof of the (h, h+ 1)-property the number mn =⌊qn+1γn·q2

⌋(see equation 7.3) will play

a decisive role. This number mn is known, when αn+1 = αn ± γnqn+1

is determined guaranteeingthe convergence of the sequence (fn)n∈N in Diff∞ (M) with the help of Lemma 7.6.4. Then wecan compute mn · αn. Let r

qn:= mn · αn mod 1. Hereby, in “case +” we define an (l · qn + i) to

be:0 if 0 ≤ l ≤ 2qn−2q

m−1n−1 − 1 and 0 ≤ i ≤ qn − 1

i · r mod qn if 2qn−2qm−1n−1 ≤ l ≤ qn−2q

mn−1 − 1 and 0 ≤ i ≤ qn − 1

i · (r + pn) mod qn if qn−2qmn−1 ≤ l ≤ 2qn−2q

mn−1 − 2qn−2q

m−1n−1 − 1 and 0 ≤ i ≤ qn − 1

0 if 2qn−2qmn−1 − 2qn−2q

m−1n−1 ≤ l ≤ 2qn−2q

mn−1 − 1 and 0 ≤ i ≤ qn − 1

In “case -” the parameter an (l · qn + i) is chosen as follows:0 if 0 ≤ l ≤ 2qn−2q

m−1n−1 − 1 and 0 ≤ i ≤ qn − 1

−i · r mod qn if 2qn−2qm−1n−1 ≤ l ≤ qn−2q

mn−1 − 1 and 0 ≤ i ≤ qn − 1

−i · (r + pn) mod qn if qn−2qmn−1 ≤ l ≤ 2qn−2q

mn−1 − 2qn−2q

m−1n−1 − 1 and 0 ≤ i ≤ qn − 1

0 if 2qn−2qmn−1 − 2qn−2q

m−1n−1 ≤ l ≤ 2qn−2q

mn−1 − 1 and 0 ≤ i ≤ qn − 1

In both cases we define an (l · qn + i) = 2 · qn−2 · qmn−1 · an (l · qn + i).

7.7 Proof of (h, h+ 1)-property

7.7.1 Towers for approximation of type (h, h+ 1)

Let c(n)0,1 be the set

- in case of dimension m = 2:⋃[k

2qn−2q2n−1 · qn

qn−1 · q2n

+j + 1q2n

− 1qn−1 · q2

qn−1+s2 · q2

qn+1+s

(2)2 · q2

qn · q2n+1

qn−1+s2 · q2

(2)2 + 1

)· q2n

− q2n

qn · q2n+1

Proof of (h, h+ 1)-property 157

where the union is taken over all j, k, l, s2, s(2)2 ∈ Z satisfying

⌈qn+1qn

⌉≤ s(2)

2 ≤ qn+1−⌈qn+1qn

⌉− 1,

0 ≤ j ≤ qn − 1 = qn2qn−2q2

n−1− 1, 0 ≤ k ≤ 2qn−2q

2n−1 − 1, 2qn−2qn−1 ≤ l ≤ qn−2q

2n−1 − 2 and

0 ≤ s2 ≤ γnqn − 1.

qn · qn+1+

qn · q2n+1

q2n · q2

qn · qn+1+

s(2)1 + 1

qn · q2n+1

− 1q2n · q2

1qn−1

+s2 · q3

qn · qn+1,

1qn−1

+(s2 + 1) · q3

qn+1− q3

qn · qn+1

1qn−1

qn−1 · q2n

qn−1+

+j + 1q2n

− 1qn−1 · q2

where the union is taken over all j, k, l, s(1)1 , s

(2)1 , s2 ∈ Z satisfying 0 ≤ k ≤ 2qn−2q

3n−1 − 1,⌈

qn+1qn−1

⌉≤ s

(1)1 ≤ qn+1 −

⌈qn+1qn−1

⌉− 1,

⌈qn+1qn

⌉≤ s

(2)1 ≤ qn+1 −

⌈qn+1qn

⌉− 1, 0 ≤ s2 ≤ γnqn − 1,

0 ≤ j ≤ qn − 1 = qn2qn−2q3

n−1− 1 and 2qn−2q

2n−1 ≤ l ≤ qn−2q

3n−1 − 2.

- in case of dimension m ≥ 4:⋃[k

qn · qn+1+

1q2n · qn+1

s1 + 1qn · qn+1

− 1q2n · qn+1

1qn−1

+s2 · qmnqn+1

qn · qn+1,

1qn−1

+(s2 + 1) · qmn

qn+1− qmnqn · qn+1

1qn−1

2qn−2qmn−1 · qn+

qn−1 · q2n

qn−1+

2qn−2qmn−1 · qn+j + 1q2n

− 1qn−1 · q2

1qn−1

qn · qn+1+

qn · q2n+1

q2n · q2

qn−1+

qn · qn+1+

s(2)4 + 1

qn · q2n+1

− 1q2n · q2

m∏i=5

qn−1+

siqn · qn+1

q2n · qn+1

qn−1+

si + 1qn · qn+1

− 1q2n · qn+1

where the union is taken over all j, k, l, s2, si, s(2)4 ∈ Z satisfying

⌈qn+1qn−1

⌉≤ si ≤ qn+1−

⌈qn+1qn−1

⌉−1

for i = 1, 4, 5, ...,m, 0 ≤ j ≤ qn − 1 = qn2qn−2qmn−1

− 1, 0 ≤ k ≤ 2qn−2qmn−1 − 1, 0 ≤ s2 ≤ γnqn − 1,

2qn−2qm−1n−1 ≤ l ≤ qn−2q

mn−1 − 2 and

⌈qn+1qn

⌉≤ s(2)

4 ≤ qn+1 −⌈qn+1qn

⌉− 1.

With these the base c(n)0,1 of the first tower is defined to be the set c(n)

0,1 := Hn−1

(c(n)0,1

Similarly c(n)0,2 is the set

qn−1q2n

+j + 1q2n

− 1qn−1q2

qn−1+s2 · q2

qn+1+s

(2)2 · q2

qn · q2n+1

qn−1+s2 · q2

(2)2 + 1

)· q2n

− q2n

qn · q2n+1

where the union is taken over all j, k, l, s2, s(2)2 ∈ Z satisfying

⌈qn+1qn

⌉≤ s(2)

2 ≤ qn+1−⌈qn+1qn

⌉− 1,

0 ≤ j ≤ qn − 1 = qn2qn−2q2

n−1− 1, 0 ≤ k ≤ 2qn−2q

2n−1 − 1, 1 ≤ l ≤ qn−2q

2n−1 − 2qn−2qn−1 − 1 and

0 ≤ s2 ≤ γnqn − 1.

- in case of dimension m = 3:

qn · qn+1+

qn · q2n+1

q2n · q2

qn · qn+1+

s(2)1 + 1

qn · q2n+1

− 1q2n · q2

1qn−1

+s2 · q3

qn · qn+1,

1qn−1

+(s2 + 1) · q3

qn+1− q3

qn · qn+1

1qn−1

2qn−2q3n−1qn

qn−1q2n

qn−1+

2qn−2q3n−1qn

+j + 1q2n

− 1qn−1q2

where the union is taken over all j, k, l, s(1)1 , s

(2)1 , s2 ∈ Z satisfying 0 ≤ k ≤ 2qn−2q

3n−1 − 1,⌈

qn+1qn−1

⌉≤ s

(1)1 ≤ qn+1 −

⌈qn+1qn−1

⌉− 1,

⌈qn+1qn

⌉≤ s

(2)1 ≤ qn+1 −

⌈qn+1qn

⌉− 1, 0 ≤ s2 ≤ γnqn − 1,

0 ≤ j ≤ qn − 1 = qn2qn−2q3

n−1− 1 and 1 ≤ l ≤ qn−2q

3n−1 − 2qn−2q

2n−1 − 1.

- in case of dimension m ≥ 4:

qn · qn+1+

1q2n · qn+1

,s1 + 1qn · qn+1

− 1q2n · qn+1

1qn−1

+s2 · qmnqn+1

qn · qn+1,

1qn−1

+(s2 + 1) · qmn

1qn−1

2qn−2qmn−1qn+

qn−1q2n

qn−1+

2qn−2qmn−1qn+j + 1q2n

− 1qn−1q2

1qn−1

qn · qn+1+

qn · q2n+1

q2n · q2

qn−1+

qn · qn+1+

s(2)4 + 1

qn · q2n+1

− 1q2n · q2

m∏i=5

qn−1+

siqn · qn+1

q2n · qn+1

qn−1+

si + 1qn · qn+1

− 1q2n · qn+1

where the union is taken over all j, k, l, s2, si, s(2)4 ∈ Z satisfying

⌈qn+1qn−1

⌉≤ si ≤ qn+1−

⌈qn+1qn−1

⌉−1

for i = 1, 4, 5, ...,m, 0 ≤ j ≤ qn − 1 = qn2qn−2qmn−1

− 1, 0 ≤ k ≤ 2qn−2qmn−1 − 1, 0 ≤ s2 ≤ γnqn − 1,

1 ≤ l ≤ qn−2qmn−1 − 2qn−2q

m−1n−1 − 1 and

⌈qn+1qn

⌉≤ s(2)

4 ≤ qn+1 −⌈qn+1qn

⌉− 1.

Then c(n)0,2 := Hn−1

(c(n)0,2

)is the base of the second tower.

Remark 7.7.1. We note that(

1− 3·2mqn−1

)· γn·q

2qn+1≤ µ

(c(n)0,i

)≤ γn·q2

2qn+1for i = 1, 2. Moreover, we

have in both cases:

diam(c(n)0,i

)≤ ‖DHn−1‖0 · diam

(c(n)0,i

)≤ ‖DHn−1‖0 ·

which is smaller than 1n because of Lemma 7.6.4, 2..

In the next step we will construct a sequence (mn)n∈N of natural numbers in such a way thatmn · (αn+1 − αn) = ±mn · γn

qn+1is approximately ± 1

mn :=⌊qn+1

γn · q2n

⌋. (7.3)

With it we define the following sets:

c(n)i,1 := f in

(c(n)0,1

)for i = 0, 1, ...,mn − 1,

c(n)i,2 := f in

(c(n)0,2

)for i = 0, 1, ...,mn.

Observe that these sets are disjoint by construction. Hence, we are able to define these twotowers:

Definition 7.7.2. The first tower B(n)1 with base c(n)

0,1 and height mn consists of the sets c(n)i,1

for i = 0, 1, ...,mn − 1. The second tower B(n)2 with base c(n)

0,2 and height mn + 1 consists of thesets c(n)

i,2 for i = 0, 1, ...,mn.

In the rest of this subsection we check that these towers satisfy the requirements in thedefinition of a (h, h+ 1)-approximation. First of all we notice that both towers are substantialbecause we have:

)= mn · µ

(c(n)0,1

)≥(qn+1

γn · q2n

− 1)·(

1− 3 · 2mqn−1

)· γn · q

2qn+1≥(

1− 6mqn−1

2− γn · q2

2qn+1,

)= (mn + 1) · µ

(c(n)0,2

)≥ qn+1

γn · q2n

1− 3 · 2mqn−1

)· γn · q

2qn+1=(

1− 3 · 2mqn−1

Using the notation from section 7.3 we consider the partial partition

ξn :=c(n)i,1 , c

(n)k,2 : i = 0, 1, ...,mn − 1; k = 0, 1, ...,mn

and have to show: ξn → ε as n → ∞. This property is fulfilled if we show that the partialpartitions ξn :=

c ∈ ξn : diam (c) < 1

satisfy µ

(⋃c∈ξn c

)→ 1 as n → ∞. For this we

examine which tower elements satisfy the condition on their diameter. Since

c(n)i,j = f in

(c(n)0,j

)= Hn−1φnRiαn+1

φ−1n H−1

(Hn−1

(c(n)0,j

))= Hn−1φnRiαn+1

φ−1n

(c(n)0,j

)we have to check for how many iterates i the set Riαn+1

φ−1n

(c(n)0,j

)is contained in the “good

area” of φn. Note that the bases of both towers are positioned in this “good area”. By the 1qn-

equivariance of the map φn and i ·αn+1 = i·pnqn± i·γn

qn+1we consider the displacement i·γn

qn+1, which

is at most 1q2nbecause of i ≤ mn ≤ qn+1

γn·q2n. So the restrictions will come from the maps φ(2)

14qn−1

qm−1n , 1

4qn−1

,...,φ(m−1)

14qn−1

. By the same observations as in Remark 7.5.2 we can estimate the

number of allowed iterates i ∈ 0, 1, ...,mn − 1 by(

1− 1qn−1

)m−1

·mn. We conclude that there

are at least 2 ·(

1− 1qn−1

)m−1

·mn partition elements c(n)i,j in ξn and this corresponds to a measure

⋃c∈ξn

≥ 2 ·(

1− 1qn−1

)m−1

·mn · µ(c(n)i,j

≥ 2 ·(

1− 1qn−1

)m−1

·(qn+1

γn · q2n

− 1)·(

1− 3 · 2mqn−1

)· γn · q

2qn+1,

which converges to 1 as n→∞.

Remark 7.7.3. Actually we have a linked approximation of type (h, h+ 1): To see this weconsider the partition

ηn :=ci := c

(n)i,1 ∪ c

(n)i,2 : 0 ≤ i ≤ mn − 1, diam (ci) <

and show limn→∞ µ(⋃

c∈ηn c)

= 1. For this purpose, we note that c(n)i,1 ∪ c

(n)i,2 is contained in one

1qn-cube, if Riαn+1

φ−1n

(c(n)0,1 ∪ c

(n)0,2

)belongs to the “good area” of the map φn and the deviation

i · |αn+1 − αn| is less than 1q2n. Hence, the calculations from above apply.

7.7.2 Speed of approximation

In this section we want to compute the speed of the approximation:∑c∈ξn

µ (f (c)4σn (c)) ≤∑c∈ξn

(µ (f (c)4fn+1 (c)) + µ (fn+1 (c)4fn (c)) + µ (fn (c)4σn (c))) .

For this we note σn|c(n)i,1

= fn|c(n)i,1

for i = 0, ...,mn − 2 and σn

(c(n)mn−1,1

(n)0,1 as well as

σn|c(n)i,2

= fn|c(n)i,2

for i = 0, ...,mn − 1, σn(c(n)mn,2

(n)0,2 .

To estimate the expression∑c∈ξn µ (fn (c)4σn (c)) we observe that fn

(c(n)mn−1,1

)and c(n)

0,1 differ

in the deviation of mn · |αn+1 − αn| from 1q2nand in that the part of c(n)

0,1 corresponding to theboth values l = qn−2q

mn−1 − 2 as well as j = qn − 1 in “case +” (resp. l = 2qn−2q

m−1n−1 and j = 0

in “case -”) is not mapped back to c(n)0,1 . The second discrepancy yields a measure difference of at

most 2 · γn·qnqn+1·(

1− 2qn

)2m−1

≤ 2·γn·qnqn+1

.Examining the first one we recall that q2

n divides qn+1 by requirement A. So qn+1 = qn+1 · q2n,

where qn+1 ∈ Z. This implies mn =⌊qn+1γn·q2

⌋=⌊qn+1γn

⌋and we can write qn+1 = mn · γn + t with

t ∈ Z, 0 ≤ t ≤ γn − 1. Then we have

mn · |αn+1 − αn| = mn ·γnqn+1

=mn · γn + t

qn+1− t

qn+1=qn+1

qn+1− t

Hence, the deviation can take the values 0, 1qn+1

,... or γn−1qn+1

. Because of this deviation some1

qn+1-stripes of φ−1

(c(n)0,1

)(i.e. parts corresponding to a fixed values of k, l, j, s2 and having a

measure of at most(

1− 2qn

)2m−1

· 1qn+1

) are shifted out of φ−1n

(c(n)0,1

). The caused measure

difference is at most 2 · γn−1qn+1

· qn−2 · qmn−1 · qn ·(

1− 2qn

. Then we have:

µ(fmnn

(c(n)0,1

)4c(n)

2 · γn · qn−2 · qmn−1 · qnqn+1

+2 · γn · qnqn+1

≤4 · γn · qn−2 · qmn−1 · qn

Analogously fn(c(n)mn,2

)and c(n)

0,2 differ in the displacing of at most γn · qn−2 · qmn−1 · qn ·(

1− 2qn

)such 1

qn+1-stripes caused by the deviation of (mn + 1) · |αn+1 − αn| from 1

q2nand in that the part

of c(n)0,2 corresponding to l = qn−2q

mn−1 − 2qn−2q

m−1n−1 − 1 as well as j = qn − 1 in “case +” (resp.

l = 1 and j = 0 in the “case -”) is not mapped back to c(n)0,2 . Thus, we get

µ(fmn+1n

(c(n)0,2

)4c(n)

4 · γn · qn−2 · qmn−1 · qnqn+1

This yields ∑c∈ξn

µ (fn (c)4σn (c)) ≤8 · γn · qn−2 · qmn−1 · qn

qn+1. (7.4)

In the next step we consider∑c∈ξn µ (fn+1 (c)4fn (c)). Thus, we have to compare the sets

φn+1Riαn+2φ−1

n+1φ−1n

(c(n)0,1

)and Riαn+1

φ−1n

(c(n)0,1

)= φn+1Riαn+1

φ−1n+1φ−1

(c(n)0,1

)(recall

that we have φn+1 R 1qn+1

= R 1qn+1

φn+1 by construction). Since Riαn+1 φ−1

n+1 φ−1n

(c(n)0,1

well as Riαn+2 φ−1

n+1 φ−1n

(c(n)0,1

)are positioned in the “good area” of the map φn+1 for i ≤ mn

by definition of c(n)0,1 , the deviation i · |αn+2 − αn+1| = i·γn+1

qn+2on the θ-axis for every of the at

most 12 ·γn ·q

2n ·qm−1

n+1 ·(

1− 2qn

)mstripes of θ-width

(1− 2

qn−1

qmn+1causes the following measure

difference:

µ(fn+1

(c(n)i,1

))≤ 2 · i · γn+1

qn+2· 1

2· γn · q2

n · qm−1n+1 ·

(1− 2

1− 2qn

)m−1

This difference occurs for every i ∈ 0, ...,mn − 1 and thus we can estimate:

mn−1∑i=0

µ(fn+1

(c(n)i,1

))≤ 2 · 1

2·γn ·q2

n ·qm−1n+1 ·m2

1− 2qn

)2m−1

· γn+1

qn+2≤ qm+1

n+1 ·γn+1

In the same spirit we estimatemn∑i=0

µ(fn+1

(c(n)i,2

))≤ 2·1

2·γn·q2

n·qm−1n+1 ·(mn + 1)2·

(1− 2

)2m−1

·γn+1

qn+2≤ qm+1

n+1 ·γn+1

Thus, we obtain ∑c∈ξn

µ (fn+1 (c)4fn (c)) ≤2 · qm+1

n+1 · γn+1

qn+2. (7.5)

Lastly we consider∑c∈ξn

µ (f (c)4fn+1 (c)) ≤∞∑

∑c∈ξn

µ (fk+1 (c)4fk (c)) .

So we compute for every c ∈ ξn using the notation c := H−1n+1 (c):

µ (fn+2 (c)4fn+1 (c)) = µ(Hn+2 Rαn+3 φ−1

n+2 (c)4Hn+1 Rαn+2 (c))

= µ(Hn+2 Rαn+3 φ−1

n+2 (c)4Hn+1 φn+2 Rαn+2 φ−1n+2 (c)

(Rαn+3 φ−1

n+2 (c)4Rαn+2 φ−1n+2 (c)

Since we have no controll on φ−1n+2 (c) for these areas of c, that do not belong to the “good area”

of the map φn+2, they will be part of the measure difference in our estimates. On the otherhand, for the part of c belonging to the “good area” of the map φn+2 the difference is causedby the deviation |αn+3 − αn+2|. Using Remark 7.5.2 the “good area” of the map φn+2 on anθ-interval of the form

qn+2, l+1qn+2

]has length at least

(1− m

)· 1qn+2

. So the measure difference

of Rαn+3 φ−1n+2 (c) and Rαn+2 φ−1

n+2 (c) on a section of the form[

, l+1qn+2

]× [0, 1]m−1 is at

qn+1· 1qn+2

+ |αn+3 − αn+2| ·(

1− 2qn+1

)m−1)≤ 4mqn+1

· 1qn+2

Moreover, we recall thatH−1n+1 (c) consists of at most 1

2 ·γn ·q2n ·qm−1

n+1 ·(

1− 2qn

)mstripes of θ-width(

1− 2qn−1

qmn+1. Since the θ-width

(1− 2

qn−1

qmn+1accords to at most

⌈(1− 2

qn−1

)qn+2qmn+1

⌉+ 2

intervals of type[

, l+1qn+2

], we have

µ (fn+2 (c)4fn+1 (c)) ≤ 12·γn ·q2

n ·qm−1n+1 ·

(1− 2

)m·(⌈(

1− 2qn−1

⌉+ 2)· 4mqn+1

· 1qn+2

Every of the (2mn + 1) elements c ∈ ξn contributes and so we obtain∑c∈ξn

µ (fn+2 (c)4fn+1 (c)) ≤ 5 ·mqn+1

Analogously we estimate the other summands µ (fk+1 (c)4fk (c)) and hence∑c∈ξn

µ (f (c)4fn+1 (c)) ≤∞∑

5 ·mqk≤ 10 ·m

qn+1. (7.6)

Using equations 7.4, 7.5 and 7.6 we conclude∑c∈ξn

µ (f (c)4σn (c)) ≤8 · γn · qn−2 · qmn−1 · qn

2 · qm+1n+1 · γn+1

10 ·mqn+1

In order to prove that this speed of approximation is of order o(

)we compute

8·γn·qn−2·qmn−1·qnqn+1

+2·qm+1

n+1 ·γn+1

qn+2+ 10·m

≤ qn+1

γn · q2n

(8 · γn · qn−2 · qmn−1 · qn

2 · qm+1n+1 · γn+1

10 ·mqn+1

=8 · qn−2 · qmn−1

2 · qm+2n+1 · γn+1

γn · q2n · qn+2

+10 ·mγn · q2

Proof of good cyclic approximation 163

Since this converges to 0 as n → ∞ (in particular because of equation 7.2), we have a goodapproximation of type (h, h+ 1).

7.8 Proof of good cyclic approximation

7.8.1 Tower for good cyclic approximation

Let d(n)0 be the set

- in case of dimension m = 2:⋃[1

qn−1 · qn+

1qn−1 · q2

qn−1 · qn+

− 1qn−1 · q2

qn−1+s2 · q2

qn+1+s

(2)2 · q2

qn · q2n+1

qn−1+s2 · q2

(2)2 + 1

)· q2n

− q2n

qn · q2n+1

,where the union is taken over all s2, s

(2)2 ∈ Z satisfying

⌈qn+1qn

⌉≤ s

(2)2 ≤ qn+1 −

⌈qn+1qn

⌉− 1,

0 ≤ s2 ≤ 2qn−1q2n − 1.

- in case of dimension m = 3:

qn · qn+1+

qn · q2n+1

q2n · q2

qn · qn+1+

s(2)1 + 1

qn · q2n+1

− 1q2n · q2

1qn−1

+s2 · q3

qn · qn+1,

1qn−1

+(s2 + 1) · q3

qn+1− q3

qn · qn+1

1qn−1

qn−1 · qn+

1qn−1 · q2

qn−1+

1qn−1 · qn

− 1qn−1 · q2

where the union is taken over all s(1)1 , s

(2)1 , s2 ∈ Z satisfying

⌈qn+1qn−1

⌉≤ s(1)

1 ≤ qn+1 −⌈qn+1qn−1

⌉− 1,⌈

qn+1qn

⌉≤ s(2)

1 ≤ qn+1 −⌈qn+1qn

⌉− 1 and 0 ≤ s2 ≤ 2qn−1q

3n − 1.

- in case of dimension m ≥ 4:⋃[s1

qn · qn+1+

1q2n · qn+1

,s1 + 1qn · qn+1

− 1q2n · qn+1

1qn−1

+s2 · qmnqn+1

qn · qn+1,

1qn−1

+(s2 + 1) · qmn

1qn−1

qn−1 · qn+

1qn−1 · q2

qn−1+

1qn−1 · qn

− 1qn−1 · q2

1qn−1

qn · qn+1+

qn · q2n+1

q2n · q2

qn−1+

qn · qn+1+

s(2)4 + 1

qn · q2n+1

− 1q2n · q2

m∏i=5

qn−1+

siqn · qn+1

q2n · qn+1

qn−1+

si + 1qn · qn+1

− 1q2n · qn+1

164 Proof of good cyclic approximation

where the union is taken over all si, s(2)4 ∈ Z satisfying

⌈qn+1qn−1

⌉≤ si ≤ qn+1 −

⌈qn+1qn−1

⌉− 1 for

i = 1, 4, 5, ...,m,⌈qn+1qn

⌉≤ s(2)

4 ≤ qn+1 −⌈qn+1qn

⌉− 1 and 0 ≤ s2 ≤ 2qn−1q

mn − 1.

With these the base d(n)0 of the tower is defined to be the set d(n)

0 := Hn−1

)and the tower

levels are the setsd

(n)i := f in

)for i = 0, ..., qn+1 − 1.

Recall the relations qn+1 = 2 · qn−1 · qmn · qn+1 as well as αn+1 = pn+1qn+1

= pn+1qn+1

, where pn+1 andqn+1 are relatively prime. Hence, the tower levels are disjoint sets of equal measure not less

than 2qn−1qmn

qn+1·(

1− 3qn−1

)2m−1

. Moreover, the associated cyclic permutation σn is given by the

description σn|d(n)i

= fn|d(n)i

for i = 0, ..., qn+1− 2 and σn(d

(n)qn+1−1

(n)0 . Since f qn+1

n = id wealso have σn|d(n)

qn+1−1= fn|d(n)

qn+1−1.

In order to show that this provides a cyclic approximation of the constructed map f we showthat the partial partition Γn :=

(n)i : i = 0, ..., qn+1 − 1

converges to the decomposition into

points. For this purpose, it suffices to show that Γn :=d

(n)i ∈ Γn : diam

satisfies

limn→∞ µ(⋃

d∈Γnd)

= 1. As in the previous chapter we have to check for how many iterates

i ∈ 0, 1, ..., qn+1 − 1 the set Riαn+1 φ−1

)is contained in the “good area” of the map

φn, which corresponding length on the θ-axis is at least(

1− 1qn−1

)mby Remark 7.5.2. Since

Riαn+1is equally distributed on S1, there are at least

(1− 2

qn−1

)m· qn+1 such iterates i. Then

we conclude

⋃d∈Γn

≥ (1− 2qn−1

)m· qn+1 · µ

1− 2qn−1

)m· qn+1

2 · qn−1 · qmn· 2 · qn−1 · qmn

qn+1·(

1− 3qn−1

)2m−1

1− 3qn−1

)3m−1

which converges to 1 as n→∞.

7.8.2 Speed of approximationAs observed in the previous section σn|d = fn|d for every d ∈ Γn. Thus, for the speed ofapproximation it holds:∑

d∈Γn

µ (f (d)4σn (d)) ≤∑d∈Γn

(µ (f (d)4fn+1 (d)) + µ (fn+1 (d)4fn (d))) .

First of all we aim for estimating the error∑d∈Γn

µ (fn+1 (d)4fn (d)). We have to compare

the tower elements φn+1 Riαn+2 φ−1

n+1 φ−1n

)and φn+1 Riαn+1

φ−1n+1 φ−1

every i ∈ 1, ..., qn+1. Since Riαn+2 φ−1

n+1 φ−1n

)as well as Riαn+1

φ−1n+1 φ−1

positioned in the “good area” of the map φn+1, the discrepancy i · |αn+2 − αn+1| = i·γn+1qn+2

on the

Proof of good cyclic approximation 165

θ-axis for every of the at most 2qn−1qmn q

m−1n+1 ·

(1− 2

)m−1

stripes causes the following measuredifference

µ(fn+1

))≤ 2 · 2qn−1q

m−1n+1 ·

(1− 2

)m−1

· i · γn+1

This difference occours for every i ∈

0, ..., qn+12qn−1qmn

and thus we can estimate

∑d∈Γn

µ (fn+1 (d)4fn (d)) ≤2 · qm+1

n+1 · γn+1

qn+2. (7.7)

In the next step we consider

∑d∈Γn

µ (f (d)4fn+1 (d)) ≤∞∑

∑d∈Γn

µ (fk+1 (d)4fk (d)) .

In order to estimate µ (fn+2 (d) ∆fn+1 (d)) we argument as in the previous section using that

H−1n+1 (d) consists of at most 2qn−1q

m−1n+1 ·

(1− 2

)m−1

stripes of θ-width(

1− 2qn−1

qmn+1.

Therewith, we obtain

µ (fn+2 (d)4fn+1 (d))

≤ 2 · 2qn−1qmn q

m−1n+1 ·

(1− 2

)m−1

·(⌈(

1− 2qn−1

⌉+ 2)· 4mqn+1

· 1qn+2

Every of the qn+1 = qn+12qn−1qmn

elements d ∈ Γn contributes and so we obtain

∑d∈Γn

µ (fn+2 (d)4fn+1 (d)) ≤ 8 ·mqn+1

Analogously we estimate the other summands µ (fk+1 (d)4fk (d)) and hence

∑d∈Γn

µ (f (d)4fn+1 (d)) ≤∞∑

8 ·mqk≤ 16 ·m

qn+1. (7.8)

Using equations 7.7 and 7.8 we conclude∑d∈Γn

µ (f (d)4σn (d)) ≤2 · qm+1

n+1 · γn+1

16 ·mqn+1

In order to prove that this speed of approximation is of order o(

)we compute

2·qm+1n+1 ·γn+1

qn+2+ 16·m

≤ qn+1

2 · qn−1 · qmn·

(2 · qm+1

n+1 · γn+1

16 ·mqn+1

=qm+2n+1 · γn+1

8 ·mqn−1 · qmn

Since this converges to 0 as n→∞ (again by equation 7.2), we have a good cyclic approximation.

166 Reduction to Proposition 7.2.1

7.9 Reduction to Proposition 7.2.1

Using the concepts and results from the previous sections we can reduce the proof of TheoremF to Proposition 7.2.1. Indeed, such a constructed diffeomorphism has the following properties:

• Since f allows a good cylic approximation, it has simple spectrum by Lemma 7.4.4. Fur-thermore, f admits a good linked approximation of type (h, h+ 1) and so we can useRemark 7.3.5 to obtain the weak convergence Uh+1

f −→w r · Uf + (1− r) · Id for somer ∈ (0, 1). Hence, we can apply Lemma 7.4.5 and conclude that f × f has homogeneousspectrum of multiplicity 2.

• With the aid of Lemma 7.3.6 and the good cyclic approximation of f we have that f isergodic. Then we can exploit the good approximation of type (h, h+ 1) and Lemma 7.3.7to see that f is even weak mixing. Due to Remark 7.4.8 there exists a strictly increasingsequence (kn)n∈N of natural numbers such that the convergence Uknf →w Pc holds in theweak operator topology as n→∞. Along this sequence we have using Remark 7.3.5

Ukn·(h+1)f −→w (r · Pc + (1− r) · Id)

for some r ∈ (0, 1). Thus, f is κ-weak mixing (with κ = r ∈ (0, 1)). Then the maximalspectral type σ of f is disjoint with its convolutions by Lemma 7.4.9.

By Proposition 7.2.1 the set of diffeomorphisms having the aimed properties is dense in Aαwith respect to the Diff∞-topology: Because of Aα = h Rα h−1 : h ∈ Diff∞ (M,µ)

itis enough to show that for every diffeomorphism h ∈ Diff∞ (M,µ) and every ε > 0 there is adiffeomorphism f with the aimed three properties such that d∞

(f , h Rα h−1

)< ε. For this

purpose, let h ∈ Diff∞ (M,µ) and ε > 0 be arbitrary. By [Om74], p. 3, resp. [KM97], Theorem43.1., Diff∞ (M) is a Lie group. In particular, the conjugating map g 7→ h g h−1 is continuouswith respect to the metric d∞. Continuity in the point Rα yields the existence of δ > 0, suchthat d∞ (g,Rα) < δ implies d∞

(h g h−1, h Rα h−1

)< ε. By Proposition 7.2.1 we can find

a diffeomorphism f with the three properties and d∞(f,Rα) < δ. Hence, f := hf h−1 satisfiesd∞

(f , h Rα h−1

)< ε and has the aimed qualities because the properties are invariant under

isomorphisms.In order to prove the genericity of the aimed properties we consider all sequences of constructeddiffeomorphisms (fn)n∈N satisfying the requirements from the previous sections. Let Un (fn) bethe subsequent neighbourhood of the diffeomorphism fn:

Un (fn) :=g ∈ Diff∞µ (M) : dkn (fn, g) <2kn,∑c∈ξn

µ (g (c)4fn (c)) <1

nmn,∑d∈Γn

µ (g (d)4fn (d)) <1

By Θn we denote the union of all neighbourhoods Un (fn) over all the n-th diffeomorphisms inthe above mentioned sequences. Since the neighbourhoods Un (fn) are open, the sets Θn areopen as well. Then

Θ :=⋂n∈N

⋃s≥n

is a Gδ-set as the countable intersection of open sets.

Reduction to Proposition 7.2.1 167

• For all the sequences (fn)n∈N the respective limit diffeomorphism f ∈ Aα belongs to Θbecause it belongs to Un (fn) for every n ∈ N by construction. So Θ contains all theconstructed diffeomorphisms with the aimed properties. Hence, it is dense in Aα due tothe above considerations.

• In the next step we want to show that f ∈ Θ admits a good linked approximation of type(h, h+ 1) as well as a good cyclic approximation:For any f ∈

⋂n∈N

⋃s≥n Θs there is a sequence (nk)k∈N with nk →∞ as k →∞, such that

f ∈ Θnk . So there is a sequence (fnk)k∈N of diffeomorphisms, at which fnk is the nk-thelement of one of the above mentioned sequences of constructed diffeomorphisms, such thatf ∈ Unk (fnk). We observe that ξnk → ε as well as Γnk → ε as k → ∞, where ξnk andΓnk are the partitions belonging to the diffeomorphism fnk . Then f admits a good linkedapproximation of type (h, h+ 1) as well as a good cyclic approximation by the definitionof the neighbourhoods Unk (fnk).

Thus, the set of diffeomorphisms in Aα admitting a good linked approximation of type (h, h+ 1)as well as a good cyclic approximation contains a dense Gδ-set. Since these types of approxima-tion imply the aimed properties, we conclude that the set of diffeomorphisms f ∈ Aα with thefollowing properties

• a good approximation of type (h, h+ 1);

• a maximal spectral type disjoint with its convolutions;

• a homogeneous spectrum of multiplicity two for the Cartesian square f × f .

is a residual subset in the C∞-topology. So Theorem F is deduced.

Chapter 8

Uniform rigidity sequences for weakmixing diffeomorphisms on T2

8.1 Introduction

In [GM89] the notion of uniform rigidity was introduced as the topological analogue of rigidityin ergodic theory:

Definition 8.1.1. 1. Let T be an invertible measure-preserving transformation of a non-atomic probability space (X,B, µ). T is called rigid if there exists an increasing sequence(kn)n∈N of natural numbers such that the powers T kn converge to the identity in the strongoperator topology as n→∞, i.e.

∥∥f T kn − f∥∥2→ 0 as n→∞ for all f ∈ L2 (X,µ). So

rigidity along a sequence (kn)n∈N implies µ(T knA ∩A

)→ µ (A) as n→∞ for all A ∈ B.

2. Let (X,B, µ) be a Lebesgue probability space, where X is a compact metric space withmetric d. A measure-preserving homeomorphism T : X → X is called uniformly rigid ifthere exists an increasing sequence (kn)n∈N in N such that du

(T kn , id

)→ 0 as n → ∞,

where du (S, T ) = d0 (S, T ) + d0

(S−1, T−1

)with d0 (S, T ) := supx∈X d (S (x) , T (x)) is the

uniform metric on the group of measure-preserving homeomorphisms on X.

Remark 8.1.2. Uniform rigidity implies rigidity ([JKLSS09], Lemma 2.3.). In [Ya], example3.1, an example of a rigid, but not uniformly rigid homeomorphism of T2 is presented. Thus,rigidity and uniform rigidity do not coincide on T2.

In [JKLSS09], Proposition 4.1., it is shown that if an ergodic map is uniformly rigid, thenany uniform rigidity sequence has zero density. Afterwards, the following question is posed:

Question 8.1.3. Which zero density sequences occur as uniform rigidity sequences for an ergodictransformation?

Under some assumptions on the sequence (kn)n∈N measure-preserving transformations thatare weak mixing and rigid along this sequence are constructed by a cutting and stacking methodin [BJLR].K. Yancey considered Ouestion 8.1.3 in the setting of homeomorphisms on T2 (see [Ya]). Given asufficient growth rate of the sequence she proved the existence of a weak mixing homeomorphismof T2 that is uniformly rigid with respect to this sequence: Let ψ (x) = xx

3. If (kn)n∈N is an

170 Introduction

increasing sequence of natural numbers satisfying kn+1kn≥ ψ (kn), there exists a weak mixing

homeomorphism of T2 that is uniformly rigid with respect to (kn)n∈N.In this chapter we start to examine this problem in the smooth category. As a starting pointwe use the construction of weak mixing diffeomorphisms on T2 undertaken in the real-analytictopology in [FS05] with the explicit conjugation maps

φn (θ, r) =(θ, r + q2

n · cos (2πqnθ)),

gn (θ, r) = (θ + [nqσn] · r, r) with some 0 < σ <12,

hn = gn φn.

Furthermore, let R = Rtt∈S1 denote the standard circle action on T2 comprising of the diffeo-morphisms Rt (θ, r) = (θ + t, r). Note that hn R pn

qn= R pn

qn hn. With the conjugation maps

Hn := h1 ... hn we will define the diffeomorphism fn = Hn Rαn+1 H−1n . The sequence of

rational numbers will beαn+1 =

qn+1= αn −

anqn · qn+1

where an ∈ Z, 1 ≤ an ≤ qn is chosen in such a way that qn+1 · pn ≡ an mod qn. Therewith, wehave |αn+1 − αn| ≤ 1

qn+1and qn+1 · αn+1 = qn+1·pn

qn− an

qn≡ 0 mod 1, which implies f qn+1

n = id.Hence, (qn)n∈N will be a rigidity sequence of f = limn→∞ fn under some restrictions on thecloseness between fn and f (see Remark 8.3.6), which depend on the norms of the conjugationmaps Hi and the distances |αi+1 − αi| ≤ 1

qi+1for every i > n. Thus, we have to estimate the

norms |||Hn|||n carefully. This will yield the subsequent requirement on the number qn+1 (seethe end of section 8.3.2):

qn+1 > ϕ1 (n) · q2·((n+2)·(n+1)n+1+1)n ,

where ϕ1 (n) := 2n · (n+ 1)! · ((n+ 2)!)(n+2)n−2·(n+1) · (2πn)(n+2)·(n+1)n+1

. This is a sufficientcondition on the growth rate of the rigidity sequence (qn)n∈N and we prove that f is weak mixingusing a criterion similar to that deduced in [FS05] (see section 8.5). Consequently we obtain:

Theorem G. Let ϕ1 (n) := 2n ·(n+ 1)! ·((n+ 2)!)(n+2)n−2·(n+1) ·(2πn)(n+2)·(n+1)n+1

. If (qn)n∈Nis a sequence of natural numbers satisfying

qn+1 ≥ ϕ1 (n) · q2·((n+2)·(n+1)n+1+1)n ,

there exists a weak mixing C∞-diffeomorphism of T2 that is uniformly rigid with respect to(qn)n∈N.

In section 8.6.1 we conclude a rougher but more handsome statement:

Corollary C. If (qn)n∈N is a sequence of natural numbers satisfying q1 ≥ 108π and qn+1 ≥ qqnn ,then there exists a weak mixing C∞-diffeomorphism of T2 that is uniformly rigid with respect to(qn)n∈N.

We note that our requirement on the growth rate is less restrictive than the mentionedcondition in [Ya], Theorem 1.5.. In fact, the proof in [Ya] shows that a condition of the formkn+1kn≥ k

4k2n+20

n is sufficient for her construction of a weakly mixing homeomorphism, which isuniformly rigid along (kn)n∈N. Our requirement on the growth rate is still weaker.By the same approach we consider the problem in the real-analytic topology. In this setting wewill deduce the following sufficient condition on the growth rate of the rigidity sequence (qn)n∈N:

Theorem H. Let ρ > 0. If (qn)n∈N is a sequence of natural numbers satisfying q1 ≥ ρ+ 1 and

qn+1 ≥ 2n · 64π2 · n2 · q14n · exp

(4π · n · q6

n · exp(2π · q4

n · (1 + n · qn))),

there exists a weak mixing Diffωρ -diffeomorphism of T2 that is uniformly rigid along the sequence(qn)n∈N.

Again, we derive from this a more convenient statement in section 8.6.2:

Corollary D. If (qn)n∈N is a sequence of natural numbers satisfying q1 ≥ (ρ+ 1) · 27 · π2 andqn+1 ≥ q15

n · exp(q7n · exp

)), then there exists a weak mixing Diffωρ -diffeomorphism of T2 that

is uniformly rigid with respect to (qn)n∈N.

In this section we will formulate a criterion for weak mixing that will be used in the smooth aswell as in the real-analytic case.

8.2.1 (γ, δ, ε)-distribution of horizontal intervals

Since we work on the manifold T2, we recall the following definitions stated in [FS05]:

Definition 8.2.1. Let η be a partial decomposition of T into intervals and consider on T2 thedecomposition η consisting of intervals in η times some r ∈ [0, 1]. Sets of this form will be calledhorizontal intervals and decompositions of this type standard partial decompositions. On theother hand, sets of the form θ × J , where J is an interval on the r-axis, are called verticalintervals.

Hereby, we can introduce the notion of (γ, δ, ε)-distribution of a horizontal interval in thevertical direction:

Definition 8.2.2. A diffeomorphism Φ : T2 → T2 (γ, δ, ε)-distributes a horizontal interval I ifthe following conditions are satisfied

• πr (Φ (I)) is an interval J with 1− δ ≤ λ (J) ≤ 1.

• Φ (I) is contained in a vertical strip [c, c+ γ]× J for some c ∈ T.

• For any interval J ⊆ J we have∣∣∣∣∣∣λ(I ∩ Φ−1

(T× J

))λ (I)

−λ(J)

λ (J)

∣∣∣∣∣∣ ≤ ε ·λ(J)

λ (J).

8.2.2 Statement of the criterion

The proof of the criterion is the same as in [FS05], section 3. The only difference occurs incomparison to Lemma 3.5., which in our case will be stated in the subsequent way:

Lemma 8.2.3. Let (ηn)n∈N be a sequence of standard partial decompositions of T2 into horizontalintervals of length less than q−2.5

n . Moreover, let gn be defined by gn (θ, r) = (θ + [nqσn] · r, r) with

172 Convergence of (fn)n∈N in Diff∞(T2)

some 0 < σ < 1 and let (Hn)n∈N be a sequence of area-preserving diffeomorphisms such that forevery n ∈ N:

‖DHn−1‖0 ≤ q0.5n . (C1)

Consider the partitions νn := Γn = Hn−1 (gn (In)) : In ∈ ηn.Then ηn → ε implies νn → ε.

Proof. For every ε > 0 we can choose n large enough such that µ(⋃

I∈ηn I)> 1 − ε (because

of ηn → ε) and there is a collection of squares Sn := Sn,i with side length between q−1.5n

and q−2n with total measure of the union Sn :=

⋃i Sn,i greater than 1 −

√ε. Then we have

µ(⋃

I∈ηn I ∩ Sn)≥ (1−

√ε) · µ (Sn), because otherwise µ

⋃I∈ηn I

)>√ε · µ (Sn) >

√ε · (1−

√ε) and so µ

⋃I∈ηn I

)>√ε − ε > ε in case of ε < 1

4 , which contradicts

µ(⋃

I∈ηn I)> 1 − ε. Since the horizontal intervals I ∈ ηn have length less than q−2.5

n , we can

approximate the squares in the above collection Sn for n sufficiently large in such a way thatµ(⋃

I∈ηn,I⊂Sn I)≥ (1− 2

√ε) · µ (Sn).

In the next step we consider the sets Cn,i := Hn−1 (gn (Sn,i)) with Sn,i ∈ Sn. For these sets Cn,iwe have:

diam (Cn,i) ≤ ‖DHn−1‖0 · ‖Dgn‖0 · diam (Sn,i) ≤ q0.5n · n · qσn ·

√2 · q−1.5

n = n ·√

2 · qσ−1n ,

which goes to 0 as n → ∞, because σ < 1. Therefore, any Borel set B can be approximatedby a union of such sets Cn,i with any prescribed accuracy if n is sufficiently large, i.e. for everyε > 0 there is N ∈ N such that for n ≥ N there is an index set Jn: µ

⋃i∈Jn Cn,i

)< ε. Now

we choose the union of these elements I ∈ ηn contained in the occurring cubes Sn,i and obtain:µ (B4

⋃Hn−1 gn (I)) ≤ µ

⋃i∈Jn Cn,i

⋃I∈ηn,I⊂Sn I

)< ε+ 2

√ε ·µ (Sn) < 3

√ε.

Thus, B gets well approximated by unions of elements of νn if n is chosen sufficiently large.

Now the criterion for weak mixing can be stated in the following way (compare with [FS05],Proposition 3.9.):

Proposition 8.2.4. Let fn = Hn Rαn+1 H−1n be diffeomorphisms constructed as explained in

the introduction with 0 < σ < 12 and such that ‖DHn−1‖0 ≤ q0.5

n holds for all n ∈ N.Suppose that the limit f := limn→∞ fn exists. If there exists a sequence (mn)n∈N of natural num-bers satisfying d0 (fmnn , fmn) < 1

2n and a sequence (ηn)n∈N of standard partial decompositionsof T2 into horizontal intervals of length less than q−2.5

n such that ηn → ε and the diffeomor-phism Φn := φn Rmnαn+1

φ−1n

(1nqσn

, 1n ,

)-distributes every interval In ∈ ηn, then the limit

diffeomorphism f is weak mixing.

Remark 8.2.5. In [FS05] it is demanded ‖DHn−1‖0 < ln (qn) instead of requirement C1. Wedid this modification because the fulfilment of the original condition would lead to stricter re-quirements on the rigidity sequence: In particular, equation A3 would require an exponentialgrowth rate.

8.3 Convergence of (fn)n∈N in Diff∞(T2)

8.3.1 Properties of the conjugation maps hn and Hn

Using the explicit definitions of the maps gn, φn we can compute

hn (θ, r) =(θ + [nqσn] · r + [nqσn] · q2

n · cos (2πqnθ) , r + q2n · cos (2πqnθ)

Convergence of (fn)n∈N in Diff∞(T2)

as well ash−1n (θ, r) =

(θ − [nqσn] · r, r − q2

n · cos (2πqn (θ − [nqσn] · r))).

Then we can estimate for every k ∈ N and every multiindex ~a ∈ N20, |~a| ≤ k:

‖D~ahn‖0 ≤ 2 · (2π)k · [nqσn] · q2+kn

and ∥∥D~ah−1n

∥∥0≤ 2 · (2π)k · [nqσn]k · q2+k

Thus, we obtain|||hn|||k ≤ 2k+1 · πk · q2+k

n · nk · qσ·kn ≤(2πnq2

)k+1. (8.1)

In the next step we want to deduce norm estimates for the conjugation map Hn = Hn−1 hn.Therefore, we have to understand the derivatives of a composition of maps:

Lemma 8.3.1. Let g, h ∈ Diff∞(T2)and k ∈ N. Then for the composition g h it holds

|||g h|||k ≤ (k + 1)! · |||g|||kk · |||h|||kk.

Proof. By induction on k ∈ N we will prove the following observation:Claim: For any multiindex ~a ∈ N2

0 with |~a| = k and i ∈ 1, 2 the partial derivative D~a [g h]iconsists of at most (k + 1)! summands, where each summand is the product of one derivative ofg of order at most k and at most k derivatives of h of order at most k.

• Start: k = 1For i1, i ∈ 1, 2 we compute:

Dxi1[g h]i (x1, x2) =

2∑j1=1

(h (x1, x2)) ·Dxi1[h]j1 (x1, x2) .

Hence, this derivative consists of 2! = 2 summands and each summand has the announcedform.

• Induction assumption: The claim holds for k ∈ N.

• Induction step: k → k + 1Let i ∈ 1, 2 and ~b ∈ N2

0 be any multiindex of order∣∣∣~b∣∣∣ = k + 1. There are j ∈ 1, 2 and

a multiindex ~a of order |~a| = k such that D~b = DxjD~a. By the induction assumption thepartial derivative D~a [g h]i consists of at most (k + 1)! summands, at which the summandwith the most factors is of the subsequent form:

D~c1 [g]i (h (x1, x2)) ·D~c2 [h]i2 (x1, x2) · ... ·D~ck+1 [h]ik+1(x1, x2) ,

where each ~ci is of order at most k. Using the product rule we compute how the derivativeDxj acts on such a summand: 2∑

Dxj1D~c1 [g]i h ·Dxj [h]j1 D~c2 [h]i2 · ... ·D~ck+1 [h]ik+1

D~c1 [g]i h ·DxjD~c2 [h]i2 · ...D~ck+1 [h]ik+1+ ...+D~c1 [g]i h ·D~c2 [h]i2 · ...DxjD~ck+1 [h]ik+1

Thus, each summand is the product of one derivative of g of order at most k+1 and at mostk+1 derivatives of h of order at most k+1. Moreover, we observe that 2+k summands ariseout of one. So the number of summands can be estimated by (k + 2) · (k + 1)! = (k + 2)!and the claim is verified.

Using this claim we obtain for i ∈ 1, 2 and any multiindex ~a ∈ N20 of order |~a| = k:

‖D~a [g h]i‖0 ≤ (k + 1)! · |||g|||k · |||h|||kk.

Applying the claim on h−1 g−1 yields:∥∥D~a [h−1 g−1]i

∥∥0≤ (k + 1)! · |||g|||kk · |||h|||k.

We conclude|||g h|||k ≤ (k + 1)! · |||g|||kk · |||h|||kk.

Using this result we compute for every k ∈ N:

|||Hn|||k ≤ (k + 1)! · |||Hn−1|||kk · |||hn|||kk. (8.2)

Hereby, we can deduce the subsequent estimate of the norm |||Hn|||k+1 under some assumptionson the growth rate of the numbers qn:

Lemma 8.3.2. Let k, n ∈ N and n ≥ 2. Assume

qn+1 ≥ 2 · π · n · q2n. (A2)

Then we have

|||Hn|||k+1 ≤ ((k + 2)!)(k+2)n−2

· (2πnqn)(k+2)·(k+1)n−1·(n+1).

Proof. Let k ∈ N be arbitrary. We proof this result by induction on n:

• Start: n = 2Using equation 8.2 and the norm estimate on hn from equation 8.1 we obtain the claim:

|||H2|||k+1 ≤ (k + 2)! · |||H1|||k+1k+1 · |||h2|||k+1

= (k + 2)! · |||h1|||k+1k+1 · |||h2|||k+1

≤ (k + 2)! ·((

2πq21

)k+2)k+1

2π · 2 · q22

)k+2)k+1

≤ (k + 2)! · q(k+2)·(k+1)2 · (2π · 2 · q2)2·(k+2)·(k+1)

≤ (k + 2)! · (2π · 2 · q2)3·(k+2)·(k+1)

= ((k + 2)!)(k+2)2−2

· (2π · 2 · q2)(k+2)·(k+1)2−1·(2+1)

• Induction assumption: The claim is true for n ∈ N, n ≥ 2.

• Induction step n→ n+ 1:Using equation 8.2, the norm estimate on hn from equation 8.1 and the induction assump-tion we compute:

|||Hn+1|||k+1 ≤ (k + 2)! · |||Hn|||k+1k+1 · |||hn+1|||k+1

≤ (k + 2)! ·(

((k + 2)!)(k+2)n−2

(2πnqn)(k+2)·(k+1)n−1·(n+1))k+1

2π (n+ 1) q2n+1

)k+2)k+1

≤ (k + 2)! · ((k + 2)!)(k+2)n−2·(k+1) · q(k+2)·(k+1)n·(n+1)n+1 · (2π · (n+ 1) · qn+1)2·(k+2)·(k+1)

≤ ((k + 2)!)(k+2)n−1

· (2π · (n+ 1) · qn+1)(k+2)·(k+1)n·(n+2),

Convergence of (fn)n∈N in Diff∞(T2)

where we used in the last step the subsequent estimation:

(k + 2) · (k + 1)n · (n+ 1) + 2 · (k + 2) · (k + 1) = (k + 2) · (k + 1) ·(

(k + 1)n−1 (n+ 1) + 2)

≤ (k + 2) · (k + 1) · (k + 1)n−1 · (n+ 2) = (k + 2) · (k + 1)n · (n+ 2) .

Remark 8.3.3. As a special case of Lemma 8.3.1 we have ‖DHn‖0 ≤ 2! · ‖DHn−1‖0 · ‖Dhn‖0.With the aid of equation 8.1 we can estimate:

‖DHn‖0 ≤ 2! · q0.5n ·

(2π · n · q2

)2= 8π2 · n2 · q4.5

where we used condition C1, i.e. ‖DHn−1‖0 ≤ q0.5n . In order to guarantee this property for DHn

we demand:qn+1 ≥ ‖DHn‖20 ≥

(8π2 · n2 · q4.5

)2= 64π4 · n4 · q9

n. (A3)

8.3.2 Proof of ConvergenceIn the proof of convergence the following result, which is more precise than [FS05], Lemma 5.6.,and Lemma 3.6.7 respectively, is useful:

Lemma 8.3.4. Let k ∈ N0 and h ∈ Diff∞(T2). Then for all α, β ∈ R we obtain:

)≤ Ck · |||h|||k+1

k+1 · |α− β| ,

where Ck = (k + 1)!.

Proof. As an application of the claim in the proof of Lemma 8.3.1 we observeFact: For any ~a ∈ N2

0 with |~a| = k and i ∈ 1, 2 the partial derivative D~a[h Rα h−1

consists of at most (k + 1)! summands, where each summand is the product of one derivative ofh of order at most k and at most k derivatives of h−1 of order at most k.Furthermore, with the aid of the mean value theorem we can estimate for any multiindex ~a ∈ N2

with |~a| ≤ k and i ∈ 1, 2:∣∣D~a [h]i(Rα h−1 (x1, x2)

)−D~a [h]i

(Rβ h−1 (x1, x2)

)∣∣ ≤ |||h|||k+1 · |α− β| .

Since(hn Rα h−1

)−1 = hn R−α h−1n is of the same form, we obtain in conclusion:

)≤ (k + 1)! · |||h|||k+1 · |||h|||kk · |α− β|≤ (k + 1)! · |||h|||k+1

k+1 · |α− β| .

Under some conditions on the proximity of αn and αn+1 we can prove convergence:

Lemma 8.3.5. We assume

|αn+1 − αn| ≤1

2n · (n+ 1)! · qn · |||Hn|||n+1n+1

. (A1)

Then the diffeomorphisms fn = Hn Rαn+1 H−1n satisfy:

• The sequence (fn)n∈N converges in the Diff∞(T2)-topology to a measure-preserving diffeo-

morphism f .

• We have for every n ∈ N and m ≤ qn+1:

d0 (fm, fmn ) <12n.

Proof. 1. According to our construction it holds hn Rαn = Rαn hn and hence we canapply Lemma 8.3.4 for every k, n ∈ N:

dk (fn, fn−1) = dk(Hn Rαn+1 H−1

n , Hn Rαn H−1n

)≤ Ck · |||Hn|||k+1

k+1 · |αn+1 − αn| .

By the assumptions of this Lemma it follows for every k ≤ n:

dk (fn, fn−1) ≤ dn (fn, fn−1) ≤ Cn · |||Hn|||n+1n+1 ·

12n · Cn · qn · |||Hn|||n+1

<12n. (8.3)

In the next step we show that for arbitrary k ∈ N (fn)n∈N is a Cauchy sequence inDiffk

(T2), i.e. limn,m→∞ dk (fn, fm) = 0. For this purpose, we calculate:

limn→∞

dk (fn, fm) ≤ limn→∞

n∑i=m+1

dk (fi, fi−1) =∞∑

dk (fi, fi−1) . (8.4)

We consider the limit processm→∞, i.e. we can assume k ≤ m and obtain from equations8.3 and 8.4:

limn,m→∞

dk (fn, fm) ≤ limm→∞

∞∑i=m+1

Since Diffk(T2)is complete, the sequence (fn)n∈N converges consequently in Diffk

(T2)for

every k ∈ N. Thus, the sequence converges in Diff∞(T2)by definition.

2. Again with the help of Lemma 8.3.4 we compute for every i ∈ N:

(fmi , f

mi−1

(Hi Rm·αi+1 H−1

i , Hi Rm·αi H−1i

)≤ |||Hi|||1 ·m · |αi+1 − αi| .

Since m ≤ qn+1 ≤ qi we conclude for every i > n:

(fmi , f

mi−1

)≤ |||Hi|||1 ·m ·

12i · (i+ 1)! · qi · |||Hi|||i+1

qi· 1

2i≤ 1

Thus, for every m ≤ qn+1 we get the claimed result:

d0 (fm, fmn ) = limk→∞

d0 (fmk , fmn ) ≤ lim

k→∞

k∑i=n+1

(fmi , f

mi−1

∞∑i=n+1

Remark 8.3.6. By definition qn+1 ≤ qn+1. Hence, the second statement of the previous Lemmaimplies d0

(fqn+1n , f qn+1

2n . Since the number αn+1 was chosen in such a way that f qn+1n = id,

we have d0

(id, f qn+1

2n , which goes to zero as n→∞. Thus, (qn)n∈N is an uniform rigiditysequence of f .

Convergence of (fn)n∈N in Diffωρ(T2)

By Lemma 8.3.2 we can satisfy the requirement A1 if we demand:

|αn+1 − αn| ≤1

2n · (n+ 1)! · qn · ((n+ 2)!)(n+2)n−2·(n+1) · (2πnqn)(n+2)·(n+1)n+1 .

Since |αn+1 − αn| = anqn·qn+1

≤ 1qn+1

this requirement can be met if we demand

qn+1 ≥ ϕ1 (n) · q(n+2)·(n+1)n+1+1n ,

at which ϕ1 (n) := 2n · (n+ 1)! · ((n+ 2)!)(n+2)n−2·(n+1) · (2πn)(n+2)·(n+1)n+1

. Hereby, the otherconditions A3 and A2 are fulfilled.Using qn = qn−1 · qn < q2

n this yields the condition

qn+1 ≥ ϕ1 (n) · q2·((n+2)·(n+1)n+1+1)n .

8.4 Convergence of (fn)n∈N in Diffωρ(T2)

Let ρ > 0 be given.

8.4.1 Properties of the conjugation maps hn and Hn

Regarded as a function hn : C2 → C2 we have

h−1n (θ, r) = h−1

n (θ1 + ı · θ2, r1 + ı · r2)

=(θ1 + ı · θ2 − [nqσn] · (r1 + ı · r2) , r1 + ı · r2 − q2

n · cos (2πqn (θ1 + ı · θ2 − [nqσn] · (r1 + ı · r2)))).

Since for (θ1 + ı · θ2, r1 + ı · r2) ∈ Aρ it holds −ρ < r2 < ρ and −ρ < θ2 < ρ, we can estimate:

infk∈Z

sup(θ,r)∈Aρ

∣∣[h−1n

+ k∣∣ = inf

√(θ1 − [nqσn] · r1 + k)2 + (θ2 − [nqσn] · r2)2

≤ (1 + [nqσn]) ·√

1 + ρ2

infk∈Z

sup(θ,r)∈Aρ

∣∣[h−1n

+ k∣∣ ≤ 2 · q2

n · exp (2πqn · ρ+ 2πqn · [nqσn] · ρ) .

Hence, it holds (note that we demand q1 ≥ ρ0 = ρ+ 1 in equation B2’):∥∥h−1n

∥∥ρ≤ 2 · q2

n · exp (2π · qn · ρ · (1 + [nqσn])) . (8.5)

We introduce the subsequent quantities:

ρn :=∥∥H−1

∥∥ρ, ρ0 := ρ

ρn := 2 · q2n · exp (2π · qn · ρn−1 · (1 + [nqσn])) , ρ0 := ρ+ 1

Using equation 8.5 we obtain

ρn =∥∥h−1

n H−1n−1

∥∥ρ≤∥∥h−1

∥∥ρn−1

≤ 2 · q2n · exp (2πqn · ρn−1 · (1 + [nqσn])) .

We state the following relation between the quantities:

178 Convergence of (fn)n∈N in Diffωρ(T2)

Remark 8.4.1. We have ρn + 1 ≤ ρn for every n ∈ N.

Proof. We prove this result by induction on n:

• Start: n = 1By the above formula we verify:

ρ1 = 2 · q21 · exp (2π · q1 · ρ0 · (1 + [qσ1 ]))

= 2 · q21 · exp (2π · q1 · ρ · (1 + [qσ1 ])) · exp (2π · q1 · (1 + [qσ1 ]))

≥ 2 · q21 · exp (2π · q1 · ρ · (1 + [qσ1 ])) + 1 ≥ ρ1 + 1.

• Induction assumption: The claim holds for n− 1.

• Induction step: n− 1→ nUsing the induction assumption we can compute in the same way:

ρn = 2 · q2n · exp (2π · qn · ρn−1 · (1 + [nqσn]))

≥ 2 · q2n · exp (2π · qn · ρn−1 · (1 + [nqσn])) · exp (2π · qn · (1 + [nqσn]))

≥ 2 · q2n · exp (2π · qn · ρn−1 · (1 + [nqσn])) + 1 ≥ ρn + 1.

We demandqn+1 ≥ ρn. (B2’)

This yields the condition: qn+1 ≥ 2 · q2n · exp (2πqn · ρn−1 · (1 + [nqσn])). And since we demand

ρn−1 ≤ qn we require for the sequence (qn)n∈N:

qn+1 ≥ 2 · q2n · exp

(2π · q2

n · (1 + nqσn)). (B2)

Furthermore, recall that

hn (θ, r) =(θ + [nqσn] · r + [nqσn] · q2

n · cos (2πqnθ) , r + q2n · cos (2πqnθ)

The occurring partial derivatives are∂ [hn]1∂θ

= 1− [nqσn] · 2π · q3n · sin (2πqnθ)

∂ [hn]1∂r

= [nqσn]

∂ [hn]2∂θ

= −2π · q3n · sin (2πqnθ)

∂ [hn]2∂r

Thus, in order to calculate ‖Dhn‖ρ, we have to examine∥∥∥∂[hn]1

∥∥∥ρ:

‖Dhn‖ρ =∥∥∥∥∂ [hn]1

∥∥∥∥ρ

≤ 1 + [nqσn] · 2π · q3n · exp (2π · qn · ρ) ≤ 4π · n · q3+σ

n · exp (2π · qn · ρ) .

Under condition B2’ we can estimate with the aid of Remark 8.4.1:

‖Dhn‖ρn+1 ≤ 4πn · q3+σn · exp (2π · qn · (ρn + 1)) ≤ 4πn · q3+σ

n · exp (2π · qn · ρn)

≤ 4π · n · q3+σn · exp

(4π · q3

n · exp(2π · q2

n · (1 + n · qσn))). (8.6)

In order to be able to apply the criterion for weak mixing 8.2.4 we have the requirement C1:qn+1 ≥ ‖DHn‖20. Using the above calculations we obtain

‖DHn‖0 ≤ 2! · ‖DHn−1‖0 · ‖Dhn‖0 ≤ 2 · q0.5n · 4π · n · q3+σ

n ≤ 8π · n · q3.5+σn .

So we demand (since 0 < σ < 12 )

qn+1 ≥ 64π2 · n2 · q8n. (B3)

8.4.2 Proof of Convergence

As a preparatory result we prove the subsequent Lemma:

Lemma 8.4.2. Let n ∈ N, m ∈ N0. Under the condition∥∥∥hn Rmαn+1 h−1

n −Rmαn∥∥∥ρn−1

2n · ‖Dh1‖ρ1+1 · ... · ‖Dhn−1‖ρn−1+1

we have dρ(fmn , f

mn−1

Proof. We introduce the functions ψn,k := hk+1 ... hn Rmαn+1 h−1

n ... h−1k+1 in case of

1 ≤ k ≤ n− 1 and ψn,n = Rmαn+1. With these we have fmn = h1 ... hk ψn,k h−1

k ... h−11 .

By induction on k ∈ N in the range 1 ≤ k ≤ n− 1 we prove:Fact: Under the condition

∥∥ψn,k H−1k − ψn−1,k H−1

∥∥ρ< 1

2n·‖Dh1‖ρ1+1·...·‖Dhk‖ρk+1we have

∥∥h1 ... hk ψn,k h−1k ... h

−11 − h1 ... hk ψn−1,k h−1

k ... h−11

∥∥ρ<

• Start: k = 1At first we note h−1

1 (Aρ) ⊆ Aρ1 . By our assumption we have

∥∥ψn,1 h−11 − ψn−1,1 h−1

∥∥ρ<

12n · ‖Dh1‖ρ1+1

So a sufficient condition for our claim is given by

‖Dh1‖ρ1+1 ·∥∥ψn,1 h−1

1 − ψn−1,1 h−11

∥∥ρ<

which is satisfied by our requirements.

• Induction hypothesis: The claim holds for 1 ≤ k − 1 ≤ n− 2.

• Induction step: k − 1→ kUsing the induction hypothesis the proximity∥∥ψn,k−1 H−1

k−1 − ψn−1,k−1 H−1k−1

∥∥ρ

=∥∥hk ψn,k H−1

k − hk ψn−1,k H−1k

∥∥ρ

2n · ‖Dh1‖ρ1+1 · ... · ‖Dhk−1‖ρk−1+1

is sufficient to prove the claim. Since∥∥ψn,k H−1

k − ψn−1,k H−1k

∥∥ρ< 1 this is fulfilled if

‖Dhk‖ρk+1 ·∥∥ψn,k H−1

∥∥ρ<

12n · ‖Dh1‖ρ1+1 · ... · ‖Dhk−1‖ρk−1+1

By our assumption on∥∥ψn,k H−1

∥∥ρthe claim is true.

In the opposite direction we show that our assumption on∥∥∥hn Rmαn+1

h−1n −Rmαn

∥∥∥ρn−1

implies

the conditions∥∥ψn,k H−1

∥∥ρ< 1

2n·‖Dh1‖ρ1+1·...·‖Dhk‖ρk+1:

180 Convergence of (fn)n∈N in Diffωρ(T2)

• Start: k = n− 1The condition on

∥∥ψn,n−1 H−1n−1 − ψn−1,n−1 H−1

∥∥ρ≤∥∥∥hn Rmαn+1

h−1n −Rmαn

∥∥∥ρn−1

is exactly the supposition of the Lemma.

• Induction hypothesis: The claim holds for 2 ≤ k ≤ n− 1.

• Induction step: k → k − 1We estimate with the aid of our induction hypothesis∥∥ψn,k−1 H−1

k−1 − ψn−1,k−1 H−1k−1

∥∥ρ

=∥∥hk ψn,k H−1

k − hk ψn−1,k H−1k

∥∥ρ

≤ ‖Dhk‖ρk+1 ·∥∥ψn,k H−1

∥∥ρ

< ‖Dhk‖ρk+1 ·1

2n · ‖Dh1‖ρ1+1 · ... · ‖Dhk‖ρk+1

2n · ‖Dh1‖ρ1+1 · ... · ‖Dhk−1‖ρk−1+1

Hence, the requiremts of the fact are met and the Lemma is proven.

Now we are able to deduce the aimed statement on convergence of (fn)n∈N in the Diffωρ(T2)-

topology:

Lemma 8.4.3. We assume that

|αn+1 − αn| <1

2n · ‖Dh1‖ρ1+1 · ... · ‖Dhn−1‖ρn−1+1 · 4πn · q4+σn · exp

(4πn · q1+σ

n · ρn−1

) .(B1’)

Then the diffeomorphisms fn = Hn Rαn+1 H−1n satisfy:

• The sequence (fn)n∈N converges in the Diffωρ(T2)-topology to a measure-preserving diffeo-

morphism f .

• We have for every n ∈ N and m ≤ qn+1:

d0 (fm, fmn ) <12n.

Proof. At first we introduce for m ∈ N the function

Tm(z) = cos (2π · qn · (z +m · αn+1))− cos (2π · qn · z)

and exploiting the relation cos (x)− cos (y) = 2 · sin(x+y

)· sin

(y−x

)we can estimate for every

s ≥ 0:

‖Tm‖s = ‖2 · sin (π · qn · (2z +m · αn+1)) · sin (π · qn ·m · αn+1)‖s

= 2 ·∥∥∥∥ 1

(eıπ·qn·(2z+m·αn+1) − e−ıπ·qn·(2z+m·αn+1)

)∥∥∥∥s

· |sin (π · qn ·m · αn+1)|

≤ 2 ·∥∥e2πı·qn·z

∥∥s· |sin (π · qn ·m · (αn+1 − αn))|

≤ 2 ·∥∥e2πı·qn·z

∥∥s· π · qn ·m · |αn+1 − αn| ,

where we made use of |sin (x)| ≤ |x| in the last step.Using this map Tm we compute

hn Rmαn+1 h−1

n (θ, r) = hn Rmαn+1

(θ − [nqσn] · r, r − q2

n · cos (2π · qn · (θ − [nqσn] · r)))

= hn(θ +m · αn+1 − [nqσn] · r, r − q2

n · cos (2π · qn · (θ − [nqσn] · r)))

= gn(θ +m · αn+1 − [nqσn] · r, r + q2

n · Tm (θ − [nqσn] · r))

=(θ +m · αn+1 + [nqσn] · q2

n · Tm (θ − [nqσn] · r) , r + q2n · Tm (θ − [nqσn] · r)

Then we have

hn Rmαn+1 h−1

n (θ, r)−Rmαn (θ, r)

=(m · αn+1 −m · αn + [nqσn] · q2

n · Tm (θ − [nqσn] · r) , q2n · Tm (θ − [nqσn] · r)

Thus, we can estimate for m ≤ qn:∥∥∥hn Rmαn+1 h−1

n H−1n−1 −Rmαn H

−1n−1

∥∥∥ρ≤∥∥∥hn Rmαn+1

h−1n −Rmαn

∥∥∥ρn−1

≤ 2 · [nqσn] · q2n · ‖Tm (θ − [nqσn] · r)‖ρn−1

≤ 2 · [nqσn] · q2n · 2 ·

∥∥∥e2πı·qn·(θ−[nqσn]·r)∥∥∥ρn−1

· π · qn ·m · |αn+1 − αn|

≤ 4 · π · n · q3+σn · e4π·n·q1+σ

n ·ρn−1 · qn · |αn+1 − αn|

2n · ‖Dh1‖ρ1+1 · ... · ‖Dhn−1‖ρn−1+1

So the prerequisites of Lemma 8.4.2 are fulfilled and we conclude dρ(fmn , f

mn−1

2n . In thesame spirit as in the proof of Lemma 8.3.5 we can show the convergence of (fn)n∈N and thesecond property.

Now we formulate the next requirement on the sequence (qn)n∈N:

qn+1 ≥ ‖Dh1‖ρ1+1 · ... · ‖Dhn−1‖ρn−1+1 · ‖Dhn‖ρn+1 . (B4’)

By the requirement ‖Dh1‖ρ1+1 · ... · ‖Dhn−1‖ρn−1+1 ≤ qn as well as equation 8.6 condition B4’is satisfied if we demand

qn+1 ≥ 4π · n · q4+σn · exp

(4π · q3

n · exp(2π · q2

n · (1 + n · qσn))). (B4)

Since |αn+1 − αn| = anqn·qn+1

≤ 1qn+1

condition B1’ yields the requirement:

qn+1 ≥ 2n · ‖Dh1‖ρ1+1 · ... · ‖Dhn−1‖ρn−1+1 · 4π · n · q4+σn · exp

(4π · n · q1+σ

n · ρn−1

Using ρn−1 ≤ qn (see condition B2’) and ‖Dh1‖ρ1+1 · ... · ‖Dhn−1‖ρn−1+1 ≤ qn (see conditionB4’) this requirement is satisfied if we demand

qn+1 ≥ 2n · 4π · n · q5+σn · exp

(4π · n · q2+σ

). (B1)

By collecting all the prerequisites B1, B2, B3, B4 on the sequence (qn)n∈N we demand:

qn+1 ≥ 2n · 64π2 · n2 · q8n · exp

(4π · n · q3

n · exp(2π · q2

n · (1 + n · qσn))).

Since qn+1 = qn+1qn

, qn = qn−1 · qn ≤ q2n and 0 < σ < 1

2 we obtain the following sufficient conditionon the growth rate of the rigidity sequence (qn)n∈N:

qn+1 ≥ 2n · 64π2 · n2 · q14n · exp

(4π · n · q6

n · exp(2π · q4

n · (1 + n · qn))).

182 Proof of weak mixing

8.5 Proof of weak mixing

By the same approach as in [FS05] we want to apply Proposition 8.2.4. For this purpose,we introduce a sequence (mn)n∈N of natural numbers mn ≤ qn+1 in subsection 8.5.1 and asequence (ηn)n∈N of standard partial decompositions in subsection 8.5.2. Finally, we show that

the diffeomorphism Φn := φn Rmnαn+1φ−1

, 0, 1n

)-distributes the elements of this partition.

8.5.1 Choice of the mixing sequence (mn)n∈N

By condition A3 resp. B3 our chosen sequence (qn)n∈N satisfies

qn+1 ≥ q8n. (C2)

Define

∣∣∣∣m · pn+1

qn+1− 1

2 · qn+

∣∣∣∣ ≤ qnqn+1

m ≤ qn+1 : m ∈ N, inf

∣∣∣∣m · qn · pn+1

qn+1− 1

∣∣∣∣ ≤ q2n

Lemma 8.5.1. The set Mn :=

∣∣∣m · qn·pn+1qn+1

− 12 + k

∣∣∣ ≤ q2n

Proof. The number αn+1 was constructed by the rule pn+1qn+1

= pnqn− an

qn·qn+1, where an ∈ Z,

1 ≤ an ≤ qn, i.e. pn+1 = pn · qn+1 − an and qn+1 = qn · qn+1. So qn·pn+1qn+1

= pn+1qn+1

and the setj · qn·pn+1

qn+1: j = 1, 2, ..., qn+1

contains qn+1

gcd(pn+1,qn+1) different equally distributed points on

S1. Hence, there are at least qn+1qn

= qn+1q2n

different such points and so for every x ∈ S1 there is aj ∈ 1, ..., qn+1 such that

infk∈Z

∣∣∣∣x− j · qn · pn+1

qn+1+ k

∣∣∣∣ ≤ q2n

In particular, this is true for x = 12 .

∆n =(mn ·

qn+1− 1

2 · qn

By the above construction of mn it holds: |∆n| ≤ qnqn+1

. By C2 we get:

|∆n| ≤1q7n

8.5.2 Standard partial decomposition ηn

In the following we will consider the set

Bn =2qn−1⋃k=0

2qn− 1

2q1.5n

12q1.5n

Proof of weak mixing 183

Our horizontal intervals belonging to the partial decomposition ηn will lie outside Bn. To showthat Φn

, 0, 1n

)-distributes the elements of this partition we will need the subsequent result

similar to the concept of “uniformly stretching” from [Fa02]:

Lemma 8.5.3. Let I = [a, b] ⊂ R be an interval and ψ : I → R be a strictly monotonicC2-function. Furthermore, we denote J := [infx∈I ψ (x) , supx∈I ψ (x)]. If ψ satisfies

supx∈I|ψ′′ (x)| · λ (I) ≤ ε · inf

x∈I|ψ′ (x)| ,

then for any interval J ⊆ J we have∣∣∣∣∣∣λ(I ∩ ψ−1

λ (I)−λ(J)

λ (J)

∣∣∣∣∣∣ ≤ ε ·λ(J)

λ (J).

Proof. We consider the case that ψ is strictly increasing (the proof in the decreasing case isanalogous), which implies ψ′ > 0 (due to our assumption supx∈I |ψ′′ (x)|·λ (I) ≤ ε·infx∈I |ψ′ (x)|)and J = [ψ (a) , ψ (b)].Let J = [ψ (c) , ψ (d)], where a ≤ c ≤ d ≤ b. By the mean value theorem there are ξ1 ∈ [a, b] andξ2 ∈ [c, d], such that ψ (b)−ψ (a) = ψ′ (ξ1) ·(b− a) resp. ψ (d)−ψ (c) = ψ′ (ξ2) ·(d− c). Applyingthe mean value theorem on ψ′ gives ξ3 ∈ [a, b] with |ψ′ (ξ2)− ψ′ (ξ1)| = |ψ′′ (ξ3)| · |ξ2 − ξ1|. Thenwe have:

|ψ′ (ξ1)− ψ′ (ξ2)| ≤ supx∈[a,b]

|ψ′′ (x)| · |b− a| = supx∈[a,b]

|ψ′′ (x)| ·λ (I) ≤ ε · infx∈[a,b]

|ψ′ (x)| ≤ ε · |ψ′ (ξ2)| .

Hereby, we obtain: ∣∣∣∣ψ′ (ξ1)ψ′ (ξ2)

− 1∣∣∣∣ ≤ ε.

Since ψ′ > 0 this implies 1− ε ≤ ψ′(ξ1)ψ′(ξ2) ≤ 1 + ε and thus:

λ(I ∩ ψ−1

λ (I)=d− cb− a

=ψ′ (ξ1) · (ψ (d)− ψ (c))ψ′ (ξ2) · (ψ (b)− ψ (a))

≤ (1 + ε) ·λ(J)

λ (J).

This impliesλ(I ∩ ψ−1

λ (I)−λ(J)

λ (J)≤ ε ·

λ (J).

Similarly we obtain the estimate from below and the claim follows.

By the explicit definitions of the conjugation maps the transformation Φn = φn Rmnαn+1φ−1

has a lift of the formΦn (θ, r) = (θ +mn · αn+1, r + ψn (θ))

withψn (θ) = q2

n · (cos (2π (qnθ +mnqnαn+1))− cos (2πqnθ)) .

We examine this map ψn:

184 Proof of weak mixing

Lemma 8.5.4. The map ψn satisfies:

infθ∈T\Bn

|ψ′n (θ)| ≥ q2.5n and sup

θ∈T\Bn|ψ′′n (θ)| ≤ 9π2q4

Proof. Using the relation cos (2πqnθ + π) = − cos (2πqnθ) and the map

σn (θ) = q2n ·(

cos (2πqn (θ +mnαn+1))− cos(

)))we can write ψn (θ) = −2q2

n · cos (2πqnθ) + σn (θ). For this map σn we compute:

σ′n (θ) = 2π · q3n ·(− sin (2πqn (θ +mn · αn+1)) + sin

(2πqn

Applying the mean value theorem on the function ϕa (ξ) := −2π · q3n · sin (2πqnξ) we obtain:

|σ′n (θ)| ≤ 2πq3n ·max

ξ∈T|−2πqn · cos (2πqnξ)| ·∆n ≤ (2π)2 · q4

n ·qnqn+1

≤ (2π)2 · q4n ·

On the other hand, on the set T \Bn it holds:

infθ∈T\Bn

|sin (2πqnθ)| = infθ=θ+ k

2qn,k∈Z,θ∈

2q1.5n, 12qn− 1

2q1.5n

] |sin (2πqnθ)|

= infθ∈[

12q1.5n

, 12qn− 1

2q1.5n

]∣∣∣sin(2πqnθ

)∣∣∣= infθ∈[

12q1.5n

, 14qn

]∣∣∣sin(2πqnθ

)∣∣∣ ≥ 12· 2πqn ·

12q1.5n

2· q−0.5n > q−0.5

with the aid of the estimate sin(x) ≥ 12x for x ∈

[0, π2

]. Thus, we have:

infθ∈T\Bn

|ψ′n (θ)| ≥ 4πq3n · inf

θ∈T\Bn|sin (2πqnθ)| − sup

θ∈T\Bn|σ′n (θ)| ≥ 4πq3

n · q−0.5n − 1 ≥ q2.5

In order to estimate ψ′′n we compute

σ′′n (θ) = (2π)2 · q4n ·(− cos (2πqn (θ +mn · αn+1)) + cos

(2πqn

)))and use the mean value theorem on ϕb (ξ) := − (2π)2 · q4

n · cos (2πqnξ):

|σ′′n (θ)| ≤ (2π)2 · q4n ·max

ξ∈T|2πqn · sin (2πqnξ)| ·∆n ≤ (2π)3 · q5

n ·qnqn+1

Then we obtain:

supθ∈T\Bn

|ψ′′n (θ)| ≤ supθ∈T\Bn

∣∣∣2 · (2π)2 · q4n · cos (2πqnθ)

∣∣∣+ supθ∈T\Bn

|σ′′n (θ)| ≤ 8π2 · q4n + 1 ≤ 9π2 · q4

Proof of weak mixing 185

The aimed standard partial decomposition ηn of T2 is defined in the following way:Let ηn =

be the partial partition of T \Bn consisting of all the disjoint intervals In,i such

that there is k ∈ Z: ψn(In,i

)= k + [0, 1). Then we define

ηn =I × r : I ∈ ηn, r ∈ T

Note that we have πr (Φn (In)) = T for every In ∈ ηn.

Lemma 8.5.5. For any partition element In ∈ ηn we have λ(In

)≤ q−2.5

n . Moreover, it holds:ηn → ε.

Proof. By Lemma 8.5.4 we have infθ∈T\Bn |ψ′n (θ)| ≥ q2.5n . Therefore, λ

)≤ q−2.5

n for anyIn ∈ ηn. Hence, the length of the elements of ηn goes to zero as n→∞. Thus, in order to proveηn → ε we have only to check that the total measure of the partial decompositions ηn goes to 1as n → ∞. Since the elements of ηn are contained in T \ Bn and have to satisfy the additionalrequirement ψn

)= k + [0, 1) for k ∈ Z, on both sides around the set Bn there is an area

without partition elements. So the total measure of ηn can be estimated as follows:∑In∈ηn

)≥ 1− λ (Bn)− 2 · 2qn · max

In∈ηnλ(In

)≥ 1− 2qn ·

(q−1.5n + 2q−2.5

)> 1− 3 · q−0.5

and this approaches 1 as n→∞.

8.5.3 Application of the criterion for weak mixingIn order to apply the criterion for weak mixing we check that the constructed diffeormorphismf = limn→∞ fn fulfil the requirements:

• By Lemma 8.3.5, 2., resp. 8.4.3, 2.: d0 (fmn , fmnn ) < 12n , because mn ≤ qn+1 by definition.

• Because of the requirement A3 resp. B3 on the number qn we have C1.

• By Lemma 8.5.5 we have ηn → ε and the length of the horizontal interval is at most q−2.5n .

Finally, the next Lemma proves the last remaining property:

Lemma 8.5.6. Let In ∈ ηn. Then Φn(

, 0, 1n

)-distributes In.

Proof. The partial partition ηn was chosen in such a way that πr (Φn (In)) = T. Hence, δ canbe taken equal to 0.Using the form of the lift Φn (θ.r) = (θ +mn · αn+1, r + ψn (θ)) we observe that Φn (In) iscontained in the vertical strip

(In +mn · αn+1

)× T for every In ∈ ηn. Since the length of In is

at most 1q2.5n

< 1nqn

by Lemma 8.5.5, we can take γ = 1nqn

Recall that an element In ∈ ηn has the form I × r for some r ∈ T and an interval I ∈ ηn

contained in T \ Bn with λ(I)≤ q−2.5

n (see Lemma 8.5.5). Then Lemma 8.5.4 implies theestimate

supθ∈I |ψ′′n (θ)|infθ∈I |ψ′n (θ)|

· λ(I)≤ 9π2q4

· q−2.5n =

186 Proof of the Corollaries

Then we can apply Lemma 8.5.3 on ψn and I with ε = 1n . Moreover, we note that for any

J ⊆ J = T the fact Φn (θ, r) ∈ T × J is equivalent to ψn (θ) ∈ J − r :=j − r : j ∈ J

Combining these both results we conclude:∣∣∣∣∣∣λ(In ∩ Φ−1

(T× J

))λ (In)

−λ(J)

λ (J)

∣∣∣∣∣∣ =

∣∣∣∣∣∣λ(I ∩ ψ−1

(J − r

))λ(I) −

λ (J)

∣∣∣∣∣∣ ≤ 1n·λ(J)

λ (J).

Thus, we can choose ε = 1n in the definition of (γ, δ, ε)-distribution.

Then we can apply Proposition 8.2.4 and conclude that the constructed diffeomorphisms areweak mixing.

8.6 Proof of the Corollaries

By some estimates we show that the assumptions of the Corollaries are enough to fulfil therequirements of the corresponding theorem.

8.6.1 Proof of Corollary C

We recall the assumptions q1 ≥ 108π and qn+1 ≥ qqnn on the sequence (qn)n∈N.Claim: Under these assumptions the numbers qn satisfy qn ≥ 4π · n · (n+ 2)n+2.Proof with the aid of complete induction:

• Start n = 1: q1 ≥ 108π = 4π · (1 + 2)1+2

• Assumption: The claim is true for n ∈ N.

• Induction step n→ n+ 1: We calculate

qn+1 ≥ qqnn ≥(

4π · n · (n+ 2)n+2)4π·n·(n+2)n+2

≥ 4π · (n · (n+ 2))π·n·(n+2)n+2

(n+ 2)n+1)3π·n·(n+2)n+2

≥ 4π · (n+ 1) · (n+ 3)n+3,

where we used the relation (n+ 2)n+1 ≥ n+ 3 in the last step.

Hereby, we have

qn+1 ≥ qqnn ≥ q4π·n·(n+2)n+2

n = qπ·n·(n+2)n+2

n · q3π·n·(n+2)n+2

4π · n · (n+ 2)n+2)π·n·(n+2)n+2

· q2·(n+2)n+2

≥ (4π · n · (n+ 2)!)(n+2)n+2

· q2·((n+2)·(n+1)n+1+1)n

≥ 2n · (2π · n · (n+ 2)!)(n+2)n+1·(n+1) · q2·((n+2)·(n+1)n+1+1)n

≥ 2n · (2πn)(n+2)·(n+1)n+1

· (n+ 2)! · ((n+ 2)!)(n+2)n−1·(n+1) · q2·((n+2)·(n+1)n+1+1)n

≥ ϕ1 (n) · q2·((n+2)·(n+1)n+1+1)n .

Hence, the requirement of Theorem G is met.

Proof of the Corollaries 187

8.6.2 Proof of Corollary D

Let (qn)n∈N be a sequence satisfying q1 ≥ (ρ+ 1) · 27 · π2 and qn+1 ≥ q15n · exp

(q7n · exp

Again we start with a proof by complete induction:Claim: The numbers qn satisfy qn ≥ 2n+6 · n2 · π2.Proof:

• Start n = 1: By assumption we have: q1 ≥ (ρ+ 1) · 27 · π2 ≥ 21+6 · 12 · π2.

• Assumption: The claim is true for n ∈ N.

• Induction step n→ n+ 1: We estimate

qn+1 ≥ q15n ·exp

(q7n · exp

))≥(2n+6 · n2 · π2

)15 ·exp(q7n · exp

))≥ 2n+7 ·(n+ 1)2 ·π2.

Then we have:

qn+1 ≥ q15n · exp

(q7n · exp

))≥ 2n+6 · n2 · π2 · q14

n · exp(2n+6 · n2 · π2 · q6

n · exp(2n+6 · n2 · π2 · q5

))≥ 2n · n2 · 64π2 · q14

n · exp(4π · n · q6

n · exp(2π · 2 · n · q5

Thus, the condition of Theorem H is fulfilled.

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Abstract

In this thesis we present a multitude of new constructions of diffeomorphisms with variousspecific ergodic, topological and spectral properties. These constructions are based on the “Con-jugation by approximation”-method developed by D.V. Anosov and A. Katok in [AK70]. In fact,on every smooth compact connected manifold M of dimension m ≥ 2 admitting a non-trivialcircle action S = Stt∈S1 preserving a smooth volume ν this method enables the construction ofsmooth diffeomorphisms with particular ergodic properties or non-standard smooth realizationsof measure preserving systems. Moreover, it allows to deduce results on the genericity of designedproperties in A (M) = h St h−1 : t ∈ S1, h ∈ Diff∞ (M,ν)

One main issue is the construction of weak mixing diffeomorphisms preserving a measurableRiemannian metric in the restricted space Aα (M) = h Sα h−1 : h ∈ Diff∞ (M,ν)

for agiven Liouvillean number α ∈ S1. In the case of the m-dimensional torus Tm, m ≥ 2, we examinethe existence of such diffeomorphisms with a prescribed number of ergodic invariant measures, inparticular uniquely ergodic ones. Likewise, we construct weak mixing diffeomorphisms preservinga measurable Riemannian metric on the manifolds with boundary D2 as well as S1 × [0, 1] withthe minimal number of three ergodic invariant measures. All these aforementioned constructionsare supplemented by structure theorems concerning the denseness or genericity of the particularset of constructed diffeomorphisms in Aα (M).Furthermore, we answer a question of B. Fayad and A. Katok ([FK04], Problem 7.11.) aboutthe existence of smooth diffeomorphisms admitting a special type of periodic approximation(namely a good approximation of type (h, h+ 1)) and possessing specific spectral propertiesaffirmatively. These mentioned spectral properties are a homogeneous spectrum of multiplicity2 for the Cartesian square and a maximal spectral type disjoint with its convolutions.In addition, we start to examine the problem of uniformly rigid and simultaneously weak mixingmaps, which is a current research topic in measurable as well as topological dynamics, in thesmooth and beyond that even in the real-analytic category: Under sufficient conditions on thegrowth rate of the rigidity sequence we are able to construct uniformly rigid and weak mixingreal-analytic as well as C∞-diffeomorphisms on T2.

Zusammenfassung

In der vorliegenden Arbeit wird eine Vielzahl an Konstruktionen von Diffeomorphismen mitverschiedenen ergodischen, topologischen sowie spektraltheoretischen Eigenschaften durchge-führt. Eine zentrale Rolle hierbei spielt die “Konstruktion durch Konjugation”-Methode vonD.V. Anosov und A. Katok ([AK70]), die sich auf beliebige glatte, kompakte und zusammen-hängende Mannigfaltigkeiten M von Dimension mindestens 2 mit nichttrivialer KreisoperationS = Stt∈S1 , die ein glattes invariantes Volumen ν besitzt, anwenden lässt. Dabei erzeugt mandurch geschickte Wahl von Parametern sukzessive eine Folge von Konjugationsabbildungen so,dass die Konjugation eines betrachteten Diffeomorphismus, der zur Kreisoperation gehört, gegeneine Abbildung mit angestrebten Merkmalen konvergiert. Mit ihrer Hilfe lassen sich zum BeispielDiffeomorphismen mit verschiedenen ergodischen Eigenschaften, deren Existenz zuvor unbekanntwar, und glatte Versionen von maßerhaltenden Abbildungen konstruieren. Des Weiteren ist manin der Lage, Aussagen über die Generizität von konstruierten Eigenschaften zu treffen.

Ein besonderes Augenmerk dieser Dissertation liegt auf der Konstruktion von schwach mi-schenden Diffeomorphismen, die eine invariante messbare Riemannsche Metrik zulassen, im re-stringierten Raum Aα (M) = h Sα h−1 : h ∈ Diff∞ (M,ν)

für vorgegebene Liouvillezahlα ∈ S1. Zudem wird im Falle des m-dimensionalen Torus Tm, m ≥ 2, die Existenz von solchenDiffeomorphismen mit vorgegebener Anzahl an ergodischen invarianten Maßen, insbesondere voneindeutig ergodischen Abbildungen, untersucht. Ebenso werden auf den Mannigfaltigkeiten mitRand D2 sowie S1 × [0, 1] schwach mischende Diffeomorphismen mit invarianter messbarer Rie-mannscher Metrik und der in diesen Fällen minimalen Anzahl von drei ergodischen invariantenMaßen konstruiert. An die Konstruktionen schließen jeweils Strukturaussagen in Aα (M) an, dieim Sinne des Baireschen Satzes die jeweilige Menge der konstruierten Diffeomorphismen charak-terisieren.Des Weiteren wird die von B. Fayad und A. Katok in [FK04], Problem 7.11., formulierte Fragenach der Existenz von Diffeomorphismen, die eine besondere Art von periodischer Approxima-tion (nämlich vom Typ (h, h+ 1)) zulassen sowie spezielle spektraltheoretische Eigenschaften(ein homogenes Spektrum der Multiplizität 2 für das kartesische Produkt sowie ein maximalerSpektraltyp disjunkt zu seinen Faltungen) besitzen, positiv beantwortet.In dieser Arbeit wird zudem die Untersuchung von gleichmäßig rigiden und gleichzeitig schwachmischenden Abbildungen, die ein aktuelles Forschungsthema sowohl der maßtheoretischen alsauch der topologischen Dynamik darstellen, im Gebiet der glatten Dynamik begonnen. So wirdunter hinreichenden Wachstumsbedingungen an die Rigiditätsfolge die Existenz von schwachmischenden und bezüglich der vorgegebenen Folge gleichmäßig rigiden reell-analytischen sowieC∞-Diffeomorphismen auf T2 gezeigt.

LebenslaufPhilipp Christopher Kundegeboren am 11.07.1985 in HamburgE-Mail: philippkunde@gmx.de

Ausbildung

Oktober 2010 - September 2014: Promotionsstudium am Fachbereich Mathe-matik, Universität Hamburg, Betreuer: Prof. Dr. R. Lauterbach.

Oktober 2005 - September 2010: Studium der Mathematik an der UniversitätHamburg. Diplom am Fachbereich Mathematik mit der Gesamtnote “sehr gut”.

Juni 2005: Abitur am Gymnasium Rahlstedt, Hamburg.

Wissenschaftliche Anstellungen

Oktober 2010 - heute: Wissenschaftlicher Mitarbeiter an der Universität Ham-burg, Lehre (Übungen) an der Technischen Universität Hamburg-Harburg und amFachbereich Mathematik der Universität Hamburg.

April 2010 - Juli 2010: Studentische Hilfskraft am Fachbereich Mathematik derUniversität Hamburg.

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