Tolerance stability conjecture revisited

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  • Topology and its Applications 131 (2003)


    We pof a coToleran1971]. 2002

    MSC: p


    1. Int

    Wesystemby TaStabilnot co(withdensemetric



    0166-86doi:10.1Tolerance stability conjecture revisited Marcin Mazur

    Uniwersytet Jagiellonski, Instytut Matematyki, Reymonta 4, 30-059 Krakw, Poland

    Received 8 July 2002


    rove that the strong tolerance stability property is generic in the space of all homeomorphismsmpact smooth manifold withC0 topology. Actually, it partially resolves Zeemans and Takensce Stability Conjecture [F. Takens, in: Lecture Notes in Math., Vol. 197, Springer-Verlag,

    Elsevier Science B.V. All rights reserved.

    rimary 37C20; secondary 37B35, 37C50

    ds: Generic property; (Discrete) dynamical system; (Strong) tolerance stability; Chain recurrence;ing


    investigate the strong tolerance stability of homeomorphisms (discrete dynamicals) of a compact smooth manifold. The notion of tolerance stability was introduced

    kens in [13] together with the topological formulation of Zeemans Toleranceity Conjecture which says that for a set D H(M), equipped with the topologyarser than that of H(M), the set of all f D having the tolerance stability propertyrespect to D) is residual in D, i.e., it includes a countable intersection of open andsubsets of D. Here H(M) denotes the space of all homeomorphisms of a compactspace M with C0 topology.

    [15] White presented the counterexample showing that the set D cannot be chosenrily. There were also proved several results in the direction of Zeemans Tolerance

    pported by KBN Grant no. 5P03A01620.mail address: (M. Mazur).

    41/02/$ see front matter 2002 Elsevier Science B.V. All rights reserved.016/S0166-8641(02)00261-4

  • 34 M. Mazur / Topology and its Applications 131 (2003) 3338

    Stability Conjecture (see [3,6,8,11,14]). In this paper we restrict our investigation to thecase when the set D is equal toH(M). To the authors best knowledge such a problem wasstudied so far only by Odani [8], who showed that for a compact (smooth) manifold Mof thestabiliof anof a mthe shhomeoelemeresidu


    2. Ba

    Letall hom



    Definexistseach gf in H

    Definthere eand ea

    ObMoreo(see [8dimension at most 3 the set of all homeomorphisms satisfying the strong tolerancety condition is residual inH(M). Our aim is an extension of this theorem to the casearbitrary dimension. The proof is based on the technique of a handle decompositionanifold, proposed by Pilyugin and Plamenevskaya [12] for proof of C0 genericity ofadowing property. Additionally, applying this method we prove that for a C0 genericmorphism the chain recurrent set is a Cantor set. We recall that the property P of

    nts of a topological space X is called generic if the set of all x X satisfying P isal in X.e results of this paper are part of authors Ph.D. Thesis [7] and have already beennced (without proofs) in [9].

    sic definitions

    M be a compact metric space with the metric d and let H(M) denote the space ofeomorphisms of M equipped with the metric 0, defined by

    0(f, g) :=max{

    maxxM d

    (f (x), g(x)

    ),maxxM d

    (f1(x), g1(x)


    induces C0 topology and makes H(M) a complete metric space. We say that ance {xi}iZ M is -traced (-set-traced) by the orbit Of (x) := {f i(x)}iZ of amorphism f H(M) if d(f i(x), xi) for every i Z ((ClOf (x), Cl{xi | i ). Here denotes the Hausdorff metric induced by d .w, following [8,13], we recall the notions of tolerance stability and strong tolerancety.

    ition 1. A homeomorphism f H(M) is tolerance stable if for every > 0 there > 0 such that for every g U(f ) each f -orbit is -set-traced by some g-orbit and-orbit is -set-traced by some f -orbit. Here U(f ) denotes the -neighborhood of(M).

    ition 2. A homeomorphism f H(M) is strongly tolerance stable if for every > 0xists > 0 such that for every g U(f ) each f -orbit is -traced by some g-orbitch g-orbit is -traced by some f -orbit.

    viously, the strong tolerance stability property implies the tolerance stability one.ver, it is also stronger than the shadowing property in the case when M is a manifold]).

  • M. Mazur / Topology and its Applications 131 (2003) 3338 35

    3. Handle decomposition

    In this section we repeat the relevant material regarding a handle decomposition of amanifdirect

    Letthe Riat a aDmr an


    x = (xA s

    is callhold:

    (1) th(2) th


    (3) ea(4) fo




    We sa

    A sub



    (1) Mold (a more complete theory may be found in [12]). We also make the first step in theion of the proof of the main result (see Remark 3).M be a compact n-dimensional smooth manifold with the metric d induced by

    emanian structure. We will denote by Dmr (a) the closed ball in Rm with the centernd the radius r (to simplify notation balls centered at the origin will be written asd the unit ball as Dm) and by Sm the m-dimensional unit sphere (S1 := ). For

    nience we consider the maximum norm on Rm, i.e., x = maxi{1,...,m} |xi | for all1, . . . , xm) Rm.equence of sets

    M: =M1 M0 Mn =Med a handle decomposition of M if for any m {0, . . . , n} the following conditions

    e set Mm is n-dimensional submanifold with boundary;e set Cl(Mm \Mm1) is a disjoint union of m-handles, i.e., sets homeomorphic tom Dnm;ch m-handle is attached to the boundary of Mm1 by the image of Sm1 Dnm;r each m-handle H , the image of

    iH :Dm Dnm M,

    ere exists an embeddingH :D

    m Dnm2 Mch that:) H |DmDnm = iH ,) H (Dm Dnm2 ) Mm1 = H (Sm1 Dnm2 ),) if G is another m-handle then the widened m-handles H := H (Dm Dnm2 )

    and G := G(Dm Dnm2 ) are disjoint.

    y that a homeomorphism f H(M) preserves a handle decompositionM iff (Mm) IntMm for all m {0, . . . , n}.set V of a handle H = iH (Dm Dnm) of the formV = iH

    (D1r1(a1) D1rn(an)


    r1, . . . , rn (0,1) and a1, . . . , an (1,1), will be called a cube in H . > 0 be fixed. By B we denote the set of all homeomorphisms f H(M) forwe can find a handle decompositionMf satisfying the following conditions:

    f has the diameter less than , i.e.,|Mf | :=max{diamH |H is a handle ofM}< ;

  • 36 M. Mazur / Topology and its Applications 131 (2003) 3338

    (2) f preservesMf ;(3) if {Hi}iZ is a sequence of handles with the property that f (Hi)Hi+1 = then there

    exists a corresponding sequence of cubes {Vi}iZ such that Vi Hi , f (Vi) Hi+1an

    Now,to BMg =

    Remaas Lemsubsetin H(M

    4. Ma



    Proofit suffihomeo


    (i) fo

    (ii) g



    i=fi (Vi) = .

    let B be the subset of B defined as follows: a homeomorphism f B belongsif there exists > 0 such that for each g U(f ) the conditions (1)(3) hold withMf (in particular g B).

    rk 3. By the results of [12], especially the definition of the set A H(M) as wellmas 1 and 4 stated there, it is easily seen that the set B :=n=1 B 1

    nis a residual

    of H(M) (note that B contains the set A which was proved to be open and dense)).

    in result

    M be a compact smooth manifold with the metric d induced by the Riemanianre.

    em 4. A generic f H(M) has the strong tolerance stability property.

    . Fix > 0. Since the set B , defined in the previous section, is residual in H(M)ces to prove that for every f B there exists > 0 such that for any pair ofmorphisms g1, g2 U(f ) each g1-orbit is -traced by some g2-orbit.

    oose f B . Let M =Mf be a corresponding handle decomposition of M .there is only finite number of handles in M we can find > 0 such that eachmorphism g U(f ) satisfies the following conditions:

    r every pair of handles (H,G) ofMg(H)G= f (H)G= dist(g(H),G)> 2;

    B withMg =M.

    y M and g1, g2 U(f ). Let Hi denote a handle ofM containing gi1(y) (i Z).y dist(g2(Hi),Hi+1) 2 and, in consequence,g2(Hi)Hi+1 = .this it follows that there exists a sequence of cubes {Vi}iZ such that Vi Hi andi=i=

    gi2 (Vi) = .

  • M. Mazur / Topology and its Applications 131 (2003) 3338 37

    Let x be an arbitrarily chosen point of the above set. Then gi2(x) Vi Hi and sod(gi2(x), g

    i1(y)) < for every i Z (we recall that |M|< ).

    By the above, we conclude that each g1-orbit is -traced by some g2-orbit, whichcompl

    5. Ge



    Proofthat fowith xnonem


    Takfor anbe fou


    for eaare sin

    RemaCR(fsays th


    [1] E.So


    19[3] P.[4] Metes the proof.

    neric asymptotic behavior

    M be a compact smooth manifold with the metric d induced by the Riemanianre. In this section we apply the technique of a handle decomposition to prove theing theorem, which extends some recent Hurleys result [5] to the case of an arbitrarysion. A different and independent proof one can find in [1].

    em 5. For a generic f H(M) the chain recurrent set CR(f ) is a Cantor set.

    . We recall that the chain recurrent set CR(f ) is a collection of all such points p Mr each > 0 there is a -chain through p, i.e., a finite sequence x0, x1, . . . , xn (n 1)0 = xn = p and with d(f (xj1), xj ) for every j {1, . . . , n}. It is a compact,pty and invariant set.the corollary to Theorem 6.1 in [5], it remains to show that CR(f ) is totallynected for a generic f H(M).e > 0 and f B . LetM=Mf be a corresponding handle decomposition. Since

    y point p M lying on the boundary of some handle ofM no -chain through p cannd when is