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Topology and its Applications 131 (2003) 33–38

www.elsevier.com/locate/topo

Tolerance stability conjecture revisited✩

Marcin Mazur

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

Received 8 July 2002

Abstract

We prove that the strong tolerance stability property is generic in the space of all homeomorof a compact smooth manifold withC0 topology. Actually, it partially resolves Zeeman’s and TakeTolerance Stability Conjecture [F. Takens, in: Lecture Notes in Math., Vol. 197, Springer-V1971]. 2002 Elsevier Science B.V. All rights reserved.

MSC: primary 37C20; secondary 37B35, 37C50

Keywords: Generic property; (Discrete) dynamical system; (Strong) tolerance stability; Chain recurrence;Shadowing

1. Introduction

We investigate the strong tolerance stability of homeomorphisms (discrete dynasystems) of a compact smooth manifold. The notion of tolerance stability was introby Takens in [13] together with the topological formulation of Zeeman’s ToleraStability Conjecture which says that for a setD ⊂ H(M), equipped with the topolognot coarser than that ofH(M), the set of allf ∈ D having the tolerance stability proper(with respect toD) is residual inD, i.e., it includes a countable intersection of open adense subsets ofD. HereH(M) denotes the space of all homeomorphisms of a commetric spaceM with C0 topology.

In [15] White presented the counterexample showing that the setD cannot be chosearbitrarily. There were also proved several results in the direction of Zeeman’s Tole

✩ Supported by KBN Grant no. 5P03A01620.E-mail address: mazur@im.uj.edu.pl (M. Mazur).

0166-8641/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0166-8641(02)00261-4

34 M. Mazur / Topology and its Applications 131 (2003) 33–38

Stability Conjecture (see [3,6,8,11,14]). In this paper we restrict our investigation to thewasldrance

aseosition

y

been

f

t a

nce

one.

case when the setD is equal toH(M). To the author’s best knowledge such a problemstudied so far only by Odani [8], who showed that for a compact (smooth) manifoMof the dimension at most 3 the set of all homeomorphisms satisfying the strong tolestability condition is residual inH(M). Our aim is an extension of this theorem to the cof an arbitrary dimension. The proof is based on the technique of a handle decompof a manifold, proposed by Pilyugin and Plamenevskaya [12] for proof ofC0 genericity ofthe shadowing property. Additionally, applying this method we prove that for aC0 generichomeomorphism the chain recurrent set is a Cantor set. We recall that the propertP ofelements of a topological spaceX is called generic if the set of allx ∈ X satisfyingP isresidual inX.

The results of this paper are part of author’s Ph.D. Thesis [7] and have alreadyannounced (without proofs) in [9].

2. Basic definitions

LetM be a compact metric space with the metricd and letH(M) denote the space oall homeomorphisms ofM equipped with the metricρ0, defined by

ρ0(f, g) := max{maxx∈M d

(f (x), g(x)

),maxx∈M d

(f−1(x), g−1(x)

)},

which inducesC0 topology and makesH(M) a complete metric space. We say thasequence{xi}i∈Z ⊂M is ε-traced (ε-set-traced) by the orbitOf (x) := {f i(x)}i∈Z of ahomeomorphismf ∈ H(M) if d(f i(x), xi) � ε for everyi ∈ Z (ρ(ClOf (x), Cl{xi | i ∈Z})� ε). Hereρ denotes the Hausdorff metric induced byd .

Now, following [8,13], we recall the notions of tolerance stability and strong tolerastability.

Definition 1. A homeomorphismf ∈ H(M) is tolerance stable if for everyε > 0 thereexistsδ > 0 such that for everyg ∈Uδ(f ) eachf -orbit isε-set-traced by someg-orbit andeachg-orbit is ε-set-traced by somef -orbit. HereUδ(f ) denotes theδ-neighborhood off in H(M).

Definition 2. A homeomorphismf ∈ H(M) is strongly tolerance stable if for everyε > 0there existsδ > 0 such that for everyg ∈ Uδ(f ) eachf -orbit is ε-traced by someg-orbitand eachg-orbit is ε-traced by somef -orbit.

Obviously, the strong tolerance stability property implies the tolerance stabilityMoreover, it is also stronger than the shadowing property in the case whenM is a manifold(see [8]).

M. Mazur / Topology and its Applications 131 (2003) 33–38 35

3. Handle decomposition

n of ain the

as

to

In this section we repeat the relevant material regarding a handle decompositiomanifold (a more complete theory may be found in [12]). We also make the first stepdirection of the proof of the main result (see Remark 3).

Let M be a compactn-dimensional smooth manifold with the metricd induced bythe Riemanian structure. We will denote byDmr (a) the closed ball inRm with the centerat a and the radiusr (to simplify notation balls centered at the origin will be writtenDmr and the unit ball asDm) and bySm them-dimensional unit sphere (S−1 := ∅). Forconvenience we consider the maximum norm onR

m, i.e.,‖x‖ = maxi∈{1,...,m} |xi | for allx = (x1, . . . , xm) ∈ R

m.A sequence of sets

M: ∅ =M−1 ⊂M0 ⊂ · · · ⊂Mn =Mis called a handle decomposition ofM if for anym ∈ {0, . . . , n} the following conditionshold:

(1) the setMm is n-dimensional submanifold with boundary;(2) the set Cl(Mm \Mm−1) is a disjoint union ofm-handles, i.e., sets homeomorphic

Dm ×Dn−m;(3) eachm-handle is attached to the boundary ofMm−1 by the image ofSm−1 ×Dn−m;(4) for eachm-handleH , the image of

iH :Dm ×Dn−m ↪→M,

there exists an embedding

ıH :Dm ×Dn−m2 ↪→M

such that:(a) ıH |Dm×Dn−m = iH ,(b) ıH (Dm ×Dn−m2 ) ∩Mm−1 = ıH (Sm−1 ×Dn−m2 ),(c) if G is anotherm-handle then the “widened”m-handlesH := ıH (Dm ×Dn−m2 )

andG := ıG(Dm ×Dn−m2 ) are disjoint.

We say that a homeomorphismf ∈H(M) preserves a handle decompositionM if

f (Mm)⊂ IntMm for all m ∈ {0, . . . , n}.A subsetV of a handleH = iH (Dm ×Dn−m) of the form

V = iH(D1r1(a1)× · · · ×D1

rn(an)

),

wherer1, . . . , rn ∈ (0,1) anda1, . . . , an ∈ (−1,1), will be called a cube inH .Let ε > 0 be fixed. ByBε we denote the set of all homeomorphismsf ∈ H(M) for

which we can find a handle decompositionMf satisfying the following conditions:

(1) Mf has the diameter less thanε, i.e.,

|Mf | := max{diamH |H is a handle ofM}< ε;

36 M. Mazur / Topology and its Applications 131 (2003) 33–38

(2) f preservesMf ;

se

n

f

(3) if {Hi}i∈Z is a sequence of handles with the property thatf (Hi)∩Hi+1 �= ∅ then thereexists a corresponding sequence of cubes{Vi}i∈Z such thatVi ⊂ Hi , f (Vi) ⊂ Hi+1and

∞⋂i=−∞

f−i (Vi) �= ∅.

Now, let Bε be the subset ofBε defined as follows: a homeomorphismf ∈ Bε belongsto Bε if there existsδ > 0 such that for eachg ∈ Uδ(f ) the conditions (1)–(3) hold withMg =Mf (in particularg ∈ Bε).

Remark 3. By the results of [12], especially the definition of the setAε ⊂ H(M) as wellas Lemmas 1 and 4 stated there, it is easily seen that the setB := ⋂∞

n=1B 1n

is a residual

subset ofH(M) (note thatBε contains the setAε which was proved to be open and denin H(M)).

4. Main result

Let M be a compact smooth manifold with the metricd induced by the Riemaniastructure.

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

Proof. Fix ε > 0. Since the setB, defined in the previous section, is residual inH(M)it suffices to prove that for everyf ∈ Bε there existsδ > 0 such that for any pair ohomeomorphismsg1, g2 ∈Uδ(f ) eachg1-orbit is ε-traced by someg2-orbit.

Choosef ∈ Bε . Let M = Mf be a corresponding handle decomposition ofM.Since there is only finite number of handles inM we can findδ > 0 such that eachhomeomorphismg ∈ Uδ(f ) satisfies the following conditions:

(i) for every pair of handles(H,G) of Mg(H)∩G= ∅ �⇒ f (H)∩G= ∅ �⇒ dist

(g(H),G

)> 2δ;

(ii) g ∈ Bε with Mg =M.

Fix y ∈M andg1, g2 ∈Uδ(f ). LetHi denote a handle ofM containinggi1(y) (i ∈ Z).Clearly dist(g2(Hi),Hi+1)� 2δ and, in consequence,

g2(Hi)∩Hi+1 �= ∅.From this it follows that there exists a sequence of cubes{Vi}i∈Z such thatVi ⊂Hi and

i=∞⋂i=−∞

g−i2 (Vi) �= ∅.

M. Mazur / Topology and its Applications 131 (2003) 33–38 37

Let x be an arbitrarily chosen point of the above set. Thengi (x) ∈ Vi ⊂ Hi and so

nve theitrary

,

nce

t

nts

tem 5.

ath.

heory, MA,

09.

2d(gi2(x), g

i1(y)) < ε for everyi ∈ Z (we recall that|M|< ε).

By the above, we conclude that eachg1-orbit is ε-traced by someg2-orbit, whichcompletes the proof. ✷

5. Generic asymptotic behavior

Let M be a compact smooth manifold with the metricd induced by the Riemaniastructure. In this section we apply the technique of a handle decomposition to profollowing theorem, which extends some recent Hurley’s result [5] to the case of an arbdimension. A different and independent proof one can find in [1].

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

Proof. We recall that the chain recurrent setCR(f ) is a collection of all such pointsp ∈Mthat for eachδ > 0 there is aδ-chain throughp, i.e., a finite sequencex0, x1, . . . , xn (n� 1)with x0 = xn = p and withd(f (xj−1), xj ) � δ for everyj ∈ {1, . . . , n}. It is a compactnonempty and invariant set.

By the corollary to Theorem 6.1 in [5], it remains to show thatCR(f ) is totallydisconnected for a genericf ∈ H(M).

Takeε > 0 andf ∈Bε . LetM =Mf be a corresponding handle decomposition. Sifor any pointp ∈M lying on the boundary of some handle ofM noδ-chain throughp canbe found whenδ is too small (note thatf preservesM), we have

CR(f )⊂⋃

{IntH |H is a handle ofM}.From this it may be concluded that each connected component ofCR(f ) does not intersecmore than one handle ofM and therefore its diameter is not greater thanε. It follows thatfor eachf ∈ B the setCR(f ) is completely disjoint (note that its connected componeare single points), which makes the proof complete.✷Remark 6. In [2,4,10] was proved that for a genericf ∈ H(M) the chain recurrent seCR(f ) is the closure of the set of all periodic orbits. So, in the other words, Theorsays thatC0 generically dynamics of a homeomorphism is, in a specific way, chaotic

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[3] P. Daszkiewicz, A note on tolerance stable dynamical systems, Coll. Math. 67 (1994) 69–72.[4] M. Hurley, On proofs of theC0 general density theorem, Proc. Amer. Math. Soc. 124 (1996) 1305–13

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