GENERICITY OF INFINITE ENTROPY FOR MAPS WITH LOW REGULARITY

WITH LOW REGULARITY

EDSON DE FARIA, PETER HAZARD, AND CHARLES TRESSER

Abstract. For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi- Lipschitz homeomorphisms, with respect to either the H¨older or the Sobolev topologies, topological entropy is generically infinite. We also prove versions of theC¹-Closing Lemma in either of these spaces. Finally, we give examples of homeomorphisms and endomorphisms with infinite topological entropy which are H¨older and/or Sobolev of every exponent.

1. Introduction

1.1. Background. In 1974 Palis, Pugh, Shub and Sullivan [27] published a list of dynamical properties satisfied by a generic homeomorphism acting on an arbitrary compact manifold. Six years later, Yano [36] submitted an extra striking property in relation totopological entropy. Recall that topological entropy is a non-negative extended real number defined for any continuous self-map of a compact space, and that this number is an invariant of topological conjugacy. It was first introduced by Adler, Konheim, and McAndrew [2] as an analogue, in the topological category, of the Kolmogorov-Sinai entropy for measure-preserving transformations. Topological entropy, whose precise definition will be given below, is a useful way of quantifying topological aspects of chaos. Yano proved the following result.

Yano’s Theorem. For homeomorphisms of compact manifolds of dimension greater than one topological entropy is generically infinite.

Here, the space of homeomorphisms is endowed with the uniform topology. Yano also states the same result in the case of endomorphisms on compact manifolds of dimension one or greater. However, in this article we will focus on the homeo- morphism case. Note that, in Yano’s result, the fact that the space being acted upon is a manifold matters: there are compact metric spaces for which a generic homeomorphism has zero topological entropy [3].

For smooth maps on compact manifolds it had already been demonstrated, by Ito [20] for homeomorphisms and soon afterwards by Bowen [7] for general endo- morphisms, that the topological entropy is always finite. Thus, in [36], Yano also obtained the following result as a consequence.

Date: May 3, 2017.

2010Mathematics Subject Classification. Primary: 37B40; Secondary: 37E99, 46E35, 26A16.

Key words and phrases. Entropy, genericity, H¨older classes, Sobolev classes.

This work has been partially supported by “Projeto Tem´atico Dinˆamica em Baixas Di- mens˜oes” FAPESP Grant 2011/16265-2, FAPESP Grant 2015/17909-7, and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS) .

1

Corollary. A generic homeomorphism of a compact manifold of dimension two or greater is not topologically conjugate to any diffeomorphism.

Let us now recall the definition of topological entropy [2]. As we will be working only in compact metric spaces, we give the reformulation in this setting due to Bowen¹. Letf be a continuous self-mapping of a compact metric space (X, d). A subsetE ofX is (n, ǫ)-separated forf if for all distinct pointsx, y∈Ethere exists a non-negative integerk < nsuch thatd(f^k(x), f^k(y))> ǫ. LetSf(n, ǫ) denote the maximal cardinality of an (n, ǫ)-separated set. Then thetopological entropy off is given by

htop(f) = lim

ǫ→0lim sup

n→∞

1

nlogSf(n, ǫ).

That is, topological entropy is the growth rate of the maximal size of (n, ǫ)-separated sets at arbitrarily small scales ǫ. Note that in some applications, one prefers to compute the metric entropy hµ(f) of f with respect to some f-invariant measure µ. However, by virtue of the variational principle [35, Theorem 8.6], topological entropy is always the supremum (and in some important cases the maximum) of the metric entropieshµ(f), whereµvaries over allf-invariant Borel measures.

As was already stated, for smooth maps on manifolds the topological entropy is finite. In fact, if X is a compact metric space of Hausdorff dimension dimH(X) andf is a self-map ofX with Lipschitz constantL, then

htop(f)≤dimH(X)·log⁺(L).

See, for instance, [22, Theorem 3.2.9]. We note that, in the case of smooth maps acting on smooth manifolds, this bound was already implicit in the work of Ito [20]

and Bowen [7].

Thus the existence of bounds onhtop(f) change dramatically when the regularity varies from just continuity to Lipschitz continuity. However, the notion of “going from” continuity to Lipschitz continuity must be treated with care. In this paper we make an initial foray into the problem of determining what occurs between these two cases by considering mappings in H¨older and Sobolev classes. These are perhaps two of the most classical ways of interpolating betweenC⁰- and Lipschitz- regularity. We note that homeomorphisms with H¨older or Sobolev regularity have been of interest recently in the study of certain PDE’s, such as the Ball-Evans Problem in nonlinear elasticity (cf. [21] and references therein). However, as α- H¨older andW^1,p-Sobolev classes are not closed under composition, this makes the study of their dynamics more complicated. This being said, spaces of such maps are closed under pre- and post-composition by Lipschitz maps, and the union of such spaces, underαandprespectively, is closed under composition. This allows us to still make local perturbations of these maps. Also note thatC¹, or even Lipschitz, is not in general dense in any H¨older or Sobolev class. Thus results concerning such classes cannot be derived from direct approximation arguments.

Below we will show that, in the closure of the space of bi-Lipschitz maps in either of these topologies, for suitable parameters of regularity, infinite entropy is a generic property. It also follows from our results that there is no “barrier”

separating infinite entropy maps from the space of Lipschitz maps (we even give

1Michel H´enon pointed out to one of the authors that this definition has a significant advantage over the original definition when trying to compute entropy for specific systems: only the forward iterates of the map must be considered, rather than the backward iterates!

explicit examples of endomorphisms and homeomorphisms with infinite entropy which are H¨older or Sobolev of every exponent – they are even in the Zygmund class).

1.2. Summary of our results. To topologise the space of H¨older or Sobolev homeomorphisms on a smooth manifold one requires (in principle) additional struc- ture: a distance function in the first case and a Riemannian structure in the second.

We take a different approach by defining topologies on function spaces which are analogous to the Whitney topology. More specifically, for 0≤α <1, let H^α(M) denote the space of homeomorphisms onM which are bi-α-H¨older continuous in all local charts. We also denote by H¹(M) the space of homeomorphisms which are bi-Lipschitz in all local charts. In Section 2.1 we define a topology onH^α(M) which we call theα-H¨older-Whitney topology. For 0≤α < β ≤1, we denote byH^βα(M) the closure ofH^β(M) with respect to theα-H¨older-Whitney topology. Recall that a property is genericin a Baire space if the set of points satisfying this property contains a residual subset (i.e., a countable intersection of open and dense subsets).

We show the following.

Theorem A (Generic Infinite Entropy for H¨older Classes). Let M be a smooth compact manifold of dimension dgreater than or equal to two. For 0≤α <1, the following holds. InH¹α(M), infinite topological entropy is a generic property.

Similarly, for 1≤p, p^∗<∞, letS^p,p∗(M) denote the space of homeomorphisms on M which in all local charts are of Sobolev class W^1,p and whose inverse is of Sobolev classW^1,p∗. In Section 3.1 we define a topology onS^p,p∗(M) which we call the (p, p^∗)-Sobolev-Whitney topology.

Theorem B (Generic Infinite Entropy for Sobolev Classes). Let M be a smooth compact manifold of dimension d.

(a) If d= 2 and1≤p, p^∗ <∞then, in S^p,p∗(M), infinite topological entropy is a generic property.

(b) If d > 2 and d−1 < p, p^∗ < ∞, then, in S^p,p∗(M), infinite topological entropy is a generic property.

Additionally, we give an alternative proof of (a) in the case whenp^∗ = 1. This proof uses a variant of the Rad´o-Kneser-Choquet theorem for p-harmonic map- pings [4, 21]. We do not know whether this approach extends to higher dimensions, though we suspect not, as there exists a counterexample to the classical Rad´o- Kneser-Choquet theorem in dimension three (see,e.g., [11, Section 3.7]).

Let us also note that we prove two versions of the Closing Lemma along the way. Namely, for both the spaces of bi-H¨older homeomorphisms and bi-Sobolev homeomorphisms stated above, we show that the analogue of Pugh’s C¹-Closing Lemma holds. It would be interesting to determine whether there is another, more direct, approach using Pugh’sC¹-result and an approximation argument, demon- strating that a homeomorphism of bi-H¨older or bi-Sobolev type is approximable by C¹-diffeomorphisms.

1.3. Structure of the paper. In Part I we investigate some properties of bi- H¨older homeomorphisms. After the preliminary Section 2.1, where a suitable H¨older-Whitney topology is given on the space of bi-H¨older homeomorphisms be- tween manifolds, the Closing Lemma for this class of maps is proved in Section 2.2.

Following this the genericity of infinite topological entropy is shown in Section 2.3.

Part II investigates bi-Sobolev homeomorphisms. The structure of Part II mir- rors that of Part I, with the exception that we also give another proof of the genericity of infinite entropy in the special case of compact surfaces. Specifically, in Section 3.1 we introduce the space of bi-Sobolev homeomorphisms together with the Sobolev-Whitney topology. We prove a Closing Lemma for maps in this class in Section 3.2. Two different proofs of the genericity of infinite topological entropy, one specific to dimension two and another for dimensions two and greater, are given in Section 3.3.

In Appendix A we give explicit examples of endomorphisms and homeomor- phisms, in dimensions one and two respectively, with infinite topological entropy which lie in all H¨older or Sobolev classes. These examples can be thought of as certain perturbations of the identity transformation. Finally, the perturbation tools used throughout the paper are collected in Appendices B and C.

1.4. Notation and terminology. Throughout this article, we use the following notation. We denote the Euclidean norm inR^dby| · |^Rd. We denote the Euclidean distance by d(·,·). Denote the open r-ball about the point x in R^d by B^d(x, r).

When the dimension is clear we will write this asB(x, r). In the special case of the unit ball inR^d centred at the origin we denote this byB^d.

Given a manifold M endowed with distance function dM(·,·) denote the open r-ball aboutξ inM, with respect todM, by BM(ξ, r). Given pointsa, b∈R^d and r >0 define

E(a, b;r) =

x∈R^d:d(x, tp+ (1−t)q)< r, somet∈[0,1]

We call such a set an elongated neighbourhood. Given subsets Ω0 and Ω1 in some metric space we denote the Hausdorff distance between Ω0and Ω1by distH(Ω0,Ω1), i.e.,

distH(Ω0,Ω1) = max

sup

x0∈Ω0

x1inf∈Ω1

d(x0, x1), sup

x1∈Ω1

x0inf∈Ω0

d(x0, x1)

and the diameter of Ω0 by diam(Ω0).

2. Part I – H¨older Mappings

2.1. Preliminaries. We recall some basic definitions and facts concerning H¨older maps. Much of what we state here is classical and proofs are left to the reader.

H¨older mappings between metric spaces. Let Ω and Ω^∗ be metric spaces. For each α∈ (0,1), let C^α(Ω,Ω^∗) denote the space of all maps f from Ω to Ω^∗ satisfying the followingα-H¨older condition

[f]α,Ω def

= sup

x,y∈Ω;x6=y

dΩ^∗(f(x), f(y)) dΩ(x, y)^α <∞.

When the domain off is clear we will write [f]αinstead of [f]α,Ω. In the case when Ω^∗=R^d, the setC^α(Ω,R^d) has a linear structure and [·]α,Ωdefines a semi-norm², which we call theC^α-semi-norm. Consequently

kfkC^α(Ω,Rd)

def= kfkC⁰(Ω,Rd)+ [f]α,Ω

2This also induces a pseudo-distance which we will call theC^α-pseudo-distance.

defines a complete norm on C^α(Ω,R^d). (Note that, in this case we will often consider the expression [f −g]α,Ω which obviously has no meaning unless Ω^∗ is contained in some linear space.)

Let H^α(Ω,Ω^∗) denote the space of invertible maps f from Ω to Ω^∗ for which f ∈C^α(Ω,Ω^∗) andf⁻¹∈C^α(Ω^∗,Ω). Thebi-α-H¨older constantoff in H^α(Ω,Ω^∗) to be positive real number max([f]α,Ω,[f⁻¹]α,Ω^∗).

H¨older mappings between manifolds. On more general spaces than Euclidean do- mains, there are several ways to define H¨older continuity. The most direct way is to endow the space with a distance function. However, this leads to difficulties in defining a topology on the space of H¨older maps. (Either we could introduce a distance function d on the range and consider [d(f, g)]α,Ω or, if δf,α(x, y) de- notes the α-H¨older difference quotient with respect to f, then we could consider sup_x6=y|δf,α(x, y)−δg,α(x, y)|. Only when the range is contained in a normed linear space and the natural distance function is used do these definitions coincide, with both expressions being equal to [f −g]α,Ω.)

Instead, as we only consider the case when the underlying spaces are manifolds, we proceed with the following construction, which is analogous to the construction of theC^r-Whitney topology [19].

Take smooth compact manifoldsM and N. We say that f ∈ C⁰(M, N) isα- H¨older continuous if, for any pair of charts (U, ϕ) on M and (V, ψ) on N, the mapψ◦f◦ϕ⁻¹ isα-H¨older continuous on the Euclidean domainϕ(U∩f⁻¹(V)).

(Note: in a given pair of charts, since any smooth metric is Lipschitz equivalent to the Euclidean metric, this definition will coincide with the definition above.) LetC^α(M, N) denote the set ofα-H¨older continuous maps fromM to N. Denote by H^α(M, N) the subspace of homeomorphisms f such that f ∈ C^α(M, N) and f⁻¹∈C^α(N, M). WhenM andN coincide we denote this subspace byH^α(M).

Spaces of bi-H¨older mappings. We define a topology onH^α(M, N) as follows. Given f ∈ H^α(M, N), takeǫ >0, charts (U, ϕ) onM and (V, ψ) onN, such thatf(U)∩ V 6=∅, and compact setsK⊂U∩f⁻¹(V),L⊂f(U)∩V, which are the closure of open sets. Denote byN^Cα(f; (U, ϕ),(V, ψ), K, L, ǫ) the set of mapsg∈ H^α(M, N) such thatg(K)⊆V,g⁻¹(L)⊆U,

kψ◦f◦ϕ⁻¹−ψ◦g◦ϕ⁻¹k^{Cα(ϕ(K),Rd)}< ǫ , and

kϕ◦f⁻¹◦ψ⁻¹−ϕ◦g⁻¹◦ψ⁻¹kC^α(ψ(L),Rd)< ǫ .

The collection of all sets defined in this way form a subbasis for a topology on H^α(M, N). We call it the (weak) α-H¨older-Whitney topology. As shorthand, we will also refer to it as the(weak)C^α-Whitney topology. Observe that the definition is analogous to the (weak) C^r-Whitney topology, the only difference being the choice of norm we use in each chart. As in theC^r-case (see,e.g., [19, Chapter 2]) this topology is Hausdorff, and one can show the following.

Proposition 2.1. For eachα∈(0,1), and each pair of smooth compact manifolds M andN (possibly with boundary), the space H^α(M, N), endowed with the (weak) C^α-Whitney topology, satisfies the Baire property.

For Lipschitz maps we can define all the objects above as in the H¨older case.

However, for clarity we will use a different notation. Namely, denote byC^Lip(Ω,Ω^∗)

the space of all Lipschitz continuous maps from Ω to Ω^∗ and denote the Lips- chitz constant by [f]Lip,Ω. Abusing notation slightly, we denote by H¹(Ω,Ω^∗) the subspace of bi-Lipschitz maps from Ω to Ω^∗. The bi-Lipschitz constantof the bi- Lipschitz mapf inH¹(Ω,Ω^∗) is the positive real number max([f]Lip,Ω,[f⁻¹]Lip,Ω^∗).

For manifoldsM andN we may also define the(weak) Lipschitz-Whitney topol- ogyon H¹(M, N), the space of bi-Lipschitz homeomorphisms fromM to N, as in the H¨older case. For 0≤α < β≤1, letH^βα(M, N) denote the closures ofH^β(M, N) with respect to theC^α-Whitney topology.

Remark 2.1. As previously mentioned, the C^α-Whitney topology does not require the existence of a distance function on the manifoldM. However, we fix now, once and for all, a distance function dM on M. This is merely to simplify notation in the construction of open sets, etc. In particular, our results do not depend on this metric.

Basic properties of H¨older mappings. In the remainder of this subsection we collect the following straightforward, though useful, results.

Lemma 2.1 (H¨older Arzela-Ascoli Principle). For α ∈ (0,1), and β ∈ (α,1) or β= Lip, the space C^β(Ω,R^d) embeds compactly intoC^α(Ω,R^d).

Proposition 2.2 (H¨older Rescaling Principle). Let 0 ≤ α < β ≤ 1. Let Ω, Ω0 and Ω1 be bounded subsets of R^d. Let f: Ω → Ω be β-H¨older continuous.

Let φ0: Ω → Ω0 and φ1: Ω → Ω1 be bi-Lipschitz continuous bijections. Let g = φ1◦f◦φ⁻₀¹: Ω0→Ω1. Then

[g]α≤[φ1]Lip[f]β[φ⁻₀¹]^β_Lipdiam(Ω0)^β−α.

Observe that the following Gluing Principles allow us to show that H¨older maps constructed by gluing with charts are H¨older in the more usual sense, when the manifold is endowed with a smooth metric.

Proposition 2.3 (First H¨older Gluing Principle). For α ∈ (0,1) the following holds. LetΩ⊂R^dbe a connected bounded open domain. LetΩ1,Ω2⊂Ωbe disjoint subdomains such that Ω1∪Ω2 = Ω. Let f1 ∈ C^α Ω1,R^d

and f2 ∈ C^α Ω2,R^d have the property that they extend to a continuous function f on Ω. Then f is α-H¨older continuous. In fact,

[f]α≤Cmax{[f1]α,[f2]α} , whereC >0 is a constant depending uponα,Ω1 andΩ2 only.

Proposition 2.4 (Second H¨older Gluing Principle). For α ∈ (0,1) the follow- ing holds. Let (Ω, d), (Ω^∗, d^∗) be connected metric spaces. Take pairwise disjoint bounded open sets Ω1,Ω2, . . . ,Ωn ⊂Ωand define Ω0 = Ω\S

1≤m≤nΩm. Assume that

κ= max1≤m≤ndiam(Ωm)

min1≤i<j≤ndistH(Ωi,Ωj) (2.1) is finite. Let f ∈ C⁰(Ω,Ω^∗) for which the restrictions f|^Ω0 and f|Ωm, for m = 1,2, . . . , n, areα-H¨older continuous. Then f isα-H¨older continuous onΩand

[f]α,Ω≤K max

0≤m≤n[f]_α,Ωm , whereK is a constant depending only uponαandκ.

A consequence of the H¨older Rescaling Principle (Proposition 2.2) is the follow- ing.

Lemma 2.2. Let0≤α < β≤1. Let(Ω, d)and(Ω^∗, d^∗)be compact metric spaces.

For any f ∈ Hα^β(Ω,Ω^∗), φ∈ H¹(Ω), and ψ ∈ H¹(Ω^∗), the map ψ◦f ◦φ lies in H^βα(Ω,Ω^∗).

The Second H¨older Gluing Principle combines with Lemma 2.2 above to give the following.

Corollary 2.1. Let 0 ≤α < β ≤ 1. Let (Ω, d) and (Ω^∗, d^∗) be compact metric spaces and letΩ1,Ω2, . . .be pairwise disjoint open subsets ofΩ. Takef ∈ H^βα(Ω,Ω^∗) and, fork= 1,2, . . ., take φk∈ H¹(Ω), supported inΩk. Define

g=

f ◦φk inΩk

f inΩ\S

kΩk . Then g lies inH^βα(Ω,Ω^∗).

2.2. The H¨older Closing Lemma. In this section we will consider spaces of homeomorphisms on compact manifolds of dimension greater than one. We prove an analogue of Pugh’sC¹-Closing Lemma [25, 26] in a subspace of bi-H¨older maps.

Recall that, given a continuous self-map f of a topological space X, a pointx in X isnon-wandering if for all neighbourhoodsU of xthere exists some positive integernfor whichfⁿ(U)∩U 6=∅.

Theorem 2.1 (H¨older Closing Lemma). Let M be a smooth compact manifold.

For 0 ≤ α < β ≤ 1, the following holds: Take f ∈ H^βα(M) and let y be a non- wandering point off. For each neighbourhoodV ofyinM and each neighbourhood N off inH^βα(M)there existsg inN and a pointxin V such thatxis a periodic point of the mapg.

The proof of the Theorem 2.1 requires the following preparatory lemma.

Lemma 2.3. For each non-wandering pointy and each sufficiently small, positive real number η the following holds: there exists a point xin B(y, η) and a positive integerk such that f^k(x)also lies in B(y, η) and, for allj = 1,2, . . . , k−1,

f^j(x)∈/ B x,³₄ρ

∪B f^k(x),³₄ρ , whereρ=dM(x, f^k(x)).

Remark 2.2. As was pointed out to us by Charles Pugh, this is, in essence, the Fundamental Lemma given in his paper[25]. However, we include this version here for completeness.

Proof of Lemma 2.3. As y is a non-wandering point, there exists a point z in B(y,₁₀^η) such that f^m1(z) also lies in B(y,₁₀^η) for some positive integer m1. Let zm=f^m(z) for each integerm and denote the orbit segment{z0, z1, . . . , zm1} by O. Also letm0= 0. Letρ1=dM(zm1, zm0). If

B zm0,³₄ρ1

∪B zm1,³₄ρ1

∩O={zm0, zm1}

holds, then we are done. Otherwise there exists a point zm2 in the orbit segment O, withm2 different fromm0 andm1, such that dM(zm2, zm1)< ³₄dM(zm1, zm0), say. Letρ2=dM(zm2, zm1). If

B zm1,³₄ρ2

∪B zm2,³₄ρ2

∩O={zm1, zm2}

holds, then we are done. Otherwise there exists a point zm3 in the orbit segment O, withm3 different fromm1 andm2, such that dM(zm3, zm2)< ³₄dM(zm2, zm1), say, etc.

Continuing in this way, we move from one pair of points, zmn and zmn−1, to the next, zmn+1 and zmn. Since there are only finitely many points in the orbit segment O, and since the distance between pairs decreases at least geometrically (which implies that there is no cyclicity, i.e., zm_a 6=zm_b fora6=b), it follows that this process must terminate. Hence there are points zm_N and zm_N−1 in the orbit segmentO such that

B zmN,³₄ρN

∪B zmN−1,³₄ρN

∩O={zmN, zmN−1},

where ρN = dM(zmN, zmN−1). Moreover, as distances between subsequent pairs of points decreases at least geometrically, the distance between the initial point z0=zm0 and the terminal pointzmN satisfies the following upper bound

dM(z0, zmN)≤

N

X

n=0

dM(zmn+1, zmn)≤

N

X

n=0

3 4

n

dM(zm0, zm1) =4 5η . Consequently the pointszmN andzmN−1also lie inD(y, η). Consider the case when mN < mN−1. Then settingx=zmN andk=mN−1−mN, so thatf^k(x) =zmN−1, we find that the point xand integer k satisfy the conclusion of the lemma. The case whenmN−1< mN is similar. Hence the lemma is shown.

We now proceed with the proof of the H¨older Closing Lemma (Theorem 2.1).

Proof of the H¨older Closing Lemma. We will prove this in the case when β = 1, i.e., for maps in theC^α-closure of bi-Lipschitz homeomorphisms. The general case follows analogously, as the only property being used here is that the mapf satisfies thelittleα-H¨older condition,i.e., for eachxinM, [f]α,B(x,r)=o(r).

Cover the compact manifoldM with a finite collection of charts{(Uk, ψk)}, each of which is κ-bi-Lipschitz for κ sufficiently close to 1, i.e., for eachk and for all x, y∈Uk,

κ⁻¹dM(x, y)≤ |ψk(x)−ψk(y)| ≤κdM(x, y).

(Conditions determiningκwill be given in Claim 2 below.) This may be done, for example, by using geodesic coordinates about each point, restricting to a sufficiently small neighbourhood of the origin,where the exponential map is close the identity, then invoking compactness. Moreover, by Lebesgue’s Lemma, there exists a positive real number δ0 such that for each x in M, the ball B(x, δ0) is contained in the domain of at least one of these charts, and similarly its image f(B(x, δ0)) is also contained in the domain of one of these charts. (That is,δ0is the Lebesgue number of the cover{Uk} ∪ {f(Uk)}.)

Let ǫn be a decreasing sequence of positive real numbers converging to zero.

Letδn be a sequence of positive real numbers satisfyingδn < δ0, such that for all suitable integerskandl, and allx∈Uk,

maxn

ψl◦f◦ψ⁻_k¹

α,ψk(B(x,δn)),

ψk◦f⁻¹◦ψ⁻_l ¹

α,ψl◦f(B(x,δn))

o< ǫn (2.2) holds for alln. Fix n. Let y be a non-wandering point off. By Lemma 2.3 there exists a pointx∈B(y,¹₅δn) and a positive integerk0 such thatf^k0(x)∈B(y,¹₅δn)

and, if we letρn=dM(x, f^k0(x)), then the ballsB(x,³₄ρn) andB(f^k0(x),³₄ρn) do not contain any pointf^k(x), for 0< k < k0. Observe that,

B x,³₄ρn

∪B f^k0(x),³₄ρn

⊂B(y, δn).

Hence the balls B(x,³₄ρn) and B(f^k0(x),³₄ρn) lie in the domain of some chart (U0, ψ0). Ifδn is sufficiently small, the imagef(B(x,³₄ρn)∪B(f^k0(x),³₄ρn)) is also contained in the domain of some chart (U1, ψ1). Observe that we may take these charts independently ofn, and hence fixed, provided thatnis sufficiently large.

Claim 1. For κ >1 sufficiently close to 1, there exists a positive real number c, independent of ρn, such that

ψ0 B x,³₄ρn

∪B f^k0(x),³₄ρn

⊇E ψ0(x), ψ0(f^k0(x)), c·ρn . Define E_n¹ =E ψ0(x), ψ0(f^k0(x));^c₄ρn

and E²_n=E ψ0(x), ψ0(f^k0(x));₂^cρn . By Lemma B.3, there exists a diffeomorphism φn, supported on E_n², which satisfies φn(ψ0(f^k0(x))) =ψ0(x). Moreover, the bound from Lemma B.3 implies that there is a positive real numberK, independent ofn, such that

maxn φn

Lip, φ⁻_n¹

Lip

o≤K .

LetEn=ψ⁻₀¹(E_n²). Define a mapgn by setting it equal tof◦ψ₀⁻¹◦φn◦ψ0 inEn

and equal tof outsideEn. It is clear from the construction that g^k_n⁰(f^k0(x)) =f^k0(φn(f^k0(x))) =f^k0(x).

It is also clear thatgnis a homeomorphism. By Corollary 2.1 it also lies inH^βα(M).

Thus it remains to estimate theC^α-pseudo-distance betweenf andgn, and theC^α- pseudo-distance between f⁻¹ andg⁻_n¹ in each chart. In any chart not containing Enthe mapsf andgn are equal. Lettingf0=ψ1◦f◦ψ⁻₀¹andgn,0=ψ1◦gn◦ψ⁻₀¹ it therefore suffices to estimate

(i) theC^α-pseudo-distance betweenf0 andgn,0 on Ω0=ψ0(U0∩f⁻¹(U1)) (ii) theC^α-pseudo-distance betweenf₀⁻¹ andg⁻_n,0¹ on Ω1=ψ1(f(U0)∩U1).

Sincef0andgn,0only differ onE_n², we only need to consider theC^α-pseudo-distance on E_n². Similarly for the inverses. However, the fact that E_n² ⊂ ψ0(B(x, δn)), together with inequality (2.2), implies that

[f0−gn,0]_α,E2 n

= [f0−fn,0◦φn]_α,E2 n

≤

ψ1◦f◦ψ⁻₀¹

α,E²_n+

ψ1◦f◦ψ₀⁻¹◦φn

α,E²_n

≤

ψ1◦f◦ψ⁻₀¹

α,ψ0(B(x_k1,δn))+

ψ1◦f◦ψ⁻₀¹

α,ψ0(B(x_k1,δn))[φn]^α_Lip

≤(1 +K^α)ǫn . A similar estimate holds for

f₀⁻¹−g⁻_n,0¹

α,f0(E_n²). This gives the result.

The proof of the H¨older Closing Lemma also yields the following corollary.

Corollary 2.2. Let M be a smooth compact manifold. For0 ≤ α < β ≤1, the following holds: Take f ∈ H^βα(M) and let y be a recurrent point of f. For each neighbourhood N off inHα^β(M)there existsg in N and a positive integer ksuch thatf^k(x)is a periodic point of the mapg.

2.3. Genericity of infinite topological entropy for H¨older mappings. In this section we prove that infinite topological entropy is a generic property in the H¨older context. Theorem A will follow from Theorem 2.2 below. Before we can state this we need to introduce the following terminology. CallB^d−1×B¹ thestandard solid cylinder in R^d. We call images under affine transformations of the standard solid cylinderrigid solid cylindersand homeomorphic images of the standard solid cylindertopological solid cylinders. Given a rigid solid cylinderC denote the axial length and the coaxial radius ofCrespectively by len(C) and rad(C). Given distinct pointsa andb inR^d andr >0, denote by C(a, b;r) the rigid solid cylinder inR^d whose axis is the line segment [a, b] and whose co-axial radius isr.

LetCbe a topological solid cylinder inR^d with disjoint marked boundary balls C⁺ and C⁻. We say that an embeddingφ, from some domain containingC into R^d,mapsC across the standard solid cylinder B^d−1×B¹if the following properties are satisfied (see Figure 1)

(1) φ(C) intersectsB^d−1×B¹,

(2) φ(C) does not intersect∂ B^d−1×B¹

\ B^d−1×∂B¹ , (3) φ(C⁻) andφ(C⁺) do not intersectB^d−1×B¹,

(4) the connected component ofφ(C)\ B^d−1×B¹

whose boundary contains φ(C^±) has closure intersectingB^d−1× {±1}but notB^d−1× {∓1}. Given a topological solid cylinderC^′inR^d, which is the image of the standard solid cylinder under the homeomorphismψ, we say thatφmaps C across C^′ ifψ⁻¹◦φ mapsC across the standard solid cylinder.

B^d−1×B¹ C

φ

Figure 1. The cylinderC maps across the standard solid cylin- derB^d−1×B¹ via the embeddingφ.

Given a positive integerN, we say that an embeddingfmaps a (topological) solid cylinder C0 across a (topological) solid cylinder C1 like an N-branched horseshoe if there exist pairwise disjoint subcylinders C0,1, C0,2, . . . , C0,N of C0 such that f mapsC0,k acrossC1 for eachk= 1,2, . . . , N. See Figure 2. For results concerning solid cylinders used in this section see Appendix C. Theorem A, stated in § 1.2, follows directly from the following result.

C0

C1

f

f(C0)

C0,1

C0,2

C0,3

Figure 2. The embedding f maps the cylinder C0 across the cylinderC1 like a 3-branched horseshoe.

Theorem 2.2. Let 0≤α <1. Assume that f ∈ H¹α(M). For each neighbourhood N off inH¹α(M)and each positive integerN there existsg∈ H¹α(M) such that

(i) g∈ N

(ii) there exists a positive integer k0, a topological solid cylinder S in M and solid sub-cylinders S1, S2, . . . , S_Nk0 such that g^k0 maps Sj across S for j= 1,2, . . . , N^k0

The second property implies that htop(g)≥logN and that this property is satisfied in an open neighbourhood ofg.

Proof. Before starting the proof let us describe the idea. Take a recurrent orbit for f. Take a segment of this orbit, of lengthk0 say, whose start- and end-points are sufficiently close. In pairwise disjoint neighbourhoods of each of the points in this orbit segment take a solid cylinder. Perturbf in each of these neighbourhoods so that the solid cylinder maps over the next solid cylinder like anN-branched horse- shoe. Finally, ‘close-up’ the orbit of the horseshoe by mapping the solid cylinder at the end of the the orbit segment across the solid cylinder prescribed at the start of the orbit segment. Observe that if h0 maps the solid cylinderC0 acrossC1 and if h1 mapsC1 acrossC2 thenh1◦h0 mapsC0 acrossC1. Thus property (ii) will be satisfied. The discussion below will therefore focus on showing that (i) is satisfied.

Now we begin the details of the proof. Since f ∈ H¹α(M) we may assume, by making an arbitrarily small perturbation or otherwise, thatf is bi-Lipschitz. Hence there exists a positive real numberκf such that, for allx, y∈M, x6=y,

κ⁻_f¹dM(x, y)≤dM(f(x), f(y))≤κfdM(x, y).

Cover M with a finite collection of charts{(Vk, ψk)} and, for each k, take open Uk compactly contained inVk so that {Uk}also coversM. As stated in the proof

of the H¨older Closing Lemma, there exists a positive real numberδ0 such that for eachxin M, the ballB(x, δ0) is contained in the domain of one of these charts.

Fixǫ0>0. Sincef is bi-Lipschitz,fandf⁻¹are both littleα-H¨older continuous.

Therefore, by shrinkingδ0if necessary, we may assume that for all suitable integers kandl, wheneverx∈Uk andδ < δ0 we have

maxn

ψl◦f◦ψ_k⁻¹

α,ψ_k(B(x,δ)),

ψk◦f⁻¹◦ψ_l⁻¹

α,ψ_l◦f(B(x,δ))

o< ǫ0. (2.3) As each chart ψk is smooth (and hence approximately linear on small balls), by shrinking δ0 again if necessary, we may also assume that, for a fixed L > 1, the following property holds: for any ballB⊂Ukof radiusδ0or less, and any elongated neighbourhoodE(x, y;ρ)⊂B, for which ₁₀¹dM(x, y)≤ρ, we have

E(ψk(x), ψk(y);L⁻¹ρ)⊂ψk(E(x, y;ρ))⊂E(ψk(x), ψk(y);Lρ). (2.4) Chooseδ < δ0/3. Compactness implies that the forward-recurrent set off is non- empty. Letx0be a forward-recurrent point and, for each integerk, letxk =f^k(x0).

Choose the smallest non-negative integersk1< k2such that for all integerskandl satisfyingk1≤k < l≤k2, the property dM(xk, xl)< δ holds if and only ifk=k1

andl=k2. Such integers exist as the orbit is recurrent. Along this orbit segment we will make the sequence of perturbations as described in the first paragraph of the proof.

Letρ0=dM(xk1, xk2). Observe that

B(xk1, ρ0)∪B(xk2, ρ0)⊂B(xk1,3ρ0)⊂B(xk1, δ0).

Therefore the neighbourhoodB(xk1, ρ0)∪B(xk2, ρ0) is contained in the domain of some chart. Similarly, for eachk1 < k < k2, the ballB(xk, ρ0) is contained in the domain of some chart. Henceforth, we assume that the charts are indexed so that B(xk, ρ0) lies inUk andB(xk1, ρ0)∪B(xk2, ρ0) lies inUk1 =Uk2.

Observe that the neighbourhoodsB(xk1, ρ0)∪B(xk2, ρ0) andB(xk, ρ0), fork1<

k < k2, are not necessarily pairwise disjoint. Therefore we shrink them. However, this must be done in a controlled way so that (i) the image under the charts ψk

contain ‘nice’ neighbourhoods ofψk(pk), and (ii) when applying the Second Gluing Principle (Proposition 2.4), the constant obtained is independent ofδ. This is done via the following claim. (The proof is elementary so we omit it.)

Claim 2. There exist real numbers ρ1, ρ2, ρ_∗ ∈ (0, ρ0), where ρ1 > ρ2, and there exists κ_∗ >0, such that ρ1, ρ2 andρ_∗ may be chosen comparable to ρ0, indepen- dently ofδ0, andκ_∗may be chosen independently ofδ0, and such that the following properties hold

(a) Let

B={B(xk1, ρ1)∪B(xk2, ρ1), B(xk1+1, ρ2), . . . , B(xk2−1, ρ2)} .

Then the sets inB are pairwise disjoint in M. Moreover, ifκdenotes the quantity defined by equation (2.1) in the Second H¨older Gluing Principle (Proposition 2.4), for this collection of sets inM, thenκ≤κ_∗.

(b) If we define

EM =ψ_k⁻₁¹(E(ψk1(xk1), ψk1(xk2);ρ_∗)) and, for k1< k < k2,

Bk,M =ψ_k⁻¹ B(ψk(xk), L⁻¹ρ2)

then

B(xk1, ρ1)∪B(xk2, ρ1)⊇EM

and, for k1< k < k2,

B(xk, ρ2)⊇Bk,M

We will say thatEM corresponds to B(xk1, ρ1)∪B(xk2, ρ1)and, for k1<

k < k2,Bk,M corresponds toB(xk, ρ2).

(c) For eachk, letB^k denote the subcollection of sets inBwhich are contained in Uk. Take the corresponding subcollection of sets, as per part (b), from the collection

EM, Bk1+1,M, Bk1+2,M, . . . , Bk2−1,M .

and denote by C^k the collection composed of images of these sets under ψk. Then the sets in C^k are pairwise disjoint. Moreover, ifκk denotes the quantity defined by equation (2.1) in the Second H¨older Gluing Principle (Proposition 2.4) for this collection of sets in ψk(Uk), thenκk≤κ_∗. Proof of Claim:

(a) Let ρ1 = √¹

2ρ0 and ρ2 = ²⁻₄^√2ρ0. Since, for all k 6= k1, k2, we have dM(xk, xk1)> ρ0 anddM(xk, xk1)> ρ0, it follows that

distH(B(xk1, ρ1)∪B(xk2, ρ1), B(xk, ρ2))≥ρ0−ρ1−ρ2= ²⁻₄^√2ρ0 . Similarly, fork, l6=k1, k2, sincedM(xk, xl)> ρ0, we have

distH(B(xk, ρ2), B(xl, ρ2))≥ρ0−2ρ2=ρ0−²⁻2^√2ρ0= ^√₂²ρ0 . But

diam (B(xk1, ρ1)∪B(xk2, ρ1))≤ρ0+ 2ρ1= 1 +√

2 ρ0 , and

diam (B(xk, ρ2))≤2ρ2=²⁻₂^√2ρ0. Consequentlyκ≤ ⁴⁽¹⁺

√2) 2−√

2 , and thus (a) is shown.

(b) This follows from the property (2.4) above.

(c) Since each ψk is smooth, there exists L0 > 1 such that each ψk is L0- bi-Lipschitz. Consequently κk ≤ L²₀κ. Thus replacing κ_∗ with L²₀κ_∗ if necessary, the stated bound holds (and is independent ofδ0).

Thus (a)–(c) have been shown and the claim is proved. //

The perturbations referred to in the first paragraph of the proof are to be made in charts about each point in the orbit segment between xk1 and xk2. Therefore define, fork1≤k < k2,

fk =ψk+1◦f◦ψk .

The property (2.3) implies that there exists a real number L1 such that eachfk is L1-bi-Lipschitz, Thus, take positive real numbersK0 and K1 as in Corollary C.1 which are admissible for all fk. (This is possible as they depend only upon the Lipschitz constants of the fk and hence are independent of δ.) Takeρ1 and ρ2 as in the preceding claim. Given a positive real number r1 we may define

E=E(ψk1(xk1), ψk1(xk2);r1)

and, fork1≤k≤k2, we define

Bk =B(ψk(xk), r1).

(Observe that E and the Bk lie in Euclidean space, whereas EM andBk,m lie in M.) Then, by Claim 2(b), if r1 was chosen sufficiently small, though comparable toρ0, we have

E⊂ψk1(B(xk1, ρ1)∪B(xk2, ρ1)) and fork1≤k≤k2, we have

Bk⊂ψk(B(xk, ρ2)).

For this choice of r1 take positive real numbers r2 and r3 as in the statement of Corollary C.1. Take isometric rigid solid cylinders C(ak, bk;̺) contained in B(ψk(xk), r3) satisfying properties (i)–(iii) of the same Corollary C.1. (Since the solid cylinders are isometric, (ii) is automatically satisfied.) Then Corollary C.1 implies that, for eachk1≤k < k2, there exists a diffeomorphismφk∈Diff¹(Uk+1), supported in Bk, such that φk◦fk maps the solid cylinder C(ak, bk;̺) across the solid cylinder C(ak+1, bk+1;̺) as an N-branched horseshoe. Moreover, since the ratio of length to radius of each solid cylinder was chosen independently ofδ, there exists a positive real numberc1, independent ofδandN, such that, fork1≤k < k2,

maxn φk

Lip, φ⁻_k¹

Lip

o≤c1N . (2.5)

By Corollary B.2, there exists φ∈Diff¹(Uk1), supported inE, such that φmaps the solid cylinder C(ak2, bk2;̺) across the solid cylinderC(ak1, bk1;̺). Moreover, there exists a positive real numberc2 such that

maxn φ

Lip, φ⁻¹

Lip

o≤c2 . (2.6)

Sincer1 and|ψk1(xk1)−ψk1(xk2)|are comparable (both are comparable toρ0),c2

is independent ofδ. Define gk=

φk◦fk◦φ k=k1

φk◦fk k1< k < k2

We defineg as follows. InS

k1<k<k2ψ_k⁻¹(Bk), when expressed in relevant charts, it is equal togk. Inψ_k⁻¹

1(E), again expressed in relevant charts, we setg equal to gk1. Elsewhere we defineg equal tof.

Consider property (i) in the statement of the Theorem. Observe that g and f differ only only on the setsψ⁻_k1¹(E) andψ_k⁻¹(Bk),k1< k < k2. Moreover the image of any of these sets undergcoincides with the image of the same set underf. Since the diameter of any of these images can be made arbitrarily small by making δ small, it follows that, in any chart, theC⁰-distance betweenf andg, and between f⁻¹ and g⁻¹ can be made arbitrarily small. Thus we just need to show that in any pair of charts the C^α-pseudo-distance can be made arbitrarily small. By the H¨older Rescaling Principle (Proposition 2.2) and inequalities (2.5) and (2.6) there exists a positive real number c3 such that

maxn gk1

α,E, g⁻_k1¹

α,gk1(E)

o≤c3ǫ0, while, fork1< k < k2,

maxn gk

α,Bk, g_k⁻¹

α,gk(Bk)

o≤c3ǫ0 .

M

ψk ψk+1

f

fk

φk

xk xk+1

ψk(xk) ψk+1(xk+1)

r3

r3

Bk Bk+1

B(xk, ρ2) B(xk+1, ρ2)

C(ak, bk;̺) C(ak+1, bk+1;̺)

fk(C(ak, bk;̺)) Figure 3. The map gk is constructed as the composition of φk◦fk and maps the cylinderC(ak, bk;̺) acrossC(ak+1, bk+1;̺) like anN-branched horseshoe.

Therefore

[fk1−gk1]_α,E ≤[fk1]_α,E+ [gk1]_α,E ≤(1 +c3)ǫ0

and, sincefk1(E) =gk1(E), we find that f_k⁻₁¹−g⁻_k1¹

α,fk1(E)≤ f_k⁻₁¹

α,fk1(E)+ g_k⁻₁¹

α,gk1(E)≤(1 +c3)ǫ0 . Similarly, fork1< k < k2,

maxn

fk−gk

α,Bk,

f_k⁻¹−g_k⁻¹

α,gk(Bk)

o≤(1 +c3)ǫ0 .

Now takekandlsuch thatf(Uk)∩Ul6=∅. IfUk supports one of the perturbations made above then, applying Claim 2(c), together with the Second H¨older Gluing Principle (Proposition 2.4) and the fact that f and the chartsψk and ψl are uni- formly bi-Lipschitz, we find that there exists a positiveK, independent ofδ0, such that

ψl◦f ◦ψ_k⁻¹−ψl◦g◦ψ_k⁻¹

α,ψk(Uk∩f⁻¹(Ul))≤K(1 +c3)ǫ0 , and

ψk◦f⁻¹◦ψ_l⁻¹−ψk◦g⁻¹◦ψ⁻_l ¹

α,ψl(f(Uk)∩Ul)≤K(1 +c3)ǫ0 .

Elsewhere,i.e., in pairs of charts with domains in the complement ofψ⁻_k1¹(E) and ψ⁻_k¹(Bk), for k1 < k < k2, the maps f and g are equal. Hence the C^α-pseudo- distance between f and g in any pair of charts can be made arbitrarily small by choosing ǫ0 sufficiently small. Consequently g may be constructed in any neigh- bourhood off in theC^α-Whitney topology. This completes the proof.

Analogous to the homeomorphism case [36] we get also get the following.

Corollary 2.3. Let M be a compact manifold of dimension at least two. Let 0 ≤α <1. A generic homeomorphism in H¹α(M) is not conjugate to any diffeo- morphism (or any bi-Lipschitz homeomorphism).

Recall that horseshoes possess (unique) measures of maximal entropy. Observe that in the proof of Theorem 2.2, the worst that can happen is that the recurrent point being used already lies in a horseshoe. However, as the perturbation being used is arbitrarily small, if the horseshoe of the original map has N branches, we may assume that the perturbed map has a horsehoe with at least N branches.

Thus, considering all possible sums of these measures over all possible horseshoes, we get the following corollary.

Corollary 2.4. Let M be a compact manifold of dimension at least two. Let 0≤α <1. A generic homeomorphism in Hα¹(M)has uncountably many measures of maximal entropy.

3. Part II – Sobolev Mappings

3.1. Preliminaries. Let us recall some basic definitions and facts about Sobolev functions and maps. For details on the material here we strongly recommend [16], [23] and [37]. Here and throughout, all open domains in Euclidean spaces will be assumed to have piecewise smooth boundaries.

Sobolev functions. Let Ω ⊆ R^d be open, and let k ∈ N and 1≤ p < ∞. Recall that a measurable function u: Ω → R is in the Sobolev class W^k,p(Ω) if u has distributional partial derivatives of all orders up to k and, for each multi-index α= (α1, α2, . . . , αd)∈N^dwith|α|=Pd

i=1αi≤k, the corresponding distributional partial derivative D^αu belongs to L^p(Ω). The space W^k,p(Ω) is a Banach space under the norm

kuk^k,p = X

|α|≤k

kD^αuk^p,

where k · k^p denotes the standardL^p-norm in Ω. It is known that every Sobolev functionuis absolutely continuous on lines (ACL),i.e., its restriction to Lebesgue almost every straightline (parallel to some coordinate axis) is absolutely continu- ous [24, Section 1.1.3]. It is also known that u is differentiable Lebesgue almost everywhere in Ω provided thatp > d. (This was proved ford= 2 by Cesari [9] and for arbitrarydby Calder´on [8].)

Sobolev maps. Let us consider a measurable map f: Ω → R^d. We say that f is a Sobolev map in the class W^k,p if, writing f = (f1, f2, . . . , fd), each component fi ∈ W^k,p(Ω). Note that such a map has a formal Jacobian matrix Df(x) = (∂x_jfi(x))1≤i,j≤d defined at Lebesgue almost every pointx∈Ω.

The space of Sobolev maps in the classW^k,p, which we denote byW^k,p(Ω,R^d), can be made into a Banach space in several equivalent ways. One natural way is