INFINITE ENTROPY IS GENERIC IN H ¨OLDER AND SOBOLEV SPACES

INFINITE ENTROPY IS GENERIC IN H ¨OLDER AND SOBOLEV SPACES

EDSON DE FARIA, PETER HAZARD, AND CHARLES TRESSER

Abstract. In 1980, Yano showed that on smooth compact manifolds, for endomorphisms in dimension one or above and homeomorphisms in dimensions greater than one, topological entropy is generically infinite. It had earlier been shown that for Lipschitz endomorphisms on such spaces, topological entropy is always finite. In this article we investigate what occurs betweenC⁰-regularity and Lipschitz regularity, focussing on two cases: H¨older mappings, and Sobolev mappings.

R´esum´e en langue Franc¸aise. En 1980, Yano a montr´e que, sur une vari´et´e diff´erentielle compacte, pour les endomorphismes en toutes dimensions et les hom´eomorphismes en dimension plus grande que un, l’entropie topologique est g´en´eriquement infinie. Il avait ´et´e auparavant montr´e que pour les endomor- phismes Lipschitz continus, l’entropie est toujours finie. Dans cette note nous investissons ce qui se passe entre la r´egularit´eC⁰et la continuit´e de type Lips- chitz en nous concentrant sur deux cas, endomorphismes et hom´eomorphismes en classes H¨older et Sobolev.

1. Introduction

For a Lipschitz self-mapf of a compact metric space with finite Hausdorff dimen- sion, the topological entropy htop(f) is always finite (see [2, Theorem 3.2.9]). By contrast, Yano [3] showed that, in the space of homeomorphisms of a smooth com- pact manifold of dimensiond≥2, and in the space of endomorphisms of a smooth compact manifold of dimensiond≥1 (both spaces endowed with theC⁰-topology), infinite topological entropy is a generic property.

A natural question for Euclidean spaces is: What happens for intermediate regu- larity between Lipschitz andC⁰? Here we investigate what happens in the H¨older classC^α, 0≤α <1, of maps satisfying the H¨older condition of exponentα, and in the Sobolev classW^1,p, 1≤p <∞, of continuous maps with weak derivatives lying in L^p. For fixed αorp, neither class is closed under composition (so the space of homeomorphisms of each class does not form a group). However, they are closed under pre- and post-composition by Lipschitz maps, and the union, over αand p respectively, ofC^α andW^1,pisclosed under composition.

Before stating our main results from [1], we construct a family of endomorphisms fa of the unit interval, depending upon the parametera∈(0,1], such that eachfa

satisfieshtop(fa) =∞and has some intermediate regularity. It will be useful to first consider an auxiliary family ga,b of interval maps defined as follows. Fix a∈(0,1]

Date: 30 January 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 supported by “Projeto Tem´atico Dinˆamica em Baixas Dimens˜oes”, FAPESP Grant 2011/16265-2.

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2 EDSON DE FARIA, PETER HAZARD, AND CHARLES TRESSER

and a positive integerb. Given an intervalJ, letAJ denote the unique increasing affine bijection fromJ to [0,1]. Subdivide the interval [0,1] intob closed intervals Jb,0, Jb,1, . . . , Jb,b−1of equal length, ordered from left to right. LetAb,k =AJb,k for eachk= 0,1, . . . , b−1. Letν denote the unique decreasing affine bijection of [0,1]

to itself, and letqa(x) =x^a. For eachk= 0,1, . . . , b−1, define

ga,b(x) =qa◦ν^k◦Ab,k(x), ∀x∈Jb,k. (1.1) Observe thatga,b is continuous on [0,1]. Also,ga,b has as a factor the unilateralb- shift. In fact,htop(ga,b) = logb(see e.g. [2, Section 3.2.c]). We now define the family fa as follows. For each positive integerndefine the intervalIn = (2⁻ⁿ,2⁻ⁿ⁺¹] and letfa be given by

fa(x) =

A⁻_In¹◦ga,2n+1◦AIn(x) x∈In, n= 1,2. . .

0 x= 0 (1.2)

Observe that, sincega,2n+1 fixes the endpoints of [0,1] and is continuous, the map fa is also continuous. Moreover, the closure of each intervalIn is totally invariant.

Since the topological entropy of a map is the supremum of the topological entropy of its restriction to all closed invariant subsets, since topological entropy is invariant under topological conjugacy (see e.g. [2, Section 3.1.b]) and, as was stated above, htop(ga,b) = logb for allb, it follows that

htop(fa)≥sup

n

htop(fa|In) = sup

n

htop(ga,2n+1) = +∞. (1.3) Theorem 1.1. For f =f1, the following holds.

(i) f has modulus of continuity ω(t) =tlog(1/t). (ii) f is in the Sobolev class W^1,p for each 1≤p <∞. (iii) htop(f) = +∞.

We have already seen in (1.3) that property (iii) holds. Properties (i)–(ii) are shown in [1]. We recall that maps with modulus of continuity tlog(1/t) are in the H¨older class C^α for every α ∈ [0,1). Moreover, the map f1 is a C^α-limit of piecewise affine maps. Hence it lies in the C^α-boundary of the space of Lipschitz maps. When a6= 1, the mapfa does not satisfy this property. More precisely, we have the following result.

Theorem 1.2. For f =fa,a∈(0,1), the following holds.

(i) f isC^α if and only ifα≤a .

(ii) f isW^1,p if and only if p <(1−a)⁻¹. (iii) htop(f) = +∞.

The proof of Theorem 1.2 could be made using the argument presented in [1] for Theorem 1.1. However, we give a more direct proof for this special case below. For that we need the following, which is a simple exercise using Jensen’s Inequality.

Proposition 1.1 (Gluing Principle). Let ω be a continuous, monotone increasing function, concave at ω(0) = 0. Let f be continuous self-mapping of the compact intervalI. LetI1, I2, . . .denote a collection of closed intervals with pairwise disjoint interiors, covering I, and with the property thatf|Ik has modulus of continuity ω, for all k. LetCk denote theω-semi-norm off|Ik. Ifsup_kCk <∞ andf|∂Ik = id for all k, then f has modulus of continuity ω with ω-semi-norm bounded by C = diam(I)/ω(diam(I)) + 2 sup_kCk.

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Proof of Theorem 1.2. We adopt the following notation. Given an interval J, de- note by [f]C^α,J and [f]_W^1,p_,J the C^α- and W^1,p-semi-norms of f|J respectively.

Define the subintervalsIn,k =A⁻_In¹(J2n+1,k) ofIn fork= 0,1, . . . ,2n.

(i) Sincega,bcannot beC^αfor anyα > a, it follows thatfaalso cannot beC^α for anyα > a. To show that f isC^a we use the following:

Observation: For all non-negative integersnandkwithk≤n−1, there exists a non-negative integer l ≤n−2, and a subinterval Kn−1,l of In−1,l such that the graph off|In,k is isometric to the graph off|Kn−1,l. Hence [f]C^a,In,k≤[f]C^a,In−1,l.

(The first part follows directly from the scale-invariance ofqa.) Therefore [f]C^a,In,k is uniformly bounded. Applying Proposition 1.1 twice we find that [f]_C^a,[0,1] is bounded and thusf isC^a.

(ii) Observe that f is differentiable on each In,k. In fact, on In,k we have f(x) =A⁻_In¹◦qa◦ν^k◦AIn,k(x). Differentiatingf and applying the change of variables formula gives

[f]_W^1,p_,In,k= (2n+ 1)^p Z

[0,1]

|q^′_a(y)|^p 2⁻ⁿ 2n+ 1dy

=







2⁻ⁿa^p(2n+ 1)^p−1

(a−1)p+ 1 if (a−1)p+ 1>0,

∞ otherwise.

Hence [f]_W^1,p_([0,1])=∞ifp≥(1−α)⁻¹ and otherwise

[f]_W^1,p,[0,1]=

∞

X

n=1 2n

X

k=0

Z

In,k

|f^′|^pdx= a^p (a−1)p−1

∞

X

j=1

(2n+ 1)^p2⁻ⁿ.

The right-hand side is a convergent series. Thus f isW^1,p for all 1≤p <

(1−a)⁻¹, as required.

(iii) This follows from inequality (1.3) above.

2. Main Results.

Let us now state the main results proved in [1]. Given a Euclidean domain Ω in R^d and a map f: Ω →R^k, let [f]C^α,Ω and [f]W^1,p,Ω denote the H¨older C^α-semi- norm and the SobolevW^1,p-semi-norms respectively. We then define the norms

kfk_C^α(Ω)=kfk_C⁰(Ω)+ [f]C^α,Ω, kfk_W^1,p(Ω)=kfk_C⁰(Ω)+ [f]_W^1,p,Ω. (2.1) In the Lipschitz case we define the normkfkLip(Ω) in a similar fashion. LetH^α(Ω) denote the space of bi-α-H¨older homeomorphisms of Ω for α ∈ (0,1) and H¹(Ω) denote the space of bi-Lipschitz homeomorphisms of Ω. For α < 1, define the distance

dα(f, g) = max{kf−gk_C^α(Ω),kf⁻¹−g⁻¹k_C^α(Ω)}. (2.2) Forα= 1, defined1(f, g) in similar fashion, replacing theC^αnorm by the Lipschitz norm. Endow H^α(Ω) with the topology induced by the metric dα. Finally, given α < β≤1, denote byH^β_α(Ω) the closure ofH^β(Ω) as a subspace ofH^α(Ω).

4 EDSON DE FARIA, PETER HAZARD, AND CHARLES TRESSER

In the Sobolev case, for 1≤p, p^∗<∞, letS^p,p∗(Ω) denote the space of homeo- morphismsf of Ω such thatf ∈W^1,p(Ω) andf⁻¹∈W^1,p∗(Ω). We endowS^p,p∗(Ω) with the topology induced by the distance

dp,p^∗(f, g) = max{kf−gk_W^1,p(Ω),kf⁻¹−g⁻¹k_W¹,p∗(Ω)}. (2.3) Theorem 2.1 (Generic Infinite Entropy). Let Ωbe a compact d-dimensional Eu- clidean domain with piecewise-smooth boundary.

• For d≥2 and 0 ≤α < β ≤1; H^β_α(Ω) contains a residual subset of maps with infinite topological entropy .

• For d= 2 and 1≤p, p^∗ <∞; or d > 2 andd−1 < p, p^∗ <∞;S^p,p∗(Ω) contains a residual subset of maps with infinite topological entropy . As a corollary of the proof of Theorem 2.1 we get the following result.

Corollary 2.1. LetΩbe a compactd-dimensional Euclidean domain with piecewise- smooth boundary.

• For d ≥ 2 and 0 ≤ α < β ≤ 1; topological entropy is continuous on a residual subset ofH^β_α(Ω).

• For d= 2and 1≤p, p^∗ <∞; ord >2and d−1< p, p^∗<∞; topological entropy is continuous on a residual subset of S^p,p∗(Ω).

As topological entropy is invariant under topological conjugacy we also find.

Corollary 2.2. LetΩbe a compactd-dimensional Euclidean domain with piecewise- smooth boundary.

• For d≥2 and 0≤α < β≤1; a generic homeomorphism in H^β_α(Ω) is not conjugate to any bi-Lipschitz homeomorphism .

• For d = 2 and 1 ≤ p, p^∗ < ∞; or d > 2 and d−1 < p, p^∗ < ∞; a generic homeomorphism in S^p,p∗(Ω) is not conjugate to any bi-Lipschitz homeomorphism .

In [1] we prove these results in more generality for homeomorphisms on smooth compact manifolds. However, the construction of the topology in the Sobolev and H¨older classes is more involved (analogous to the construction of the weak C^k- Whitney topology).

References

[1] E. de Faria, P. Hazard, and C. Tresser.Genericity of infinite entropy for maps with low regularity. preprint, 2016.

[2] A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems Encyclopedia of Mathematics and its Applications,54, Cambridge University Press, 1995.

[3] K. Yano.A remark on the topological entropy of homeomorphisms.Inventiones Mathemat- icæ,59(1980), 215–220.

Edson de Faria, Instituto de Matem´atica e Estat´ıstica, USP, S˜ao Paulo, SP, Brazil E-mail address: edson@ime.usp.br

Peter Hazard, Instituto de Matem´atica e Estat´ıstica, USP, S˜ao Paulo, SP, Brazil E-mail address: pete@ime.usp.br

Charles Tresser, Aperio, MATAM Scientific Industrial Ctr., 9 A. Sakharov St., Ha¨ıfa, 3508409, Israel

E-mail address: charles@aperio-system.com