Dimensions of C1-average conformal hyperbolic sets

This paper introduces the concept of average conformal hyperbolic sets, which admit only one positive and one negative Lyapunov exponents for any ergodic measure. For an average conformal hyperbolic set of a C1 diffeomorphism, utilizing the techniques in sub-additive thermodynamics formalism and some geometric arguments with unstable/stable manifolds, a formula of the Hausdorff dimension and lower (upper) box dimension is given in this paper, which are exactly the sum of the dimensions of the restriction of the hyperbolic set to a stable and unstable manifolds. Furthermore, the dimensions of an average conformal hyperbolic set varies continuously with respect to the dynamics.


Introduction
The dimension of invariant sets is one of their important characteristics, it plays an important role in various problems in dynamics, see the books [3,4,12,21,22]. Despite many interesting and non-trivial developments in the dimension theory of dynamical systems, only the case of conformal dynamics is completely understood. Indeed, Bowen [7] and Ruelle [24] found that the Hausdorff dimension of a C 1+γ conformal repeller was a solution of an equation involving topological pressure. The smoothness is relaxed to C 1 in [14]. The study of dimension of hyperbolic sets is analogous. Using techniques in thermodynamic formalism, in [18] MaCluskey and Manning obtained a formula of the Hausdorff dimension of a two dimensional hyperbolic set of a C 1+γ diffeomorphism; using a different and more geometric method, Palis and Viana relaxed the smoothness to C 1 in [20]. Takens [25] proved that the same formula also holds for lower and upper box dimensions. Using the techniques of Markov partition and thermodynamic formalism, the same formula was obtained for the C 1+γ conformal hyperbolic set in higher dimension, see the books [21] and [3] for detailed description.
For the non-conformal case, the study of dimension is substantially more complicated and to approach it. Only upper and lower bounds of dimension of repellers are obtained, see [2,13,29] for details, different version of Bowen's equation involving topological pressure are useful in estimating the dimensions of a non-conformal repeller. Finally, in [1], using thermodynamic formalism for sub-additive potentials developed in [8], the authors showed that the zero of the sub-additive topological pressure gives an upper bound of the Hausdorff dimension of repellers, and furthermore, that the upper bounds obtained in the previous works [1,13,29] are all equal. See Climenhaga's paper [11] for Bowen's equation in estimating Hausdorff dimension in the case of very general non-uniform setting. Recently, in [9] the authors introduced the super-additive topological pressure, and showed that the zero of super-additive topological pressure gives a lower bound of the Hausdorff dimension of repellers. We refer the reader to [10] and [5] for a detailed description of the recent progress in dimension theory of dynamical systems.
In [1], the authors introduced a concept of C 1 average conformal repellers which posses only one positive Lyapunov exponent for any ergodic measure. An example is given in [30] to show that such a repeller is indeed non-conformal. The dimension of an average conformal repeller is given by the zero of sub-additive topological pressure, see [1] for details.
In this paper, we introduce a concept of C 1 average conformal hyperbolic sets in higher dimension. Roughly speaking, an average conformal hyperbolic set admits only one positive and one negative Lyapunov exponents for any ergodic measure. We obtain a dimension formula of such hyperbolic sets, which can be described as the sum the dimensions of the restriction of the hyperbolic set to a stable and unstable manifolds. Furthermore, the dimension of a C 1 average conformal hyperbolic set varies continuously with respect to the dynamics. Now we recall some definitions and known results in hyperbolic dynamics. A compact invariant subset Λ ⊂ M is called a hyperbolic set if there exists a continuous splitting of the tangent bundle T Λ M = E s ⊕E u , and constants C > 0, 0 < λ < 1 such that for every Here λ is called the skewness of the hyperbolicity. Given a point x ∈ Λ, for each small β > 0, the local stable and unstable manifolds are defined as follows: The global unstable and stable sets of x ∈ Λ are given as follows: Let d u be the metric induced by the Riemannian structure on the unstable manifold W u and d s the metric induced by the Riemannian structure on the stable manifold W s . For any ρ > 0, let B u (x, ρ) (respectively, B s (x, ρ)) be the ball in the unstable (respectively, stable) manifold of radius ρ centered at x, and n}, where i ∈ {u, s} and n ∈ N. A hyperbolic set is called locally maximal, if there exists a neighbourhood U of Λ such that Λ = n∈Z f n (U ).
Let Diff 1 (M ) be the set of all C 1 diffeomorphisms from M to M , and U ⊂ Diff 1 (M ) be a neighbourhood of f such that, for each g ∈ U, Λ g = n∈Z g n (U ) is a locally maximal hyperbolic set for g and there is a homeomorphism h g : Λ → Λ g which conjugates g| Λg and f | Λ , i.e., be the hyperbolic splitting of Λ g . The local unstable and stable sets of z ∈ Λ g are denoted by W u β (g, z) and W s β (g, z) respectively. These are embedded Dimension of Conformal Hyperbolic Sets. Roughly speaking, a hyperbolic set is called conformal, if the derivative of the map is a multiple of an isometry along the stable and unstable directions (see Definition in [21]).
If Λ is a hyperbolic horseshoe of a C 1+γ surface diffeomorphism f , for every x ∈ Λ, in [18] MaCluskey and Manning proved that where t s and t u are the roots of P Λ (f, t log Df | E s ) = 0, P Λ (f, −t log Df | E u ) = 0 respectively (here P (·) denotes the topological pressure ). Since dim E s = dim E u = 1, the local product structure is a Lipschitz homeomorphism with Lipschitz inverse. Therefore dim H Λ = t s + t u . (1. 2) The equality between the Hausdorff dimension and the lower and upper box dimensions is due to Takens [25]. Palis and Viana relaxed the smoothness to C 1 in [20]. Their proof used Hölder conjugancies between nearby hyperbolic invariant sets and Hölder stable and unstable foliations with Hölder exponents close to one.
In the case of higher dimensional conformal hyperbolic sets, Pesin [21] and Barreira [3] studied the dimension of a locally maximal hyperbolic invariant sets of C 1+γ conformal dynamical systems. Using techniques in thermodynamic formalism, they proved the Hausdorff dimension, lower and upper box dimensions all agree for the restriction of the hyperbolic invariant set to local stable (unstable) manifolds. In this case, the formula (1.2) also holds.
1.3. Statement of Main Result. In this paper, we introduce the concept of average conformal hyperbolic set, i.e., it admits only one positive and one negative Lyapunov exponents for any ergodic measure (see Definition 2.2). Using MaCluskey and Manning's thermodynamic formalism techniques [18] and Palis and Viana's geometric methods [20], a formula of dimension of locally maximal average conformal hyperbolic sets of a C 1 diffeomorphism is obtained. We also give the estimations of the dimensions of the restriction of C 1 non-conformal hyperbolic invariant set to stable and unstable manifolds (see Lemma 3.5 and 3.6). Furthermore, the dimension of a C 1 average conformal hyperbolic set varies continuously with respect to the dynamics.
The following theorem gives a formula of dimension of locally maximal average conformal hyperbolic sets of a C 1 diffeomorphism. It extends Palis and Viana's result [20] to the case of average conformal hyperbolic sets in higher dimension. It relaxed the smoothness of the results in Pesin's book [21] (see also [3]) to C 1 . Of course, it extends MaCluskey and Manning's result in [18] to both higher dimension and C 1 diffeomorphisms. Furthermore, it gives the continuity of the dimension of average conformal hyperbolic sets, which implies the continuity of the dimension of conformal hyperbolic sets.
Theorem A. Let Λ be a locally maximal average conformal hyperbolic invariant set of a C 1 diffeomorphism f , such that f is transitive on Λ. Then for every x ∈ Λ, The paper is organized as follows. In Section 2, we recall definitions of dimension, topological pressure, and introduce the concept of average conformal hyperbolic sets. In Section 3, we give the detailed proof of the main result.

Definitions and Preliminaries
In this section, we recall the definitions of dimension, entropy and topological pressure. Particularly, we give the definition of average conformal hyperbolic sets and some useful preliminary results.
This limit exists, though the limiting value can be 0 or ∞. We call H s (X) the s−dimensional Hausdorff measure of X. where N (X, δ) denotes the least number of balls of radius δ that are needed to cover the set X.

2.2.
Average Conformal Hyperbolic Sets. Let (M, f ) and Λ be the same as in Section 1.1. We say a diffeomorphism f on Λ is u−conformal (respectively, s−conformal ) if there exists a continuous function a u (x) (respectively, a s (x)) on where Isom x denotes an isometry of E u (x) (respectively, E s (x)). A diffeomorphism f on Λ is called conformal if it is u−conformal and s−conformal, in this case, we also call Λ a conformal hyperbolic set of f ; otherwise, we say that Λ is a non-conformal hyperbolic set of f . Following the idea in [1], we introduce the concept of average conformal hyperbolic sets which are non-conformal case. The average conformal concept was a generalization of quasi-conformal and weakly conformal concept in [2,21]. By the Oseledec multiplicative ergodic theorem (see [19]), there exists a total measure set O ⊂ Λ such that, for each x ∈ O and each invariant measure µ supported on Λ there exist positive integers Here we call the numbers {λ i (x)} the Lyapunov exponents of (f, µ). In the case that µ is an invariant ergodic measure on Λ, the numbers p(x), {m i (x)} and {λ i (x)} are constants. We denote them simply as p, Definition 2.2. A hyperbolic set Λ is called average conformal if it has two unique Lyapunov exponents, one positive and one negative. That is, for any invariant ergodic measure µ on Λ, the Lyapunov exponents are λ 1 (µ) = λ 2 (µ) = · · · = λ k (µ) > 0 and λ k+1 (µ) = λ k+2 (µ) = · · · = λ m (µ) < 0 for some 0 < k < m.
Following the same proof of Theorem 4.2 in [1], we get the following result. and uniformly on Λ, for i ∈ {u, s}.

Entropy and Pressure. We next recall Bowen's definition of a topological
entropy h(f, Y ) for a subset Y of a compact metric space X and a continuous map f : X → X (see [6] for more details). It is defined in a way that resembles Hausdorff dimension. Let A be a finite open cover of X and write E ≺ A if E is contained in some member of A. Denote n A (E) the largest non-negative integer such that and is called the topological entropy of f on the subset Y .
Let f : X → X be a continuous transformation on a compact metric space (X, d), and φ : X → R continuous function on X. In the following, we recall the definition of topological pressure. A subset F ⊂ X is called an (n, ε)−separated set with respect to f if for any x, y ∈ F, x = y, we have d n (x, y) :

Furthermore, a sequence of continuous functions
Definition 2.4. Let Z be a subset of X, and Φ = {φ n } n≥1 a sub-additive/superadditive potential on X, put The following quantity is called the upper sub-additive/super-additive topological pressure of Φ (with respect to f ) on the set Z.
The common value is denoted by P Z (f, Φ), which is called the subadditive/super-additive topological pressure of Φ (with respect to f ) on Z. See [2,21] for proofs.
Let M(X) be the space of all Borel probability measures on X endowed with the weak* topology. Let M f (X) denote the subspace of of M(X) consisting of all f −invariant measures. For µ ∈ M f (X), let h µ (f ) denote the entropy of f with respect to µ, and let Φ * (µ) denote the following limit The existence of the above limit follows from a sub-additive argument. The authors in [8] proved the following variational principle.
Theorem 2.1. Let f : X → X be a continuous transformation on a compact metric space X, and Φ = {φ n } n≥1 a sub-additive potential on X, we have In general, it is still an open question that whether the super-additive topological pressure satisfies the variational principle. However, in the case of average conformal hyperbolic setting, following the same proof of Theorem 5.1 in [1], on can prove the following theorem.
for any t ≥ 0.

Proof of Main Result
This section provides the proof of the main result stated in Section 1.3.
The following theorem shows that the conjugacy map h g in Section 1.1 restricted to local unstable and stable manifolds are Hölder continuous.
Theorem 3.1. Let f : M → M be a C 1 diffeomorphism, and Λ ⊆ M a locally maximal average conformal hyperbolic set. Then for any r ∈ (0, 1), there is C > 0 (depending on r) and a neighborhood U f r of f in Diff 1 (M ) such that, for any g ∈ U f r and any For any r ∈ (0, 1), there exists ε > 0 such that τ e −4ε ≥ τ r . Since f is average conformal on Λ, by Lemma 2.1 there exists a positive integer N (ε) such that for any n ≥ N (ε) and Since Λ is a locally maximal hyperbolic set for f , Λ is also a locally maximal hyperbolic set for F , and the above inequality shows that F satisfies Recall that d u denote the metric induced by the Riemannian structure on the unstable foliation W u and let D y F | E u (y) := D y F | TyW u β (F,x) denote the derivative of F in the unstable direction for any y ∈ W u β (F, x), x ∈ Λ. For the above ε > 0, there exists δ > 0 such that the following is true for all x ∈ Λ, Take U F r a small neighborhood of F in Diff 1 (M ) such that for all G ∈ U F r and x ∈ Λ, we have ≤ e N ε for every y ∈ W u β (F, x).
Since F satisfies (3.1) on Λ, for every y ∈ W u β (F, x), x ∈ Λ. By the following Claim 3.1, there exists C > 0 (depending only on r) such that One may choose a sufficiently small open neighborhood U f r of f in Diff 1 (M ) such that each g ∈ U f r satisfies that g N ∈ U F r . Put G := g N . Note that h g = h G , Λ G = Λ g and so W u β (f,x) Λ ) −1 are (C, r)−Hölder continuous for any g ∈ U f r and any x ∈ Λ.
Claim 3.1. For the above r, F and U F r , there is C > 0 (depending only on r) Proof. Let x ∈ Λ and y, z ∈ W u β (F, x) Λ with d u (y, z) ≤ δ. For any integer n > 0, where ξ j , η j are between F j y and F j z. Let M ≥ 0 be the smallest integer such that Case I: If M = 0, then by (2) we have Case II: If M ≥ 1, let θ u j := θ u G,F j (x) for j ≥ 0, and by (2) we have 3) for j = 0, 1, · · · , M . On the other hand, (3.3) and the positions of ξ j and τ j , we have that It follows from (1), (3) and (3.2) that Hence,
Recall the holonomy maps of unstable and stable foliations which are Lipschitz or Hölder continuous. Let F u , F s be the unstable and stable foliations of hyperbolic dynamical system (f, Λ). For x, y ∈ Λ with x close to y, let F s loc (f, x) and F s loc (f, y) be the local stable foliations of x and y. Define the map h : F s loc (f, x) → F s loc (f, y), sending z to h(z) by sliding along the leaves of F u . The map h is called the holonomy map of F u . The map h is Lipschitz continuous if where z 1 , z 2 ∈ F s loc (f, x) and d x , d y are natural metrics on F s loc (f, x), F s loc (f, y), path metrics with respect to a fixed Riemannian structure on M . The constant L is the Lipschitz constant, and it is independent of the choice of F s . The map h is α−Hölder continuous if where H is the Hölder constant. Similarly we can define the holonomy map of F s . In [15], authors prove the regularity of foliations for C 2 −diffeomorphism. Define four quantities:  , the corresponding holonomy maps are locally uniformly C 1 . Thus the corresponding holonomy maps are Lipschitz continuous. For more information about the regularity of unstable and stable foliations, we refer to [15,16,23] for detailed description.
The following two results are well-known in the field of fractal geometry, e.g., see Falconer's book [12] for proofs. Lemma 3.2. Let X and Y be metric spaces. For any r ∈ (0, 1), Φ : X → Y is an onto, (c, r)-Hölder continuous map for some c > 0.
Corollary 3.1. Let X and Y be metric spaces, and Φ : X → Y is an onto, Lipschitz continuous map. Then Using Theorem 3.1 and the transitive property, the following theorem holds.
To prove Theorem 3.2, we need two lemmas as follow. Take a small number ε > 0 with λe 2ε < 1, where λ is the skewness of the hyperbolicity defined in Section 1.1. Since f is average conformal on Λ, by Lemma 2.1 there exists a positive integer N (ε) such that for any n ≥ N (ε) and x ∈ Λ Fixing any n ≥ N (ε), let F := f n . Then F satisfies (3.1). Since Λ is a locally maximal hyperbolic set for f , Λ is also a locally maximal hyperbolic set for F . Then we have Lemma 3.3 and 3.4. Proof. Since F satisfies (3.1), The desired result follows from Lemma 3.1 and Remark 3.1 immediately. For any r ∈ (0, 1), pick a C 2 diffeomorphism G that is C 1 -close to F such that for all x ∈ Λ G (where Λ G is a locally maximal hyperbolic set of G), As in the proof of Lemma 3.3, we can get ≤ e 2nε λ n < 1.
Therefore the holonomy maps of stable foliation F s and unstable foliation F u for G are Lipschitz. Let x 0 ∈ Λ G be a transitive point. We claim that dim H W u β (G, G j x 0 ) ∩ Λ G is independent of j ≥ 0 and small β > 0. In fact, since G j is a C 2 diffeomorphism, there exists some small β ′ > 0 such that Take M ≥ 1 such that G M (x 0 ) is sufficiently close to x 0 . Since G M is Lipschitz and the holonomy map of F s is Lipschitz, by Corollary 3.1, Moreover, by taking M arbitrarily large, we can suppose that β 0 is close to β and β ′ 0 is arbitrarily small. Therefore dim The claim now immediately follows.
Take any x ∈ Λ G and choose j ≥ 0 such that G j x 0 is close to x. Since the holonomy map of F s is Lipschitz, )∩Λ G and its inverse are (C, r)−Hölder continuous for some C > 0. Notice that r can be arbitrarily close to 1. By Lemma 3.2 and the above argument, we have that dim H W u β (F, x) ∩ Λ is independent of β and x. Similarly, dim B W u β (F, x) ∩ Λ and dim B W u β (F, x) ∩ Λ are independent of β and x.
Proof of Theorem 3.2. We only prove the statements for dimensions of unstable manifolds, since the other statements for dimensions of stable manifolds can be proven in a similar fashion.
First of all, we prove the continuity of the dimensions with respect to f . For any r ∈ (0, 1), take a C 1 diffeomorphism g that is C 1 −close to f , and let Λ g be a locally maximal hyperbolic set for g. By Theorem 3.1, the map h g : W u β (f, x)∩Λ → W u β (g, h g (x)) ∩ Λ g and its inverse are (C, r)−Hölder continuous for some C > 0. It follows from Lemma 3.2 that To prove that dim H (W u β (f, x) ∩ Λ) is independent of β and x. Take ε ∈ (0, − 1 2 log λ), where λ is the skewness of the hyperbolicity. Since f is average conformal on Λ, by Lemma 2.1, we choose a positive integer 2 k ≥ N (ε) such that for any here F := f 2 k . In fact Λ is a locally maximal hyperbolic set for f , Λ is also a locally maximal hyperbolic set for F . Notice that Finally, we prove the last statement that By (3.5), Lemmas 3.5 and 3.6 below, for each x ∈ Λ, where t k u is the unique root of the equation This completes the proof of Theorem 3.2.
Remark 3.2. Using the same arguments as the proof of Theorem 6.3 in [1], one can show that the limit point t u in (3.6) is exactly the unique solution of the following equation . Similarly, for any small β > 0 and every x ∈ Λ one can prove that where t s is the unique solution of the following equation : µ ∈ M f (Λ) (by Lemma 2.1) : µ ∈ M f (Λ) (by Lemma 2.1) where M f (Λ) is the space of all f −invariant measures on Λ. By considering f −1 , one can similarly show that : µ ∈ M f (Λ) .
, and let µ be an ergodic equilibrium state of the topological pressure P Λ (f, −t * φ u (x)) and By the variational principle of topological pressure, we have that Given ε > 0. Let A be a covering of Λ by open sets on each of which φ u (x) varies by at most ε. Denote λ u := φ u dµ. Let l be a Lebesgue number for A. Take any ball W in any unstable manifold and choose m so large that for every r ≥ 1. For each r, we can choose a cover U r of f m W ∩ G µ,r by open set in f m W satisfying For each U ∈ U r , define U * = x∈U∩Λ W s 1 2 l (f, x). For any integer n ≥ 0, if diamf n U < l, then diamf n U * < l. Hence, f n U * is contained in some element of A. For each U ∈ U r , ∃ y ∈ U and z ∈ U ∩ G µ,r such that We may choose U r fine enough so that n A (U * ) > r. By the definition of U * and n A (U * ), we have l ≤ diamf nA(U * ) U . Otherwise, f nA(U * ) U * is contained in some element of A, which contradicts with the definition of n A (U * ). Hence Thus h A (f, G µ ) ≤ (d + ε)(λ u + 2ε). It follows that Since h µ (f ) = h(f, G µ ) (see [6] for the proof), we have that Since ε is arbitrary, we have that Proof. Denote d := dim B W u β (f, x) ∩ Λ , assume that d > 0, otherwise there is nothing to prove. For a small number η > 0 with d − 3η > 0, from the definition of upper box dimension, for each sufficiently large l, there exists 0 < r l < 1/l such that where N W u β (f, x) ∩ Λ, r l denotes the minimal number of balls of radius r l that are needed to cover W u This implies that For a small ε > 0, there exists ρ > 0 such that Hence n i ≤ log 2ρ On the other hand, since f ni+1 : Thus we have r l log C 1 + 1. We now think of having N balls and B baskets.
Then there exists a basket containing at least N B balls. This implies that there exists a positive integer log 2ρ l the last inequality holds since l is sufficiently large. Since B u ni+2 (x i , ρ) ⊂ B u (x i , r l /2) and the balls {B u (x i , r l /2)} N i=1 are disjoint, we have that the set E := {x i : n i = n} is an (n + 2, ρ)−separated subset of W u β (f, x) ∩ Λ . Hence ). It immediately follows that Thus t * ≥ d − 3η. The arbitrariness of η yields that t * ≥ d.
Proof. Since f is average conformal on Λ, by Lemma 2.1 , for any ε ∈ (0, − log λ) we choose a positive integer N ≥ N (ε) such that here F := f N . In fact Λ is a locally maximal hyperbolic set for f , Λ is also a locally maximal hyperbolic set for F . Thus Therefore π s is also a map from W u β (F, x) ∩ Λ to W u β (F, x ′ ) ∩ Λ, and π u is also a map from W s β (F, x) ∩ Λ to W s β (F, x ′′ ) ∩ Λ as follows: π s (y) = W s β (F, y) ∩ W u β (F, x ′ ) and π u (z) = W u β (F, z) ∩ W s β (F, x ′′ ). For any γ ∈ (0, 1), let U F γ be a small C 1 neighborhood of F . Taking G ∈ U F γ ∩ Diff 2 (M ), by Claim 3.1, Lemma 3.3 and Remark 3.1 we have Hölder continuous. Using the same arguments, one can prove that (π s ) −1 , π u and (π u ) −1 are also (D γ , γ)−Hölder continuous.
We proceed to prove Theorem A.
Proof of Theorem A. Step 1. We claim that ∩ Λ Thus the claim holds.
Step 3. We prove the last assertion that the dimensions of an average conformal hyperbolic set varies continuous with respect to f . Since f is average conformal on Λ, for any ε ∈ (0, − 1 2 log λ), by Lemma 2.1 we choose a positive integer N ≥ N (ε) such that for any here F := f N . Since Λ is a locally maximal hyperbolic set for f , Λ is also a locally maximal hyperbolic set for F . Then there exists a neighborhood U F of F in Diff 1 (M ) such that for any G ∈ U F , < e N ε and 1 ≤ D x G| E s (x) m(D x G| E s (x) ) < e N ε for any x ∈ Λ G ,