Dimension approximation in smooth dynamical systems

Abstract

For a non-conformal repeller $\Lambda $ of a $C^{1+\alpha }$ map f preserving an ergodic measure $\mu $ of positive entropy, this paper shows that the Lyapunov dimension of $\mu $ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ of positive entropy, if $(f, \mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$, then the Hausdorff or lower box or upper box dimension of $\mu $ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda _n\}$. The same statement holds true if f is a $C^1$ diffeomorphism with a dominated Oseledet’s splitting with respect to $\mu $.

1 Introduction

In smooth dynamical systems, a fundamental approximation result asserts that a $C^{1+\alpha }$ diffeomorphism f which preserves a hyperbolic ergodic measure $\mu $ of positive entropy can be approximated gradually by compact invariant locally maximal hyperbolic sets–horseshoes $\{\Lambda _n\}$ , in the sense that dynamical quantities on the horseshoes such as the topological entropy and pressure, Lyapunov exponents, and averages of continuous functions are approaching those of the measure $\mu $ .

This type of result is widely referenced to the landmark work by Katok [20] or Katok and Hasselblatt (see [21]). Misiurewicz and Szlenk [28] earlier proved a related result for continuous and for piecewise monotone maps of the interval. Przytycki and Urbański [34] obtained corresponding properties for holomorphic maps in the case of a measure with only positive Lyapunov exponent. A related setting of dyadic diophantine approximations was established by Persson and Schmeling in [32]. For a general $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ with positive entropy, assume that $\mu $ has $\ell $ different Lyapunov exponents $\{\unicode{x3bb} _j\}_{j=1}^{\ell }$ . On each approaching horseshoe $\Lambda _n$ , Avila, Crovisier, and Wilkinson [2] obtained a continuous splitting

$$ \begin{align*}T_{\Lambda_n}M=E_1 \oplus E_2 \oplus \cdots \oplus E_\ell \end{align*} $$

and showed that the exponential growth of $D_xf^n|_{E_i}$ is roughly $\unicode{x3bb} _i$ for each $i=1, 2, \ldots , \ell $ . A corresponding statement for $C^{1+\alpha }$ non-conformal transformations (that is, non-invertible maps) was shown in [11]. See [12], [16, 17], and [44] for other results related to Katok’s approximation construction of $C^{1+\alpha }$ maps.

A natural question is how large is that part of the dynamics described by these horseshoes. So, it is interesting to estimate the Hausdorff dimension of the stable and/or unstable Cantor sets of a horseshoe. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure (that is, a measure with a particular absolute continuity property on unstable manifolds; see [6] for precise definitions), it was shown in [36] that $\mu $ can be approximated by ergodic measures supported on horseshoes with arbitrarily large unstable dimensions, which generalized Mendoza’s result in [26] for diffeomorphisms in a higher dimensional manifold. The approach in [36] was based on Markov towers that can be described by horseshoes with infinitely many branches and variable return times. However, there is an essential mistake in the proof of the key proposition [36, Proposition 5.1]. The authors in [42] proved the same result by a different method. They used the u-Gibbs property of the conditional measure of the equilibrium measure and the properties of the uniformly hyperbolic dynamical systems. Furthermore, in [42], the authors proved that the Hausdorff dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of the horseshoes $\{\Lambda _n\}$ provided that the stable direction is one dimension. See also [24, 25, 27, 29, 37, 38] that represent works close to this topic.

In this work, our main task is to compare the dimension of the horseshoes $\{\Lambda _n\}$ and the given hyperbolic ergodic measure $\mu $ of a $C^{r}$ ( $r\ge 1$ ) diffeomorphism in a more general setting where $\mu $ may not be an SRB measure. For a non-conformal repeller $\Lambda $ of a $C^{1+\alpha }$ map, using the approximation result in [11], we show that the Lyapunov dimension (see equation (3.1) for the definition) of an f-invariant ergodic measure $\mu $ supported on $\Lambda $ can be approximated gradually by the Carathéodory singular dimension (see equation (3.6) for the definition) of the horseshoes $\{\Lambda _n\}$ . For a $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ of positive entropy, if $(f, \mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ , then the Hausdorff or lower box or upper box dimension of $\mu $ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda _n\}$ . The same statement holds true if f is a $C^1$ diffeomorphism with a dominated Oseledec’s splitting with respect to $\mu $ .

We arrange the paper as follows. In §2, we give some basic notions and properties about topological and measure theoretic pressures, and dimensions of sets and measures. Statements of our main results will be given in §3. In §4, we will give the detailed proofs of the main results.

2 Definitions and preliminaries

In this section, we recall the definitions of topological pressure and various dimensions of subsets and/or of invariant measures.

2.1 Topological and measure theoretic pressures

Let $f: X\to X$ be a continuous transformation on a compact metric space X equipped with metric d. A subset $F\subset X$ is called an $(n, \epsilon )$ -separated set with respect to f if for any two different points $x,y\in F$ , we have $d_n(x,y):=\max _{0\leq k\leq n-1}d(f^k(x), f^k(y))>\epsilon .$ A sequence of continuous functions $\Phi =\{\phi _n\}_{n\ge 1}$ is called sub-additive if

$$ \begin{align*}\phi_{m+n}\leq\phi_n+\phi_m\circ f^n\quad\text{for all } n,m\in \mathbb{N}.\end{align*} $$

Furthermore, a sequence of continuous functions $\Psi =\{\psi _n\}_{n\ge 1}$ is called super-additive if $-\Psi =\{-\psi _n\}_{n\ge 1}$ is sub-additive.

2.1.1 Topological pressure defined via separated sets

Given a sub-additive potential $\Phi =\{\phi _n\}_{n\ge 1}$ on X, put

$$ \begin{align*}P_n(f, \Phi, \epsilon)=\sup\bigg\{\sum_{x\in F}e^{\phi_n(x)}| F\ \mbox{is an}\ (n, \epsilon)\mbox{-separated subset of } X\bigg\}.\end{align*} $$

Definition 2.1. We call the quantity

(2.1) $$ \begin{align} P_{\mathrm{top}}(f,\Phi)=\lim_{\epsilon\to 0}\limsup_{n\to\infty}\frac{1}{n}\log P_n(f,\Phi,\epsilon) \end{align} $$

the sub-additive topological pressure of $(f,\Phi )$ .

Remark 2.2. If $\varPhi =\{\varphi _n\}_{n\geq 1}$ is additive in the sense that $\varphi _n(x)=\varphi (x)+\varphi (fx)+\cdots +\varphi (f^{n-1}x)\triangleq S_n\varphi (x)$ for some continuous function $\varphi : X\to \mathbb {R}$ , we simply denote the topological pressure $P_{\mathrm {top}}(f, \varPhi )$ as $P_{\mathrm {top}}(f, \varphi )$ .

Let $\mathcal {M}_f(X)$ denote the space of all f-invariant measures on X. For $\mu \in \mathcal {M}_f(X)$ , let $h_\mu (f)$ denote the metric entropy of f with respect to $\mu $ (see Walters’ book [39] for details of metric entropy), and let

$$ \begin{align*}\mathcal{L}_*(\Phi,\mu)=\lim_{n\to\infty}\frac 1n \int\phi_n\,d\mu.\end{align*} $$

The existence of the above limit follows from a sub-additive argument. In [9], the authors proved the following variational principle.

Theorem 2.3. Let $f:X\to X$ be a continuous transformation on a compact metric space X, and $\Phi =\{\phi _n\}_{n\ge 1}$ a sub-additive potential on X, then we have

$$ \begin{align*}P_{\mathrm{top}}(f,\Phi)=\sup\{h_\mu(f)+\mathcal{L}_*(\Phi,\mu): \mu\in\mathcal{M}_f(X),~\mathcal{L}_*(\Phi,\mu)\neq -\infty\}.\end{align*} $$

Here we take the convention that $\sup \emptyset =-\infty $ .

Although it is unknown whether the variational principle holds for super-additive topological pressure, Cao, Pesin, and Zhao gave an alternative definition via variational principle in [11]. Given a sequence of super-additive continuous potentials $\Psi =\{\psi _n\}_{n\geq 1}$ on a compact dynamical system $(X, f)$ , the super-additive topological pressure of $\Psi $ is defined as

$$ \begin{align*}P_{\mathrm{var}}(f, \Psi) := \sup\{h_\mu(f)+\mathcal{L}_*(\Psi,\mu): \mu\in\mathcal{M}_f(X)\},\end{align*} $$

where

$$ \begin{align*}\mathcal{L}_*(\Psi,\mu)=\lim_{n\to\infty}\frac 1n \int\psi_n\,d\mu=\sup_{n\geq1}\frac 1n \int\psi_n\,d\mu.\end{align*} $$

The second equality is due to the standard sub-additive argument.

2.1.2 Measure theoretic pressure

We first follow the approach in [33] to give the definitions of topological pressures on arbitrary subsets. Given a sub-additive potential $\Phi =\{\phi _n\}_{n\ge 1}$ on X, a subset $Z\subset X$ , and $\alpha \in \mathbb {R}$ , let

$$ \begin{align*} \begin{aligned} M(Z,\Phi,\alpha,N,\epsilon)=\inf\bigg\{&\sum_i \exp\Big(-\alpha n_i+\sup_{y\in B_{n_i}(x_i,\epsilon)}\phi_{n_i}(y)\Big):\\ &\bigcup_{i}B_{n_i}(x_i,\epsilon)\supset Z, \, x_i\in X \text{ and } n_i\ge N \text{ for all } i\bigg\}. \end{aligned} \end{align*} $$

Since $M(Z,\Phi ,\alpha ,N,\epsilon )$ is monotonically increasing with N, let

(2.2) $$ \begin{align} m(Z,\Phi,\alpha,\epsilon) :=\lim_{N\rightarrow\infty}M(Z,\Phi,\alpha,N,\epsilon). \end{align} $$

We denote the jump-up point of $m(Z,\Phi ,\alpha ,\epsilon )$ by

$$ \begin{align*} P_{Z}(f,\Phi,\epsilon) =\inf \{ \alpha: m(Z,\Phi,\alpha,\epsilon)=0 \} =\sup\{ \alpha: m(Z,\Phi,\alpha,\epsilon)=+\infty\}. \end{align*} $$

Definition 2.4. We call the quantity

$$ \begin{align*} P_{Z}(f,\Phi)=\liminf_{\epsilon\to 0} P_{Z}(f,\Phi,\epsilon) \end{align*} $$

the topological pressure of $(f, \Phi )$ on the set Z (see [14] for the weighted version of this quantity).

Similarly, for $\alpha \in \mathbb {R}$ and $Z\subset X$ , define

$$ \begin{align*} R(Z,\Phi,\alpha,N,\epsilon)=\inf\bigg\{&\sum_i \exp\Big(-\alpha N+\sup_{y\in B_{N}(x_i,\epsilon)} \phi_N(y) \Big):\\ &\bigcup_{i}B_{N}(x_i,\epsilon)\supset Z,\,x_i\in X\bigg\}. \end{align*} $$

We set

$$ \begin{align*} \begin{aligned} \underline{r}(Z,\Phi,\alpha,\epsilon) &=\liminf_{N\rightarrow\infty}R(Z,\Phi,\alpha,N,\epsilon), \\ \overline{r}(Z,\Phi,\alpha,\epsilon) &=\limsup_{N\rightarrow\infty}R(Z,\Phi,\alpha,N,\epsilon) \end{aligned} \end{align*} $$

and define the jump-up points of $\underline {r}(Z,\Phi ,\alpha ,\epsilon )$ and $\overline {r}(Z,\Phi ,\alpha ,\epsilon )$ as

$$ \begin{align*} \begin{aligned} \underline{CP}_{Z}(f,\Phi,\epsilon)&=\inf\{\alpha: \underline{r}(Z,\Phi,\alpha,\epsilon)=0\} =\sup\{ \alpha: \underline{r}(Z,\Phi,\alpha,\epsilon)=+\infty \},\\ \overline{CP}_{Z}(f,\Phi,\epsilon)&=\inf \{ \alpha: \overline{r}(Z,\Phi,\alpha,\epsilon)=0 \} =\sup\{ \alpha: \overline{r}(Z,\Phi,\alpha,\epsilon)=+\infty\}, \end{aligned} \end{align*} $$

respectively.

Definition 2.5. We call the quantities

$$ \begin{align*} \underline{CP}_{Z}(f,\Phi)=\liminf_{\epsilon\to 0} \underline{CP}_{Z}(f,\Phi,\epsilon)\,\,\text{and}\,\, \overline{CP}_{Z}(f,\Phi)=\liminf_{\epsilon\to 0} \overline{CP}_{Z}(f,\Phi,\epsilon) \end{align*} $$

the lower and upper topological pressures of $(f,\Phi )$ on the set Z, respectively.

Given an f-invariant measure $\mu $ , let

$$ \begin{align*} \begin{aligned} P_{\mu}(f,\Phi,\epsilon) =\inf\{P_Z(f,\Phi,\epsilon)\colon\mu (Z)=1\} \end{aligned} \end{align*} $$

and then we call the quantity

$$ \begin{align*} P_{\mu}(f,\Phi):=\liminf_{\epsilon\to 0}P_{\mu}(f,\Phi,\epsilon) \end{align*} $$

the measure theoretic pressure of $(f,\Phi )$ with respect to $\mu $ . Let further

$$ \begin{gather*} \underline{CP}_{\mu}(f,\Phi,\epsilon) =\lim_{\delta\to 0}\inf\{\underline{CP}_{Z}(f,\Phi,\epsilon) \colon\mu (Z)\ge 1-\delta\}, \\ \overline{CP}_{\mu}(f,\Phi,\epsilon) =\lim_{\delta\to 0}\inf\{\overline{CP}_{Z}(f,\Phi,\epsilon) \colon\mu (Z)\ge 1-\delta\}. \end{gather*} $$

We call the quantities

$$ \begin{align*} \underline{CP}_{\mu}(f,\Phi)=\liminf_{\epsilon\to 0} \underline{CP}_{\mu}(f,\Phi,\epsilon),\ \ \ \overline{CP}_{\mu}(f,\Phi)=\liminf_{\epsilon\to 0} \overline{CP}_{\mu}(f,\Phi,\epsilon) \end{align*} $$

the lower and upper measure theoretic pressures of $(f,\Phi )$ with respect to $\mu $ , respectively. It is proved in [10, Theorem A] that

(2.3) $$ \begin{align} P_{\mu}(f,\Phi)=\underline{CP}_{\mu}(f,\Phi)=\overline{CP}_{\mu}(f,\Phi)=h_{\mu}(f)+ \mathcal{L}_*(\Phi,\mu) \end{align} $$

for any f-invariant ergodic measure $\mu $ with $\mathcal {L}_*(\Phi ,\mu )\neq -\infty $ .

Remark 2.6. In fact, one can show that

$$ \begin{align*} \mathcal{P}_\mu(f,\Phi)=\inf\{\mathcal{P}_Z(f,\Phi):\mu(Z)=1\}, \end{align*} $$

where $\mathcal {P}$ denotes either P or $\underline {CP}$ or $\overline {CP}$ , see [46] for a proof.

2.2 Dimensions of sets and measures

Now we recall the definitions of Hausdorff and box dimensions of subsets and measures. Given a subset $Z\subset X$ , for any $s\ge 0$ , let

$$ \begin{align*} \mathcal{H}_{\delta}^{s}(Z)=\inf\bigg\{\sum\limits_{i=1}^{\infty}(\mbox{diam}\, U_{i})^{s}:\{U_{i}\}_{i\ge1}\ \mbox{is a cover of } Z \ \mbox{with } \mbox{diam}\,U_{i}\le \delta, \text{ for all } i\ge1 \bigg\} \end{align*} $$

and

$$ \begin{align*} \mathcal{H}^{s}(Z)=\lim\limits_{\delta\rightarrow 0}\mathcal{H}_{\delta}^{s}(Z). \end{align*} $$

The above limit exists, although the limit may be infinity. We call $\mathcal {H}^{s}(Z)$ the s-dimensional Hausdorff measure of Z.

Definition 2.7. The following jump-up value of $\mathcal {H}^{s}(Z)$

$$ \begin{align*}\dim_{H}{\kern-1pt}Z=\inf\{s:\mathcal{H}^{s}(Z)=0\}=\sup\{s:\mathcal{H}^{s}(Z)=\infty\}\end{align*} $$

is called the Hausdorff dimension of Z. The lower and upper box dimension of Z are defined respectively by

$$ \begin{align*}\underline{\dim}_{B}{\kern1pt}Z=\liminf\limits_{\delta\to0}\frac{\log N(Z,\delta)}{-\log\delta}\quad \text{and}\quad \overline{\dim}_{B}{\kern1pt}Z=\limsup\limits_{\delta\to0}\frac{\log N(Z,\delta)}{-\log\delta},\end{align*} $$

where $N(Z,\delta )$ denotes the least number of balls of radius $\delta $ that are needed to cover the set Z. If $\underline {\dim }_BZ=\overline {\dim }_BZ$ , we will denote the common value by $\dim_{B}{\kern-1pt}Z$ and call it the box dimension of Z.

The following two results are well known in the field of fractal geometry, e.g. see Falconer’s book [13] for proofs.

Lemma 2.8. Let X and Y be metric spaces. For any $r\in (0, 1)$ , $\Phi : X \to Y$ is an onto, $(C, r)$ -H $\ddot {o}$ lder continuous map for some $C>0$ . Then

$$ \begin{align*}\dim_{H}{\kern-1pt}Y \leq r^{-1} \dim_{H}{\kern-1pt}X,\quad \underline{\dim}_{B}{\kern1pt}Y \leq r^{-1} \underline{\dim}_{B}{\kern1pt}X \quad \text{and} \quad \overline{\dim}_{B}{\kern1pt}Y \leq r^{-1} \overline{\dim}_{B}{\kern1pt}X.\end{align*} $$

Corollary 2.9. Let X and Y be metric spaces, and let $\Phi : X \to Y$ be an onto, Lipschitz continuous map. Then

$$ \begin{align*}\dim_{H}{\kern-1pt}Y \leq \dim_{H}{\kern-1pt}X,\quad \underline{\dim}_{B}{\kern1pt}Y \leq \underline{\dim}_{B}{\kern1pt}X \quad \text{and} \quad \overline{\dim}_{B}{\kern1pt}Y \leq \overline{\dim}_{B}{\kern1pt}X.\end{align*} $$

Given a Borel probability measure $\mu $ on X, the quantity

$$ \begin{align*} \begin{aligned} \dim_{H}{\kern-1pt}\mu &=\inf\{\dim_{H}{\kern-1pt}Z: Z\subset X \text{ and }\mu(Z)=1\}\\ &=\lim_{\delta\to 0} \inf\{\dim_{H}{\kern-1pt}Z: Z\subset X \text{ and }\mu(Z)\ge1-\delta\} \end{aligned} \end{align*} $$

is called the Hausdorff dimension of the measure $\mu $ . Similarly, we call the two quantities

$$ \begin{align*} \underline{\dim}_{B}{\kern1pt}\mu=\lim_{\delta\to 0} \inf\{\underline{\dim}_{B}{\kern1pt}Z: Z\subset X \text{ and }\mu(Z)\ge1-\delta\} \end{align*} $$

and

$$ \begin{align*} \overline{\dim}_{B}{\kern1pt}\mu=\lim_{\delta\to 0} \inf\{\overline{\dim}_{B}{\kern1pt}Z: Z\subset X \text{ and }\mu(Z)\ge1-\delta\} \end{align*} $$

the lower box dimension and upper box dimension of $\mu $ , respectively.

If $\mu $ is a finite measure on X and there exists $d\geq 0$ such that

$$ \begin{align*}\lim_{r\to0}\frac{\log\mu(B(x, r))}{\log r} = d\end{align*} $$

for $\mu $ -almost every $x\in X$ , then

$$ \begin{align*}\dim_{H}{\kern-1pt}\mu = \underline{\dim}_{B}{\kern1pt}\mu = \overline{\dim}_{B}{\kern1pt}\mu=d. \end{align*} $$

This criterion was established by Young in [45].

3 Statements of main results

In this section, we will give the statements of the main results in this paper, and the proof will be postponed to the next section.

3.1 Dimension approximation for uniformly expanding systems

Let $f:M\to M$ be a smooth map of an $m_0$ -dimensional compact smooth Riemannian manifold M, and $\Lambda $ a compact f-invariant subset of M. Let $\mathcal {M}_f(\Lambda )$ and $\mathcal {E}_f(\Lambda )$ denote respectively the set of all f-invariant measures and ergodic measures on $\Lambda $ .

3.1.1 Definitions of repeller and Lyapunov dimension

We call $\Lambda $ a repeller for f or f is expanding on $\Lambda $ if:

1. (1) there exists an open neighborhood U of $\Lambda $ such that $\Lambda =\{x\in U: f^n(x)\in U \text {for all}\,\, n\ge 0\}$ ;

2. (2) there is $\kappa> 1 $ such that

   $$ \begin{align*} \|D_xf (v) \| \ge \kappa \|v\| \quad \text{for all } x \in \Lambda \text{ and } v \in T_xM, \end{align*} $$
   where $\|\cdot \|$ is the norm induced by the Riemannian metric on M, and $D_xf:T_xM\rightarrow T_{f(x)}M$ is the differential operator.

Given an f-invariant ergodic measure $\mu $ supported on the repeller $\Lambda $ , let $\unicode{x3bb} _1(\mu )\ge \unicode{x3bb} _2(\mu )\ge \cdots \ge \unicode{x3bb} _{m_0}(\mu )$ and $h_{\mu }(f)$ denote the Lyapunov exponents and the measure theoretic entropy of $(f,\mu )$ , respectively. We refer the reader to [6, 39] for detailed descriptions of Lyapunov exponents and the measure theoretic entropy. We further define the Lyapunov dimension of $\mu $ as follows:

(3.1)

where $\ell =\max \{i: \unicode{x3bb} _{m_0}(\mu )+\cdots +\unicode{x3bb} _{m_0-i +1}(\mu )\le h_{\mu }(f)\}$ .

The original definition of Lyapunov dimension in [1, 19, 22] is defined only for hyperbolic systems as follows: assume that $\nu $ is an ergodic measure of a smooth diffeomorphism f with Lyapunov exponents $\unicode{x3bb} _1\ge \cdots \ge \unicode{x3bb} _u>0\ge \unicode{x3bb} _{u+1}\ge \cdots \ge \unicode{x3bb} _{m_0}$ , then the the Lyapunov dimension is

where $\ell =\max \{i:\unicode{x3bb} _1+\cdots +\unicode{x3bb} _i\ge 0\}$ . Assume further that $\nu $ is an SRB measure, then $h_\nu (f)=\unicode{x3bb} _1+\cdots +\unicode{x3bb} _u$ . As a consequence,

and $\ell =\max \{i:-\unicode{x3bb} _{u+1}-\cdots -\unicode{x3bb} _i\ge h_\nu (f)\}$ . Hence, the definition in equation (3.1) is a reasonable substitute. For a $C^1$ expanding map f, Feng and Simon [15] defined the Lyapunov dimension of an ergodic measure as the zero of the measure theoretic pressure $P_\mu (f,\Phi _f(t))=0$ (see equation (3.4)). In this paper, we will prove that the unique solution of the equation $P_\mu (f,\Phi _f(t))=0$ is indeed our definition of Lyapunov dimension (see Theorem A). Furthermore, this paper shows that the Lyapunov dimension of an ergodic measure defined in equation (3.1) is equal to its Carathéodory singular dimension (see Proposition 3.2), so the Carathéodory singular dimension (see §3.1.3 for the detailed definition) can be regarded as a geometric explanation of the Lyapunov dimension.

3.1.2 Singular valued potentials

Let $\Lambda $ be a repeller of a smooth map $f:M\to M$ . Given $x\in \Lambda $ and $n\ge 1$ , consider the differentiable operator $D_xf^n: T_xM\to T_{f^n(x)}M$ and denote the singular values of $D_xf^n$ (square roots of the eigenvalues of $(D_xf^n)^*D_xf^n$ ) in the decreasing order by

(3.2) $$ \begin{align} \alpha_1(x,f^n)\ge\alpha_2(x,f^n)\ge\dots\ge\alpha_{m_0}(x,f^n). \end{align} $$

For $t\in [0,m_0]$ , set

(3.3) $$ \begin{align} \varphi^{t}(x,f^n): =\sum_{i=m_0-[t]+1}^{m_0}\log\alpha_i(x,f^n) +(t-[t])\log\alpha_{m_0-[t]}(x,f^n). \end{align} $$

Since f is smooth, the functions $x\mapsto \alpha _i(x,f^n)$ , $x\mapsto \varphi ^t(x,f^n)$ are continuous for any $n\ge 1$ . It is easy to see that for all $n,\ell \in \mathbb {N}$ ,

$$ \begin{align*} \varphi^t(x,f^{n+\ell})\ge\varphi^t(x,f^n)+\varphi^t(f^n(x),f^\ell). \end{align*} $$

It follows that the sequence of functions

(3.4) $$ \begin{align} \Phi_f(t):=\{-\varphi^t(\cdot,f^n)\}_{n\ge 1} \end{align} $$

is sub-additive, which is called the sub-additive singular valued potentials.

3.1.3 Carathéodory singular dimension

We recall the definition of Carathéodory singular dimension of a repeller which is introduced in [11].

Let $\Phi _f(t)=\{-\varphi ^{t}(\cdot ,f^n)\}_{n\ge 1}$ . Given a subset $Z\subseteq \Lambda $ , for each small number $r>0$ , let

$$ \begin{align*} m(Z,t,r):=\lim_{N\to\infty}\inf\bigg\{\sum_i\exp\Big(\sup_{y\in B_{n_i}(x_i,r)}-\varphi^{t}(y,f^{n_i})\Big)\bigg\}, \end{align*} $$

where the infimum is taken over all collections $\{B_{n_i}(x_i,r)\}$ of Bowen’s balls with $x_i\in \Lambda $ , $n_i\ge N$ that cover Z. It is easy to see that there is a jump-up value

(3.5) $$ \begin{align} \dim_{C,r}{\kern-1pt}Z:=\inf\{t: m(Z,t,r)=0\}=\sup\{t: m(Z,t,r)=+\infty\}. \end{align} $$

The quantity

(3.6) $$ \begin{align} \displaystyle{\dim_{C}{\kern-1pt}Z:=\liminf_{r\to 0}\dim_{C,r}{\kern-1pt}Z }\end{align} $$

is called the Carathéodory singular dimension of Z. Particularly, the Carathéodory singular dimension of the repeller $\Lambda $ is independent of the parameter r for small values of $r>0$ (see [11, Theorem 4.1]).

For each f-invariant measure $\mu $ supported on $\Lambda $ , let

$$ \begin{align*} \dim_{C,r}{\kern-1pt}\mu:=\inf\{\dim_{C,r}{\kern-1pt}Z: \mu(Z)=1 \}, \end{align*} $$

and the quantity

$$ \begin{align*} \dim_{C}{\kern-1pt}\mu:=\liminf_{r\to 0}\dim_{C,r}{\kern-1pt}\mu \end{align*} $$

is called the Carathéodory singular dimension of the measure $\mu $ .

3.1.4 Approximation of Carathéodory singular dimension of repellers

Given a repeller $\Lambda $ of a $C^{1+\alpha }$ map f, the following result shows that the zero of the measure theoretic pressure function is exactly the Lyapunov dimension of an ergodic measure $\mu \in \mathcal {E}_f(\Lambda )$ , and the Lyapunov dimension of an ergodic measure of positive entropy can be approximated by the Carathéodory singular dimension of a sequence of invariant sets. Recall that $ \Phi _f(t):=\{-\varphi ^t(\cdot ,f^n)\}_{n\ge 1} $ is the sub-additive singular valued potentials with respect to f (see the definition in equation (3.4)).

The following result gives a measure theoretic version of Bowen’s equation, that is, the unique zero of the measure theoretic pressure is exactly the Lyapunov dimension of an ergodic measure.

Proposition 3.1. Let $f: M\to M$ be a $C^{1}$ map of an $m_0$ -dimensional compact smooth Riemannian manifold M, and $\Lambda $ a repeller of f. For every f-invariant ergodic measure $\mu $ supported on $\Lambda $ , we have that

$$ \begin{align*}\dim_{\mathrm{L}}{\kern-1pt}\mu=s_{\mu},\end{align*} $$

where $s_{\mu }$ is the unique root of the equation $P_\mu (f, \Phi _f(t))=0$ .

For an ergodic measure supported on a repeller with positive entropy, one can find a sequence of compact invariant sets whose Carathéodory singular dimension gradually approaches the Lyapunov dimension of the measure.

Theorem A. Let $f: M\to M$ be a $C^{1+\alpha }$ map of an $m_0$ -dimensional compact smooth Riemannian manifold M and $\Lambda $ a repeller of f, and let $\mu $ be an f-invariant ergodic measure on $\Lambda $ with $h_{\mu }(f)>0$ . For any $\varepsilon>0$ , there exists an f-invariant compact subset $\Lambda _\varepsilon \subset \Lambda $ such that $\dim _C{\kern-1pt} \Lambda _\varepsilon \rightarrow \dim _{\mathrm {L}}{\kern-1pt}\mu $ as $\varepsilon $ approaches zero.

Some comments on the previous theorem are in order. First, the map of higher smoothness $C^{1+\alpha }$ is crucial as it allows us to use some powerful results of Pesin theory. Second, if f is a local diffeomorphism preserving an ergodic expanding measure $\mu $ of positive entropy, that is, $(f,\mu )$ has only positive Lyapunov exponent, in this case, one can also obtain an approximation result as in [11] so that we can obtain the second statement in the previous theorem in this setting. In [37], for a $C^2$ interval map f with finitely many non-degenerate critical points, the author proved that the Hausdorff dimension of an expanding measure $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of repellers.

For each f-invariant ergodic measure $\mu $ supported on $\Lambda $ , the following result shows that the Carathéodory singular dimension of $\mu $ is exactly its Lyapunov dimension.

Proposition 3.2. Let $f: M\to M$ be a $C^{1}$ map of an $m_0$ -dimensional compact smooth Riemannian manifold M, and $\Lambda $ a repeller for f. Then the following statements hold:

1. (1) for each subset $Z\subset \Lambda $ , we have that

   $$ \begin{align*} \dim_C{\kern-1pt} Z=t_Z, \end{align*} $$
   where $t_Z$ is the unique root of the equation $P_Z(f, \Phi _f(t))=0$ ;

2. (2) for each f-invariant ergodic measure $\mu $ supported on $\Lambda $ , we have that

   $$ \begin{align*} \dim_C{\kern-1pt}\mu=\dim_{\mathrm{L}}{\kern-1pt}\mu. \end{align*} $$

3.2 Dimension approximation in non-uniformly hyperbolic systems

In this section, we first recall an approximation result in non-uniformly hyperbolic systems that is proved by Avila, Crovisier, and Wilkinson [2], then we give the statement of our dimension approximation result in non-uniformly hyperbolic systems.

3.2.1 Lyapunov exponents and holonomy maps

Let $f: M\to M$ be a diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M. By Oseledec’s multiplicative ergodic theorem (see [30]), there exists a total measure set $\mathcal {O}\subset M$ such that for each $x \in \mathcal {O}$ and each invariant measure $\mu $ , there exist positive integers $d_1(x),d_2(x),\ldots , d_{p(x)}(x)$ , numbers $\unicode{x3bb} _1(x)>\unicode{x3bb} _2(x)>\cdots >\unicode{x3bb} _{p(x)}(x)$ , and a splitting

$$ \begin{align*}T_xM=E_1(x)\oplus E_2(x)\oplus\cdots \oplus E_{p(x)}(x),\end{align*} $$

which satisfy that:

1. (1) $D_xf E_i(x)=E_i(f(x))$ for each i and $\sum _{i=1}^{p(x)}d_i(x)=m_0$ ;

2. (2) for each $0\neq v\in E_i(x)$ , we have that

   

Here we call the numbers $\{\unicode{x3bb} _i(x)\}_{i=1}^{p(x)}$ the Lyapunov exponents of $(f,\mu )$ . In the case that $\mu $ is an f-invariant ergodic measure, the numbers $p(x)$ , $\{d_i(x)\}$ , and $\{\unicode{x3bb} _i(x)\}$ are constants almost everywhere. We denote them simply by p, $\{d_i\}_{i=1}^p$ , and $\{\unicode{x3bb} _i\}_{i=1}^p$ .

A compact invariant subset $\Lambda \subset M$ is called a hyperbolic set if there exists a continuous splitting of the tangent bundle $T_\Lambda M = E^{s}\oplus E^{u}$ , and constants $C> 0,\ 0 < \unicode{x3bb} < 1$ such that for every $x \in \Lambda $ :

1. (1) $D_xf(E^s(x)) = E^s(f(x)),\ D_xf(E^u(x)) = E^u(f(x))$ ;

2. (2) for all $n \geq 0, \|D_xf^n(v)\|\leq C\unicode{x3bb} ^n\|v\|$ if $v \in E^s(x)$ , and $\|D_xf^{-n}(v)\|\leq C\unicode{x3bb} ^n\|v\|$ if $v \in E^u(x)$ .

Given a point $x \in \Lambda $ , for each small $\beta>0$ , the local stable and unstable manifolds are defined as follows:

$$ \begin{align*} W_\beta^s(f,x)&=\{y \in M : d(f^n(x),f^n(y)) \leq \beta\ \text{ for all } n \geq 0\},\\ W_\beta^u(f,x)&=\{y \in M : d(f^{-n}(x),f^{-n}(y)) \leq \beta\ \text{ for all } n \geq 0\}. \end{align*} $$

The global stable and unstable sets of $x\in \Lambda $ are given as follows:

$$ \begin{align*}W^s(f,x)=\bigcup_{n\geq0}f^{-n}(W^s_\beta(f,f^n(x))),\quad W^u(f,x)=\bigcup_{n\geq0}f^n(W^u_\beta(f,f^{-n}(x))).\end{align*} $$

A hyperbolic set is called locally maximal if there exists a neighborhood U of $\Lambda $ such that $\Lambda =\bigcap _{n\in \mathbb {Z}}f^n(U)$ . Recall that a horseshoe for a diffeomorphism f is a transitive, locally maximal hyperbolic set that is totally disconnected and not finite.

Let $W^u$ and $W^s$ be the unstable and stable foliations of a hyperbolic dynamical system $(f, \Lambda )$ . For $x, y\in \Lambda $ with x close to y, let $W_\beta ^u(f, x)$ and $W_\beta ^s(f, x)$ be the local stable foliations of x and y. Define the map $h: W_\beta ^s(f, x) \to W_\beta ^s(f, y)$ sending z to $h(z)$ by sliding along the leaves of $W^u$ . The map h is called the holonomy map of $W^u$ . The map h is Lipschitz continuous if

$$ \begin{align*}d_y(h(z_1), h(z_2)) \leq L d_x(z_1, z_2),\end{align*} $$

where $z_1, z_2 \in W_\beta ^s(f, x)$ and $d_x, d_y$ are natural path metrics on $W_\beta ^s(f, x)$ , $W_\beta ^u(f, y)$ with respect to a fixed Riemannian structure on M. The constant L is the Lipschitz constant, and it is independent of the choice of $W^s$ . The map h is $\alpha $ -Hölder continuous if

$$ \begin{align*}d_y(h(z_1), h(z_2)) \leq H d_x(z_1, z_2)^\alpha,\end{align*} $$

where H is the Hölder constant. Similarly, we can define the holonomy map of $W^s$ .

3.2.2 Approximation of Lyapunov exponents and entropy

For a $C^{1+\alpha }$ diffeomorphism $f:M\rightarrow M$ , Katok [20] showed that an f-invariant ergodic hyperbolic measure (a measure has no zero Lyapunov exponents) with positive metric entropy can be approximated by horseshoes. However, Katok’s result does not explicitly mention a control of the Oseledets splitting over the horseshoes. Recently, Avila, Crovisier, and Wilkinson [2] showed that there is a dominated splitting over the horseshoes, with approximately the same Lyapunov exponents on each sub-bundle of the splitting.

Recall that $Df$ -invariant splitting on a compact f-invariant subset $\Lambda $ :

$$ \begin{align*} T_\Lambda M=E_1\oplus E_2\oplus \cdots \oplus E_{\ell},~~(\ell\ge 2), \end{align*} $$

is a dominated splitting if there exists $N\ge 1$ such that for every $x\in \Lambda $ , any unit vectors $v,w\in T_xM$ :

$$ \begin{align*} v\in E_i(x), w\in E_j(x) \quad \text{with}~~i<j\Longrightarrow \|D_xf^N(v)\|\ge 2\|D_xf^N(w)\|. \end{align*} $$

We write $E_1{\kern-1pt}\succeq{\kern-1pt} E_2 {\kern-1pt}\succeq{\kern-1pt} \cdots {\kern-1pt}\succeq{\kern-1pt} E_{\ell }$ . Furthermore, if there are numbers $\unicode{x3bb} _1{\kern-1pt}>{\kern-1pt}\unicode{x3bb} _2{\kern-1pt}>{\kern-1pt}\cdots >\unicode{x3bb} _{\ell }$ , constants $C>0$ , and ${0<\varepsilon < \min _{1\leq i < \ell } ({\unicode{x3bb} _{i}-\unicode{x3bb} _{i+1}})/{100}}$ such that for every $x\in \Lambda $ , $n\in \mathbb {N}$ , $1\leq j \leq \ell $ and each unit vector $u\in E_j(x)$ , it holds that

then we say that

$$ \begin{align*} T_\Lambda M=E_1\oplus E_2\oplus \cdots \oplus E_{\ell},~~(\ell\ge 2) \end{align*} $$

is a $\{\unicode{x3bb} _j\}_{1\leq j \leq \ell }$ -dominated splitting.

For the reader’s convenience, we recall Avila, Crovisier, and Wilkinson’s approximation results in the following, see [2] for more details.

Theorem 3.3. Let $f:M\to M$ be a $C^{1+\alpha }$ diffeomorphism, and $\mu $ an f-invariant ergodic hyperbolic measure with $h_\mu (f)>0$ . For each $\varepsilon>0$ and a weak- $*$ neighborhood $\mathcal {V}$ of $\mu $ in the space of f-invariant probability measures on M, there exists a compact set $\Lambda _\varepsilon ^*\subset M$ and a positive integer N such that the following properties hold:

1. (1) $\Lambda _\varepsilon ^*$ is a locally maximal hyperbolic set and topologically mixing with respect to $f^N$ ;

2. (2) $h_{\mu }(f){\kern-1pt}-{\kern-1pt}\varepsilon {\kern-1pt}<{\kern-1pt}h_{top}(f,\Lambda _\varepsilon ){\kern-1pt}<{\kern-1pt}h_{\mu }(f)+\varepsilon $ , where $\Lambda _\varepsilon {\kern-1pt}={\kern-1pt}\Lambda _\varepsilon ^* \cup f(\Lambda _\varepsilon ^*) \cup \cdots f^{N-1}(\Lambda _\varepsilon ^*)$ ;

3. (3) $\Lambda _\varepsilon $ is $\varepsilon $ -close to the support of $\mu $ in the Hausdorff distance;

4. (4) each invariant probability measure supported on the horseshoe $\Lambda _\varepsilon $ lies in $\mathcal {V}$ ;

5. (5) if $\unicode{x3bb} _1>\unicode{x3bb} _2>\cdots >\unicode{x3bb} _{\ell }$ are the distinct Lyapunov exponents of $(f,\mu )$ , with multiplicities $d_1, d_2,\ldots , d_{\ell }$ , then there exists a $\{\unicode{x3bb} _j\}_{1\le j<\ell }$ -dominated splitting $T_{\Lambda _\varepsilon } M=E_1\oplus E_2\oplus \cdots \oplus E_{\ell }$ with $\dim E_i=d_i$ for each i, and for each $x\in \Lambda _\varepsilon $ , $k\geq 1$ and each vector $v\in E_i(x)$

   

Remark 3.4. In the second statement, the original result does not give the inequality of the right-hand side. However, only a slight modification can give the upper bound of the topological entropy of f on the horseshoe.

Remark 3.5. By the estimation in the fifth statement of the above theorem, one can further show that there exists a constant $C>0$ such that

for all $x\in \Lambda _\varepsilon $ , $v\in E_i(x)$ , and $n\ge 1$ . Without loss of generality, one can assume that $C=1$ by considering an equivalent norm.

3.2.3 Statements of results

Let $f: M\to M$ be a $C^{1+\alpha }$ diffeomorphism of a compact Riemannian manifold M, and let $\mu $ be a hyperbolic ergodic f-invariant probability measure with positive entropy. Suppose that $(f,\mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ . Ledrappier, Young [23] and Barreira, Pesin, Schmeling [7] proved that

(3.7)

where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ . Our strategy used to prove the dimension approximation in this setting is as follows. It follows from Theorem 3.3 that $h_\mu (f)$ can be approximated by the topological entropies of a sequence of horseshoes $\{\Lambda _\varepsilon \}_{\varepsilon>0}$ . Using well-established properties of dimension theory in uniform hyperbolic systems, one can show that

for $i=u, s$ and every $x\in \Lambda $ . Burns and Wilkinson [8] prove that the holonomy maps of the stable and unstable foliations for $(f, \Lambda _\varepsilon )$ are Lipschitz continuous. Consequently, one can show that

$$ \begin{align*} \begin{aligned} &\dim_{H}{\kern-1pt}(\Lambda_\varepsilon \cap W^u_\beta(f, x)) + \dim_{H}{\kern-1pt}(\Lambda_\varepsilon \cap W^s_\beta(f, x))\\ &\quad\leq \mathrm{Dim}\Lambda_\varepsilon\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon \cap W^u_\beta(f, x)) + \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon \cap W^s_\beta(f, x)) \end{aligned} \end{align*} $$

for every $x\in \Lambda _\varepsilon $ . Hence, $\mathrm {Dim}\mu $ is approximately equal to $\mathrm {Dim}\Lambda _\varepsilon $ . The detailed proofs will be given in the next section.

Theorem B. Let $f:M\to M$ be a $C^{1+\alpha }$ diffeomorphism, and $\mu $ be an f-invariant ergodic hyperbolic measure with $h_\mu (f)>0$ . Assume that $(f,\mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ . For each $\varepsilon>0$ , there exists a horseshoe $\Lambda _\varepsilon $ such that

$$ \begin{align*} |\mathrm{Dim}\ \Lambda_\varepsilon- \mathrm{Dim}\ \mu|<\varepsilon, \end{align*} $$

where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ .

In [41], the authors relaxed the smoothness of Theorem 3.3 to $C^1$ under the additional condition that Oseledec’s splitting $E^u\oplus E^s$ of $(f,\mu )$ is dominated. In this setting, one does not have Lipschitz continuity of the holonomy map in general. However, using Palis and Viana’s method [31], one can show that for every $\gamma \in (0, 1)$ , there is some $D_\gamma>0$ such that the holonomy maps of the stable and unstable foliations for the hyperbolic dynamical system $(f, \Lambda _\varepsilon )$ (see Lemma 4.2) are $(D_\gamma , \gamma )$ -Hölder continuous. Since $\gamma $ is arbitrary, using the ideas in [43], one can prove the following theorem.

Theorem C. Let $f:M\to M$ be a $C^{1}$ diffeomorphism, and let $\mu $ be an f-invariant ergodic hyperbolic measure with $h_\mu (f)>0$ . Assume that $(f,\mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ and the corresponding Oseledec’s splitting $E^u\oplus E^s$ is dominated. For each $\varepsilon>0$ , there exists a horseshoe $\Lambda _\varepsilon $ such that

$$ \begin{align*} |\mathrm{Dim} \Lambda_\varepsilon- \mathrm{ Dim} \mu|<\varepsilon, \end{align*} $$

where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ .

4 Proofs

In this section, we provide the proof of the main results presented in the previous section.

4.1 Proof of Proposition 3.1

Given an f-invariant ergodic measure $\mu $ , let $P(t):=P_\mu (f|_\Lambda , \Phi _f(t))$ , it is easy to see that the function $t\mapsto P(t)$ is continuous and strictly decreasing on the interval $[0,m_0]$ . It follows from equation (2.3) that $P(0)=h_\mu (f)\ge 0$ , and $P(m_0)\le 0$ by Margulis-Ruelle’s inequality. Consequently, there exists a unique root $s_\mu $ of the equation $P_\mu (f|_\Lambda , \Phi _f(t))=0$ .

If $h_\mu (f)=0$ , it is easy to see that $h_\mu (f)=s_\mu =0$ . Hence, $\dim _{L}{\kern-1pt}\mu =s_\mu $ .

If $0<h_\mu (f)< \unicode{x3bb} _{m_0}(\mu )$ , then $P(0)>0$ and $P(1)<0$ . This implies that $s_\mu \in (0,1)$ and $0=P(s_\mu )=h_\mu (f)-s_\mu \unicode{x3bb} _{m_0}(\mu )$ . As a consequence, we have that

If $h_\mu (f)\ge \unicode{x3bb} _{m_0}(\mu )$ , note that

Hence,

However, since $t\mapsto P(t)$ is strictly decreasing in t, we have that

This yields that

$$ \begin{align*} s_\mu=\dim_{\mathrm{L}}{\kern-1pt}\mu. \end{align*} $$

This completes the proof of the proposition.

4.2 Proof of Theorem A

By [11, Theorem 5.1], for each f-invariant ergodic measure $\mu $ with positive entropy and for each $\varepsilon>0$ , there exists an f-invariant compact subset $\Lambda _\varepsilon \subset \Lambda $ such that the following statements hold:

1. (i) $h_{\text {top}}(f|_{\Lambda _\varepsilon })\ge h_{\mu }(f)-\varepsilon $ ;

2. (ii) there is a continuous invariant splitting $T_x M=E_1(x)\oplus E_2(x)\oplus \cdots \oplus E_\ell (x)$ over $\Lambda _\varepsilon $ and a constant $C>0$ so that

   
   for any unit vector $u\in E_j(x)$ , where $\unicode{x3bb} _1(\mu )<\dots <\unicode{x3bb} _\ell (\mu )$ are distinct Lyapunov exponents of f with respect to the measure $\mu $ .

By modifying the arguments in [11, Theorem 5.1], one may improve the estimate in statement (i) as follows:

1. (i)′ $h_{\mu }(f)+\varepsilon \ge h_{\text {top}}(f|_{\Lambda _\varepsilon })\ge h_{\mu }(f)-\varepsilon $ .

Since $\Lambda _{\varepsilon }$ is a repeller of f, one can choose an f-invariant ergodic measure $\mu _\varepsilon $ on $\Lambda _\varepsilon $ so that $h_{\mu _\varepsilon }(f)=h_{\text {top}}(f|_{\Lambda _\varepsilon })$ yields that

$$ \begin{align*} \begin{aligned} P_{\mathrm{top}}(f|_{\Lambda_\varepsilon}, \Phi_f(t))&\ge h_{\mu_\varepsilon}(f)+\mathcal{L}_*(\Phi_f(t), \mu_\varepsilon)\\ &\ge h_{\mu}(f)+\mathcal{L}_*(\Phi_f(t), \mu)-(t+1)\varepsilon\qquad (\text{by statement (ii)})\\ &\ge P_{\mu}(f|_{\Lambda}, \Phi_f(t))-(m_0+1)\varepsilon. \end{aligned} \end{align*} $$

However, since f is expanding, by the variational principle, there exists an f-invariant ergodic measure $\widetilde {\mu }_\varepsilon $ on $\Lambda _\varepsilon $ so that

$$ \begin{align*} \begin{aligned} P_{\mathrm{top}}(f|_{\Lambda_\varepsilon}, \Phi_f(t))&=h_{\widetilde{\mu}_\varepsilon}(f)+\mathcal{L}_*(\Phi_f(t), \widetilde{\mu}_\varepsilon)\\ &\le h_\mu(f)+\mathcal{L}_*(\Phi_f(t), \mu)+(t+1)\varepsilon\\ &\le P_{\mu}(f|_{\Lambda}, \Phi_f(t))+(m_0+1)\varepsilon. \end{aligned} \end{align*} $$

Hence,

$$ \begin{align*} |P_{\mathrm{top}}(f|_{\Lambda_\varepsilon}, \Phi_f(t))-P_{\mu}(f|_{\Lambda}, \Phi_f(t))|\le(m_0+1)\varepsilon. \end{align*} $$

By [11, Theorem 4.1], the Carathéodory singular dimension $\dim _{C}{\kern-1pt} \Lambda _\varepsilon $ of $\Lambda _\varepsilon $ is given by the unique root of the following equation:

$$ \begin{align*} P_{\mathrm{top}}(f|_{\Lambda_\varepsilon}, \Phi_f(t))=0. \end{align*} $$

This, together with Proposition 3.1, yields that

$$ \begin{align*} \begin{aligned} K|\dim_{C}{\kern-1pt}\Lambda_\varepsilon-\dim_{\mathrm{L}}{\kern-1pt}\mu|&\le |P_{\mu}(f|_{\Lambda}, \Phi_f(\dim_{C}{\kern-1pt}\Lambda_\varepsilon))-P_{\mu}(f|_{\Lambda}, \Phi_f(\dim_{\mathrm{L}}{\kern-1pt}\mu)) |\\ &=|P_{\mu}(f|_{\Lambda}, \Phi_f(\dim_{C}{\kern-1pt} \Lambda_\varepsilon))-P_{\mathrm{top}}(f|_{\Lambda_\varepsilon}, \Phi_f(\dim_{C}{\kern-1pt} \Lambda_\varepsilon)) |\\ &\le(m_0+1)\varepsilon, \end{aligned} \end{align*} $$

where $K=\min _{x\in \Lambda }\log m(D_xf)$ and $m(\cdot )$ denotes the minimum norm of an operator. Consequently, we have that $\dim _{C}{\kern-1pt} \Lambda _\varepsilon \rightarrow \dim _{\mathrm {L}}{\kern-1pt}\mu $ as $\varepsilon $ approaches zero.

4.3 Proof of Proposition 3.2

Given a subset $Z\subset \Lambda $ , since $P_Z(f|_{\Lambda },\Phi _f(t))$ is continuous and strictly decreasing in t, let $t_Z$ denote the unique root of the equation $P_Z(f|_{\Lambda },\Phi _f(t))=0$ . For every $t<t_Z$ , we have that $P_Z(f|_{\Lambda },\Phi _f(t))>0$ . Fix such a number t, and take $\beta>0$ so that $P_Z(f|_{\Lambda },\Phi _f(t))-\beta>0$ . Since

$$ \begin{align*}\displaystyle{P_Z(f|_{\Lambda},\Phi_f(t))=\liminf_{r\to 0} P_Z(f|_{\Lambda},\Phi_f(t),r)},\end{align*} $$

there exists $r_0>0$ such that for each $0<r<r_0$ , one has

$$ \begin{align*} P_Z(f|_{\Lambda},\Phi_f(t),r)>P_Z(f|_{\Lambda},\Phi_f(t))-\beta. \end{align*} $$

Fix such a small $r>0$ . By the definition of topological pressure on arbitrary subsets, one has

$$ \begin{align*} m(Z,\Phi_f(t),P_Z(f|_{\Lambda},\Phi_f(t))-\beta,r)=+\infty. \end{align*} $$

Hence, for each $\xi>0$ , there exists $L\in \mathbb {N}$ so that for any $N>L$ , we have that

$$ \begin{align*} \begin{aligned} &\exp(-N(P_Z(f|_{\Lambda},\Phi_f(t))-\beta))\inf \bigg\{ \sum_{i}\exp \Big(\sup_{y\in B_{n_i}(x_i,r)}-\varphi^t(y, f^{n_i})\Big) \bigg\} \\ &\quad\ge\inf\bigg\{ \sum_{i}\exp (-(P_Z(f|_{\Lambda},\Phi_f(t))-\beta)n_i +\sup_{y\in B_{n_i}(x_i,r)}-\varphi^t(y, f^{n_i})) \bigg\}>\xi, \end{aligned} \end{align*} $$

where the infimum is taken over all collections $\{B_{n_i}(x_i,r)\}$ of Bowen’s balls with $n_i\ge N$ , which covers Z. This yields that

$$ \begin{align*} \inf \bigg\{ \sum_{i}\exp \big(\sup_{y\in B_{n_i}(x_i,r)}-\varphi^t(y, f^{n_i})\big) \bigg\}>\xi \exp(N(P_Z(f|_{\Lambda},\Phi_f(t))-\beta)). \end{align*} $$

Letting $N\to \infty $ , we have that

$$ \begin{align*} m(Z,t, r)=+\infty. \end{align*} $$

Hence,

$$ \begin{align*} \dim_{C,r}{\kern-1pt}Z\ge t \end{align*} $$

for all $0<r<r_0$ . Consequently, since $t<t_Z$ is arbitrary, we have that

(4.1) $$ \begin{align} \dim_{C}{\kern-1pt}Z\ge t_Z. \end{align} $$

However, for each $t>t_Z$ , one has that $P_Z(f|_{\Lambda },\Phi _f(t))<0$ . Fix such a number t, and take $\widetilde {\beta }>0$ so that $P_Z(f|_{\Lambda },\Phi _f(t))+\widetilde {\beta }<0$ . By the definition of topological pressure on arbitrary subsets, for any $R>0$ , there exists $0<r<R$ such that

$$ \begin{align*} P_Z(f|_{\Lambda},\Phi_f(t),r)<P_Z(f|_{\Lambda},\Phi_f(t))+\widetilde{\beta}. \end{align*} $$

For such a small $r>0$ , one has

$$ \begin{align*} m(Z,\Phi_f(t),P_Z(f|_{\Lambda},\Phi_f(t))+\widetilde{\beta},r)=0. \end{align*} $$

Hence, for each small $\widetilde {\xi }>0$ , there exists $\widetilde {L}\in \mathbb {N}$ so that for any $N>\widetilde {L}$ , we have that

$$ \begin{align*} \begin{aligned} &\exp(-N(P_Z(f|_{\Lambda},\Phi_f(t))+\widetilde{\beta}))\inf \bigg\{ \sum_{i}\exp \Big(\sup_{y\in B_{n_i}(x_i,r)}-\varphi^t(y, f^{n_i})\Big) \bigg\} \\ &\quad\le\inf\bigg\{ \sum_{i}\exp \Big(-(P_Z(f|_{\Lambda},\Phi_f(t))+\widetilde{\beta})n_i +\sup_{y\in B_{n_i}(x_i,r)}-\varphi^t(y, f^{n_i})\Big) \bigg\}\le \widetilde{\xi}, \end{aligned} \end{align*} $$

where the infimum is taken over all collections $\{B_{n_i}(x_i,r)\}$ of Bowen’s balls with $n_i\ge N$ , which covers Z. This yields that

$$ \begin{align*} \inf \bigg\{ \sum_{i}\exp \big(\sup_{y\in B_{n_i}(x_i,r)}-\varphi^t(y, f^{n_i})\big) \bigg\}\le \widetilde{\xi} \exp(N(P_Z(f|_{\Lambda},\Phi_f(t))+\widetilde{\beta})). \end{align*} $$

Letting $N\to \infty $ , one has

$$ \begin{align*} m(Z, t,r)=0. \end{align*} $$

Consequently, for such $r>0$ , one has

$$ \begin{align*} \dim_{C, r}{\kern-1pt} Z\le t. \end{align*} $$

Hence, we have that

(4.2) $$ \begin{align} \dim_{C}{\kern-1pt} Z=\liminf_{r\to 0}\dim_{C, r}{\kern-1pt} Z\le t_Z. \end{align} $$

It follows from equations (4.1) and (4.2) that

$$ \begin{align*}\dim_{C}{\kern-1pt}Z=t_Z.\end{align*} $$

To show the second statement, for a given f-invariant ergodic measure $\mu $ supported on $\Lambda $ , and a subset $Z\subset \Lambda $ with $\mu (Z)=1$ , we have that

$$ \begin{align*} P_Z(f|_{\Lambda}, \Phi_f(t))\ge P_\mu(f|_{\Lambda}, \Phi_f(t)). \end{align*} $$

By item (1) of Theorem A and the first statement, one has

$$ \begin{align*} \dim_{C}{\kern-1pt} Z\ge \dim_{L}{\kern-1pt}\mu. \end{align*} $$

By the definition of Carathéodory singular dimension of arbitrary subsets, one has

$$ \begin{align*} \dim_{C,r}{\kern-1pt} Z\ge \dim_{L}{\kern-1pt}\mu \end{align*} $$

for all sufficiently small $r>0$ . Consequently, we have that

$$ \begin{align*} \dim_{C}{\kern-1pt}\mu=\liminf_{r\to 0}\dim_{C,r}{\kern-1pt}\mu =\liminf_{r\to 0}\inf\{\dim_{C,r}{\kern-1pt} Z: \mu(Z)=1 \}\ge \dim_{L}{\kern-1pt}\mu. \end{align*} $$

To prove that $\dim _{C}{\kern-1pt}\mu =\dim _{L}{\kern-1pt}\mu $ , we assume that $\dim _{C}{\kern-1pt}\mu>\widetilde {t}>\dim _{L}{\kern-1pt}\mu $ . By the first statement in Theorem A, we have that

$$ \begin{align*} P_\mu(f|_{\Lambda},\Phi(\widetilde{t}))<0. \end{align*} $$

By the definition of measure theoretic pressure, for each $n\in \mathbb {N}$ , there exists $0<r_n< 1/n$ so that

$$ \begin{align*} \inf\{P_Z(f|_{\Lambda},\Phi(\widetilde{t}),r_n): \mu(Z)=1\}<0. \end{align*} $$

Hence, there exists a subset $Z_n\subset \Lambda $ with $\mu (Z_n)=1$ so that

$$ \begin{align*} P_{Z_n}(f|_{\Lambda},\Phi(\widetilde{t}),r_n)<0. \end{align*} $$

Put $\widetilde {Z}:=\bigcap _{n\ge 1} Z_n$ , then $\mu (\widetilde {Z})=1$ and

$$ \begin{align*} \begin{aligned} P_{\widetilde{Z}}(f|_{\Lambda},\Phi(\widetilde{t}))&=\liminf_{r\to 0}P_{\widetilde{Z}}(f|_{\Lambda},\Phi(\widetilde{t}),r)\\ &\le \liminf_{n\to \infty}P_{Z_n}(f|_{\Lambda},\Phi(\widetilde{t}),r_n)\le 0. \end{aligned} \end{align*} $$

It follows from the first statement and the definition of Carathéodory singular dimension of $\mu $ that

$$ \begin{align*} \dim_{C}{\kern-1pt}\mu=\liminf_{r\to 0} \dim_{C,r}{\kern-1pt}\mu \le \liminf_{r\to 0} \dim_{C,r}{\kern-1pt} \widetilde{Z}=\dim_C{\kern-1pt}\widetilde{Z}\le \widetilde{t}, \end{align*} $$

which yields a contraction. Hence, we have that $\dim _C{\kern-1pt}\mu =\dim _L{\kern-1pt}\mu $ .

4.4 Proof of Theorem B

Ledrappier, Young [23] and Barreira, Pesin, Schmeling [7] proved that

where $\mathrm { Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ . Fix a small number $\varepsilon>0$ . By Theorem 3.3, there exists a horseshoe $\Lambda _\varepsilon $ such that:

1. (i) $|h_{\mathrm {top}}(f,\Lambda _\varepsilon ) - h_{\mu }(f)|<\varepsilon $ ;

2. (ii) there exists a dominated splitting $T_{\Lambda _\varepsilon } M=E^u\oplus E^s$ with $\dim E^i=d_i\, (i=u,s)$ , and for each $x\in \Lambda _\varepsilon $ , every $n\geq 1$ , and each vector $v\in E^i(x) (i=s,u)$ ,

   
   The above estimation can be proven in similar fashion as Remark 3.2.

Fixed any $k\in \mathbb {N}$ and denote $F=f^{2^k}$ . Since $\Lambda _\varepsilon $ is a locally maximal hyperbolic set for f, $\Lambda _\varepsilon $ is also a locally maximal hyperbolic set for F. Notice that

$$ \begin{align*} W^u_\beta(F, x)\cap\Lambda_\varepsilon = W^u_\beta(f, x)\cap\Lambda_\varepsilon \quad \text{and} \quad W^s_\beta(F, x)\cap\Lambda_\varepsilon = W^s_\beta(f, x)\cap\Lambda_\varepsilon.\end{align*} $$

Let $\|\cdot \|$ and $m(\cdot )$ denote the maximal and minimal norm of an operator. For every $x\in \Lambda _\varepsilon $ , Barreira [4] proved that

$$ \begin{align*} \underline{t}_u^k \leq \dim_H(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \overline{t}_u^k, \end{align*} $$

where $\underline {t}_u^k$ , $\overline {t}_u^k$ are the unique solutions of

$$ \begin{align*}P_{\mathrm{top}}(F, -t\log\|D_xF|_{E^u(x)}\|)=0 \quad\text{and}\quad P_{\mathrm{top}}(F, -t\log m(D_xF|_{E^u(x)}))=0,\end{align*} $$

respectively. Using the same arguments as in the proof of [3, Theorems 6.2 and 6.3], one can prove that the sequences $\{\underline {t}_u^k\}$ and $\{\overline {t}_u^k\}$ are monotone. Furthermore, setting

$$ \begin{align*}\underline{t}_u:=\lim_{k\to\infty}\underline{t}_u^k \quad\text{and}\quad \overline{t}_u:= \lim_{k\to\infty}\overline{t}_u^k,\end{align*} $$

one can show that $\underline {t}_u$ , $\overline {t}_u$ are the unique solutions of the following equations:

$$ \begin{align*}P_{\mathrm{var}}(f, -t\{\log\|D_xf^n|_{E^u}\|\})=0, \quad P_{\mathrm{top}}(f, -t\{\log m(D_xf^n|_{E^u})\})=0,\end{align*} $$

respectively.

Consequently, we have that

$$ \begin{align*} \underline{t}_u \leq \dim_H{\kern-1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \underline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \overline{t}_u \end{align*} $$

and

$$ \begin{align*}\begin{aligned} \underline{t}_u &=\sup\bigg\{\frac{h_\nu(f)}{\displaystyle{\lim_{n\to\infty}} \frac 1n \int \log\|D_xf^n|_{E^u}\|\,d\nu}: \nu\in\mathcal{M}_f(\Lambda_\varepsilon) \bigg\},\\ \overline{t}_u &=\sup\bigg\{\frac{h_\nu(f)}{\displaystyle{\lim_{n\to\infty}} \frac 1n \int \log m(D_xf^n|_{E^u})\,d\nu}: \nu\in\mathcal M_f(\Lambda_\varepsilon) \bigg\}. \end{aligned} \end{align*} $$

Combining with items (i) and (ii), one has

(4.3)

for every $x\in \Lambda _\varepsilon $ , where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ . One can show in a similar fashion that

(4.4)

for every $x\in \Lambda _\varepsilon $ .

Lemma 4.1. The holonomy maps of the stable and unstable foliations for $(f, \Lambda _\varepsilon )$ are Lipschitz continuous.

Proof. Fix a positive integer N, put $F:=f^N$ and $\Lambda =\Lambda _\varepsilon $ . Since $\Lambda $ is a locally maximal hyperbolic set for f, so is $\Lambda $ for F. Notice that

(4.5) $$ \begin{align} W^u_\beta(F, x)\cap\Lambda = W^u_\beta(f, x)\cap\Lambda \quad \text{and} \quad W^s_\beta(F, x)\cap\Lambda = W^s_\beta(f, x)\cap\Lambda. \end{align} $$

Let

$$ \begin{align*}a_F=\|DF^{-1}|_{E^u}\|,\quad b_F=\|DF|_{E^s}\|, \quad c_F=\|DF|_{E^u}\|, \quad d_F=\|DF^{-1}|_{E^s}\|.\end{align*} $$

It follows from item (ii) that

$$ \begin{align*} 1<\frac{\|D_xF|_{E^i(x)}\|}{m(D_xF|_{E^i(x)})}<e^{2\varepsilon N} \quad\text{for every } x\in \Lambda\text{ and } i\in\{s,u\}. \end{align*} $$

Hence,

provided that $\varepsilon>0$ is sufficiently small such that $\unicode{x3bb} _s(\mu )+3\varepsilon <0$ . By [8, Theorem 0.2], we have that the holonomy map of the stable foliation for $(F, \Lambda )$ is $C^1$ . Similarly, note that

provided that $\varepsilon>0$ is sufficiently small such that $\unicode{x3bb} _u(\mu )-3\varepsilon>0$ . It follows from [8, Theorem 0.2] that the holonomy map of the unstable foliation for $(F, \Lambda )$ is $C^1$ . Combining with equation (4.5), one has the holonomy maps of the stable and unstable foliations for $(f, \Lambda )$ are Lipschitz continuous.

By Lemma 4.1 and the fact f is topologically mixing on $\Lambda _\varepsilon $ , one has $\dim _H{\kern-1pt}(\Lambda _\varepsilon \cap W^u_\beta (f, x))$ , $\underline {\dim }_B(\Lambda _\varepsilon \cap W^u_\beta (f, x))$ , and $\overline {\dim }_B(\Lambda _\varepsilon \cap W^u_\beta (f,x))$ are independent of $\beta $ and x (see the proof of [5, Theorem 4.3.2] for more details). Let

$$ \begin{align*}A_{\varepsilon, x}=(\Lambda_\varepsilon \cap W^u_\beta(f,x)) \times (\Lambda_\varepsilon \cap W^s_\beta(f,x)).\end{align*} $$

By the properties of dimension (e.g. see [13, 33]), one has

(4.6) $$ \begin{align} \begin{aligned} & \dim_H{\kern-1pt}(\Lambda_\varepsilon\cap W^u_\beta(f, x)) + \dim_H{\kern-1pt}(\Lambda_\varepsilon\cap W^s_\beta(f, x))\\ &\quad\leq \dim_H{\kern-1pt} A_{\varepsilon, x}\\ &\quad\leq \underline{\dim}_{B}{\kern1pt}A_{\varepsilon, x}\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}A_{\varepsilon, x}\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^u_\beta(f, x)) + \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^s_\beta(f, x)). \end{aligned} \end{align} $$

Let $\Phi : A_{\varepsilon ,x} \to \Lambda _\varepsilon $ be given by

$$ \begin{align*} \Phi(y,z)=W^s_\beta(f, y) \cap W^u_\beta(f, z).\end{align*} $$

It is easy to see $\Phi $ is a homeomorphism onto a neighborhood $V_x$ of x in $\Lambda _\varepsilon $ . It follows from Lemma 4.1 that $\Phi $ and $\Phi ^{-1}$ are Lipschitz continuous (see [5, Theorem 4.3.2] for detailed proofs). It follows from Corollary 2.9 that

$$ \begin{align*} \mathrm{Dim}\ V_x = \mathrm{Dim}\ A_{\varepsilon, x}, \end{align*} $$

where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ . Since $\{V_x: x\in \Lambda _\varepsilon \}$ is an open cover of $\Lambda _\varepsilon $ , one can choose a finite open cover $\{V_{x_1}, V_{x_2}, \ldots , V_{x_k}\}$ of $\Lambda _\varepsilon $ . It follows from equation (4.6) that

$$ \begin{align*} \begin{aligned} & \dim_H{\kern-1pt}(\Lambda_\varepsilon \cap W^u_\beta(f, x)) + \dim_H{\kern-1pt}(\Lambda_\varepsilon \cap W^s_\beta(f, x))\\ &\quad\leq \dim_H{\kern-1pt} \Lambda_\varepsilon = \max_{1\leq i \leq k} \dim_H{\kern-1pt} V_{x_i}\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}\Lambda_\varepsilon =\max_{1\leq i\leq k}\overline{\dim}_{B}{\kern1pt}V_{x_i}\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon \cap W^u_\beta(f, x)) + \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon \cap W^s_\beta(f, x)), \end{aligned} \end{align*} $$

for every $x\in \Lambda _\varepsilon $ . Combining equations (4.3) and (4.4), we obtain

where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ . This completes the proof of Theorem B.

4.5 Proof of Theorem C

For every pair $(f,\mu )$ satisfying the assumptions, Wang and Cao [40, Corollary 1] proved that

Fix a small number $\varepsilon>0$ . Wang, Cao, and Zou [41, Theorem 1.1] proved that there exists a horseshoe $\Lambda _\varepsilon $ such that:

1. (i) $|h_{\mathrm {top}}(f,\Lambda _\varepsilon ) - h_{\mu }(f)|<\varepsilon $ ;

2. (ii) there exists a dominated splitting $T_{\Lambda _\varepsilon } M=E^u\oplus E^s$ with $\dim E^i=d_i\, (i=u,s)$ , and for each $x\in \Lambda _\varepsilon $ , every $n\geq 1$ , and each vector $v\in E^i(x) (i=s, u)$ ,

   
   The above estimation can be obtained by item (v) of [41, Theorem 1.1] and Remark 3.2.

Fix a positive integer $k\in \mathbb {N}$ , denote $F=f^{2^k}$ . Since $\Lambda _\varepsilon $ is a locally maximal hyperbolic set for f, so is $\Lambda _\varepsilon $ for F. Notice that

$$ \begin{align*} W^u_\beta(F, x)\cap\Lambda_\varepsilon = W^u_\beta(f, x)\cap\Lambda_\varepsilon \quad \text{and} \quad W^s_\beta(F, x)\cap\Lambda_\varepsilon = W^s_\beta(f, x)\cap\Lambda_\varepsilon.\end{align*} $$

For every $x\in \Lambda _\varepsilon $ , it follows from [43, Lemmas 3.5 and 3.6] that

$$ \begin{align*} \underline{t}_u^k \leq \dim_H{\kern-1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \overline{t}_u^k, \end{align*} $$

where $\underline {t}_u^k$ , $\overline {t}_u^k$ are the unique roots of

$$ \begin{align*}P_{\mathrm{top}}(F, -t\log\|D_xF|_{E^u(x)}\|)=0, \quad P_{\mathrm{top}}(F, -t\log m(D_xF|_{E^u(x)}))=0,\end{align*} $$

respectively. Using the same arguments as in the proof of [3, Theorems 6.2 and 6.3], one can prove that the sequences $\{\underline {t}_u^k\}$ and $\{\overline {t}_u^k\}$ are monotone. Set

$$ \begin{align*}\underline{t}_u:=\lim_{k\to\infty}\underline{t}_u^k \quad\text{and}\quad \overline{t}_u:= \lim_{k\to\infty}\overline{t}_u^k,\end{align*} $$

where $\underline {t}_u$ , $\overline {t}_u$ are the unique solutions of the following equations:

$$ \begin{align*}P_{\mathrm{var}}(f, -t\{\log\|D_xf^n|_{E^u(x)}\|\})=0, \quad P_{\mathrm{top}}(f, -t\{\log m(D_xf^n|_{E^u(x)})\})=0,\end{align*} $$

respectively. Hence, we have that

$$ \begin{align*} \underline{t}_u \leq \dim_H{\kern-1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \underline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^u_\beta(f,x)) \leq \overline{t}_u. \end{align*} $$

Since

$$ \begin{align*} \underline{t}_u &=\sup\bigg\{\frac{h_\nu(f)}{ \displaystyle{\lim_{n\to\infty}}\frac 1n \int \log\|D_xf^n|_{E^u(x)}\|\,d\nu}: \nu\in\mathcal{M}_f(\Lambda_\varepsilon) \bigg\} \end{align*} $$

and

$$ \begin{align*} \overline{t}_u &=\sup\bigg\{\frac{h_\nu(f)}{\displaystyle{\lim_{n\to\infty}} \frac 1n \int \log m(D_xf^n|_{E^u(x)})\,d\nu}: \nu\in\mathcal{M}_f(\Lambda_\varepsilon) \bigg\}, \end{align*} $$

using items (i) and (ii), one can show that

(4.7)

for every $x\in \Lambda _\varepsilon $ , where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ . Similarly, we obtain that

(4.8)

for every $x\in \Lambda _\varepsilon $ .

Lemma 4.2. Let $\Lambda $ be a locally maximal hyperbolic set of a $C^1$ diffeomorphism such that f is topologically mixing on $\Lambda $ . Assume that the diffeomorphism $f|_\Lambda $ possesses a $\{\unicode{x3bb} _u(\mu ), \unicode{x3bb} _s(\mu )\}$ -dominated splitting $T_\Lambda M=E^u\oplus E^s$ with $E^u\succeq E^s$ and $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ . Then for every $\gamma \in (0, 1)$ , there exists $D_\gamma>0$ such that the holonomy maps of the stable and unstable foliations for f are $(D_\gamma , \gamma )$ -H $\ddot {o}$ lder continuous.

Proof. Since the diffeomorphism $f|_\Lambda $ possesses a $\{\unicode{x3bb} _u(\mu ), \unicode{x3bb} _s(\mu )\}$ -dominated splitting $T_\Lambda M=E^u\oplus E^s$ with $E^u\succeq E^s$ and $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ , there exists a constant $C>0$ such that for every $x\in \Lambda $ , $n\in \mathbb {N}$ and each unit vector $v\in E^i(x)$ ( $i=u, s$ ),

As illustrated in Remark 3.2, for simplicity, we assume that $C=1$ in the rest of the proof. Fix a positive integer N and put $F:=f^N$ . This implies that for every $x\in \Lambda $ ,

$$ \begin{align*} 1 \leq \frac{\|D_xF|_{E^u}\|}{m(D_xF|_{E^u})} \leq e^{2N\varepsilon},\quad 1 \leq \frac{\|D_xF|_{E^s}\|}{m(D_xF|_{E^s})} \leq e^{2N\varepsilon}. \end{align*} $$

Notice that $\Lambda $ is also a locally maximal hyperbolic set for F and

$$ \begin{align*} W^u_\beta(F, x)\cap\Lambda = W^u_\beta(f, x)\cap\Lambda \quad \text{and} \quad W^s_\beta(F, x)\cap\Lambda = W^s_\beta(f, x)\cap\Lambda\end{align*} $$

for every $x\in \Lambda $ .

Let $\pi ^s$ and $\pi ^u$ be the holonomy maps of stable and unstable foliations for f, that is, for any $x\in \Lambda $ , $x'\in W^s_\beta (f, x)$ , and $x"\in W^u_\beta (f, x)$ close to x,

$$ \begin{align*}\pi^s:\ W^u_\beta(f,x)\cap\Lambda\to W^u_\beta(f,x')\cap\Lambda\ \text{with}\ \pi^s(y)=W^s_\beta(f,y)\cap W^u_\beta(f,x')\end{align*} $$

and

$$ \begin{align*}\pi^u:\ W^s_\beta(f,x)\cap\Lambda\to W^s_\beta(f,x")\cap\Lambda\ \text{with}\ \pi^u(z)=W^u_\beta(f,z)\cap W^s_\beta(f,x").\end{align*} $$

Therefore, $\pi ^s$ is also a map from $W^u_\beta (F,x)\cap \Lambda $ to $W^u_\beta (F, x')\cap \Lambda $ and $\pi ^u$ is also a map from $W^s_\beta (F,x)\cap \Lambda $ to $W^s_\beta (F, x")\cap \Lambda $ .

Let $U\subset M$ be an open subset such that $\Lambda =\bigcap _{n\in \mathbb {Z}}f^n(U)$ , and $\mathcal {U}\subset \mbox {Diff}^{1}(M)$ be a neighborhood of f such that for each $g \in \mathcal {U}$ , $\Lambda _g = \bigcap _{n\in \mathbb {Z}} g^n(U)$ is a locally maximal hyperbolic set for g and there is a homeomorphism $h_g: \Lambda \rightarrow \Lambda _g$ that satisfies $g\circ h_g=h_g\circ f$ , with $h_g\ C^0$ -close to identity if g is $C^1$ -close to f. For $g \in \mathcal {U}$ , let $T_{\Lambda _g}M=E^u_g\oplus E^s_g$ denote the hyperbolic splitting over $\Lambda _g$ . For $i\in \{u,s\}$ , $\{W^i_\beta (g,z):\ z\in \Lambda _g\}$ is continuous on g in the following sense: there is $\{\theta _{g,x}^i:\ x\in \Lambda \}$ , where $\theta _{g,x}^i:\ W^i_\beta (f,x)\to W^i_\beta (g,h_g(x))$ is a $C^1$ diffeomorphism with $\theta _{g,x}^i(x)=h_g(x)$ , such that if g is $C^1$ -close to f then, for all $x\in \Lambda $ , $\theta _{g,x}^i$ is uniformly $C^1$ -close to the inclusion of $W^i_\beta (f,x)$ in M.

For any $\gamma \in (0, 1)$ , let $\mathcal {U}_\gamma ^F$ be a small $C^1$ neighborhood of F (recall $F=f^N$ ). Taking $G\in \mathcal {U}_\gamma ^F \cap \mathrm {Diff}^2(M)$ such that for every $x\in \Lambda _G$ (here $\Lambda _G$ is a locally maximal hyperbolic set for G), $n\in \mathbb {N}$ , and $i=u, s$ ,

(4.9)

Claim 4.3. The following properties hold:

1. (a) $h_G|_{W^u_\beta (F, x)\cap \Lambda }$ and $(h_G|_{W^u_\beta (F, x)\cap \Lambda })^{-1}$ are $(C_\gamma , \gamma )$ -Hölder continuous for some $C_\gamma>0$ ;

2. (b) the stable and unstable foliations

   $$ \begin{align*}\{W^s(G, z): z\in\Lambda_G\}, \quad \{W^u(G, z): z\in\Lambda_G\}\end{align*} $$
   are $C^1$ and invariant for G. Thus the holonomy maps
   $$ \begin{align*} &\pi^s_G: W^u_\beta(G,h_G(x))\cap\Lambda_G\to W^u_\beta(G,h_G(x'))\cap\Lambda_G \quad \text{with}\\& \qquad\,\qquad \pi^s_G(y)=W^s_\beta(G,y)\cap W^u_\beta(G,h_G(x')), \end{align*} $$
   and
   $$ \begin{align*} &\pi^u_G: W^s_\beta(G,h_G(x))\cap\Lambda_G\to W^s_\beta(G,h_G(x"))\cap\Lambda_G \quad \text{with}\\& \qquad\ \ \qquad \pi^u_G(z)=W^u_\beta(G,z)\cap W^s_\beta(G,h_G(x")) \end{align*} $$
   are Lipschitz continuous.

Proof. (a) See [43, Claim 3.1].

(b) Since G satisfies equation (4.9), we conclude

provided that $\unicode{x3bb} _s(\mu )+6\varepsilon <0$ . By [18, Theorem 6.3], the stable foliation is $C^1$ . A similar argument shows that the unstable foliation is also $C^1$ . Then the corresponding maps are uniformly $C^1$ (see [35, pp. 540–541] for more details), which implies the desired result.

We proceed to prove Lemma 4.2. For any $y\in W^u_\beta (F, x)\cap \Lambda $ ,

$$ \begin{align*} \begin{aligned} h_G(\pi^s(y))&=h_G( W^s_\beta(F,y)\cap W^u_\beta(F, x') )\\ &=W^s_\beta(G, h_G(y)) \cap W^u_\beta(G, h_G(x'))\\ &=\pi^s_G(h_G(y)). \end{aligned} \end{align*} $$

For the above $\gamma $ , by Claim 4.3, there exists $D_\gamma>0$ such that

$$ \begin{align*}\pi^s=h_G^{-1} \circ \pi^s_G \circ h_G \end{align*} $$

is $(D_\gamma , \gamma )$ -Hölder continuous. Using the same arguments, one can prove $(\pi ^s)^{-1}$ , $\pi ^u$ , and $(\pi ^u)^{-1}$ are also $(D_\gamma , \gamma )$ -Hölder continuous.

By Lemma 4.2 and the fact that f is topologically mixing on $\Lambda _\varepsilon $ , one has $\dim _H{\kern-1pt}(\Lambda _\varepsilon \cap W^u_\beta (f, x))$ , $\underline {\dim }_B(\Lambda _\varepsilon \cap W^u_\beta (f, x))$ , and $\overline {\dim }_B(\Lambda _\varepsilon \cap W^u_\beta (f,x))$ are independent of $\beta $ and x (see the proof of [43, Lemma 3.4] for more details). Let

$$ \begin{align*}A_{\varepsilon, x}=(\Lambda_\varepsilon \cap W^u_\beta(f,x)) \times (\Lambda_\varepsilon \cap W^s_\beta(f,x))\end{align*} $$

be a product space. By the properties of dimension (see [33, Theorem 6.5] for details), one has

(4.10) $$ \begin{align} \begin{aligned} & \dim_H{\kern-1pt}(\Lambda_\varepsilon\cap W^u_\beta(f, x)) + \dim_H{\kern-1pt}(\Lambda_\varepsilon\cap W^s_\beta(f, x))\\ &\quad\leq \dim_H{\kern-1pt} A_{\varepsilon, x}\\ &\quad\leq \underline{\dim}_{B}{\kern1pt}A_{\varepsilon, x}\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}A_{\varepsilon, x}\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^u_\beta(f, x)) + \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon\cap W^s_\beta(f, x)). \end{aligned} \end{align} $$

Let $\Phi : A_{\varepsilon ,x} \to \Lambda _\varepsilon $ be given by

$$ \begin{align*} \Phi(y,z)=W^s_\beta(f, y) \cap W^u_\beta(f, z).\end{align*} $$

It is easy to see that $\Phi $ is a homeomorphism onto a neighborhood $V_x$ of x in $\Lambda _\varepsilon $ . For any $\gamma \in (0, 1)$ , by Lemma 4.2, there is $E_\gamma>0$ such that $\Phi $ and $\Phi ^{-1}$ are $(E_\gamma , \gamma )$ -Hölder continuous (see Step $2$ in the proof of [43, Theorem A] for more details). By Lemma 2.8 and the arbitrariness of $\gamma $ , one has

$$ \begin{align*} \mathrm{ Dim}\ V_x = \mathrm{ Dim}\ A_{\varepsilon, x}, \end{align*} $$

where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ . Since $\{V_x: x\in \Lambda _\varepsilon \}$ is an open cover of $\Lambda _\varepsilon $ , one can choose a finite open cover $\{V_{x_1}, V_{x_2}, \ldots , V_{x_k}\}$ of $\Lambda _\varepsilon $ . It follows from equation (4.10) that

$$ \begin{align*} \begin{aligned} & \dim_H{\kern-1pt}(\Lambda_\varepsilon \cap W^u_\beta(f, x)) + \dim_H{\kern-1pt}(\Lambda_\varepsilon \cap W^s_\beta(f, x))\\ &\quad\leq \dim_H{\kern-1pt} \Lambda_\varepsilon = \max_{1\leq i \leq k} \dim_H{\kern-1pt} V_{x_i}\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}\Lambda_\varepsilon =\max_{1\leq i\leq k}\overline{\dim}_{B}{\kern1pt}V_{x_i}\\ &\quad\leq \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon \cap W^u_\beta(f, x)) + \overline{\dim}_{B}{\kern1pt}(\Lambda_\varepsilon \cap W^s_\beta(f, x)), \end{aligned} \end{align*} $$

for every $x\in \Lambda _\varepsilon $ . This together with equations (4.7) and (4.8) yields that

where $\mathrm {Dim}$ denotes either $\dim _H$ or $\underline {\dim }_B$ or $\overline {\dim }_B$ . This completes the proof of Theorem C.