Measures of maximal and full dimension for smooth maps

Abstract

For a $C^1$ non-conformal repeller, this paper proves that there exists an ergodic measure of full Carathéodory singular dimension. For an average conformal hyperbolic set of a $C^1$ diffeomorphism, this paper constructs a Borel probability measure (with support strictly inside the repeller) of full Hausdorff dimension. If the average conformal hyperbolic set is of a $C^{1+\alpha }$ diffeomorphism, this paper shows that there exists an ergodic measure of maximal dimension.

1 Introduction

Let $f:M\to M$ be a $C^{1+\alpha }$ expanding map on a compact smooth Riemannian manifold M with a conformal repeller $\Lambda $ . Let $\mu $ be the unique equilibrium measure corresponding to the Hölder continuous function $-s\log \|D_xf\|$ , where s is the unique solution of Bowen’s equation

$$ \begin{align*} P(f|_\Lambda, -s\log \|D_xf\|)=0, \end{align*} $$

then the following properties hold:

1. (1) $\dim _H \Lambda =\dim _B\Lambda =s$ ;

2. (2) the s-Hausdorff dimension of $\Lambda $ is positive and finite, moreover, it is equivalent to the equilibrium measure $\mu $ ;

3. (3) $\dim _H\mu =\dim _H\Lambda $ ,

where ${\dim _H}$ and $\dim _B$ denote the Hausdorff dimension and the box dimension, respectively. The first property of Hausdorff dimension was first established by Bowen in a special case [8]. Ruelle showed the general case in [24], where his proof consists of showing the second property. Falconer [15] obtained the equality between Hausdorff dimension and box dimension. Later, the smoothness $C^{1+\alpha }$ was eventually relaxed to $C^1$ by Gatzouras and Peres [18]. The third property is clear from the variational principle of topological pressure. Such a measure is called the measure of full dimension.

By the variational principle of topological pressure, there exists an equilibrium measure provided that the entropy map $\mu \mapsto h_\mu (f)$ is upper semi-continuous. The existence of measures of full dimension could be regarded as a dimensional version of the existence of an equilibrium state. However, the map $\mu \mapsto \dim _H\mu $ enjoys no continuity property even if the entropy map is upper semi-continuous. This is the crucial difference between dimension and pressure/entropy.

As one can see, a $C^1$ conformal repeller admits a measure of full dimension. How about the non-conformal and hyperbolic case? The answer is usually negative, although a certain special non-conformal repeller–average conformal repeller, which is introduced in [2], does have an ergodic measure of full dimension (see [10, Theorem E]). So it is natural to generalize the question into two parts: one is to consider a general quantity of dimension type; the other one is to consider the existence of measures of maximal dimension, that is, try to find an invariant measure which attains the supremum of the following quantity:

$$ \begin{align*} \delta(f)=\sup\{\operatorname{dim}_{H} \mu: \mu\,\text{is}\, f\text{-invariant}\}. \end{align*} $$

This quantity was introduced by Denker and Urbański [14] in the context of one-dimensional complex dynamics, where they considered the supremum over the ergodic measures of positive entropy. Later, this quantity has been intensively studied in one-dimensional complex dynamics (see [25] for more details).

For non-conformal repellers, we consider a substitute quantity of dimension type called Carathéodory singular dimension (see §2.3.2 for a detailed definition) which was introduced by Cao, Pesin, and Zhao [11]. They proved its continuity for $C^{1+\alpha }$ maps under $C^1$ topology. Later, following the approach described in [21], the authors introduced the Carathéodory singular dimension of invariant measures in [12], and proved that the unique zero of the measure theoretic pressure function equals the Carathéodory singular dimension of ergodic measures. For a general $C^1$ non-conformal repeller, in this paper, we will prove that there exists an ergodic measure of full Carathéodory singular dimension.

For the existence of measures of maximal dimension for hyperbolic diffeomorphisms, it was shown by Barreira and Wolf [5] that if $f: M\to M$ is a $C^{1+\alpha }$ surface diffeomorphism and $\Lambda $ is a topological mixing locally maximal hyperbolic set, then there exists an ergodic measure of maximal dimension, see [4, Ch. 5] for the hyperbolic conformal case. In [22], Rams proved the existence of a measure of maximal dimension for piecewise linear horseshoe maps by computing $\delta (f)$ explicitly. In [30], Wolf showed that there exist finitely many measures of maximal dimension for polynomial automorphisms of $C^2$ . However, these are not guaranteed to be measures of full dimension unless the automorphism is volume preserving. It should be noted that only a minimal lack of hyperbolicity may yield no measure of maximal dimension. Urbański and Wolf [26] constructed nonlinear horseshoes of a surface, which are hyperbolic except at one parabolic fixed point, and which do not have any measure of maximal dimension.

For an average conformal hyperbolic (ACH for short) set $\Lambda $ of a $C^{1}$ diffeomorphism, which is introduced in [29], we construct a Borel probability measure $\mu $ (not necessarily invariant) on $\Lambda $ and show that this measure has full Hausdorff dimension, that is, $\dim _H\mu =\dim _H\Lambda $ . Our method consists of using the weak Gibbs measure of a continuous function. The desired measure is the product of two weak Gibbs measure with respect to the future and past behavior of the derivative continuous function along unstable and stable directions. The measure constructed in this way has support strictly inside the repeller. If $\Lambda $ is an average conformal hyperbolic set of a $C^{1+\alpha }$ diffeomorphism, following Barreira and Wolf’s approach in [5], this paper proves that there exists an ergodic measure of maximal dimension, which extends the result [5] for average conformal hyperbolic sets.

The paper is organized as follows. In §2, we recall some necessary concepts, such as average conformal hyperbolic set and Carathéodory singular dimension, and give the statement of the main result in this paper. Section 3 presents the detailed proofs of the results in the previous section. Namely, we prove that there exists an ergodic measure of full Carathéodory singular dimension on a $C^1$ non-conformal repeller; and we construct a Borel probability measure of full Hausdorff dimension on a $C^1$ average conformal hyperbolic set; finally, we show that there exists an ergodic measure of maximal dimension on a $C^{1+\alpha }$ average conformal hyperbolic set.

2 Preliminaries and statements

In this section, we recall some notions in dimension theory and smooth dynamical systems, and give the statement of the main results in this paper. The proofs will be postponed to the next section.

2.1 Topological pressure

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

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

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 following quantity

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

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

Remark 2.1. If $\Phi =\{\phi _n\}_{n\geq 1}$ is additive in the sense that $\phi _n(x)=\phi (x)+\phi (fx)+\cdots +\phi (f^{n-1}x)$ for some continuous function $\phi : X\to \mathbb {R}$ , we simply write $P(f, \Phi )$ as $P(f, \phi )$ . If it is clear from the context of the dynamics, we will simplify the topological pressure as $P(\phi )$ .

Cao, Feng, and Huang [9] proved the following variational principle:

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

where $\mathcal {M}_f(X)$ denotes the space of all f-invariant measures on X, $h_\mu (f)$ denotes the metric entropy of f with respect to $\mu $ (see [27] for details of metric entropy), and

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

for every $\mu \in \mathcal {M}_f(X)$ . The previous limit is well defined. A standard sub-additive argument yields the existence of this limit. A measure $\mu \in \mathcal {M}_f(X)$ that attains the supermum in equation (2.2) is called an equilibrium state of the topological pressure $P(f,\Phi )$ .

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

Definition 2.2. The following jump-up value of $\mathcal {H}^{s}(Z)$

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

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

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

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 $\dim _BZ$ and call it the box dimension of Z.

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

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

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

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

and

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

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

If $\mu $ is a finite measure on X, the following quantities

$$ \begin{align*}\bar{d}_{\mu}(x)=\limsup _{r \rightarrow 0} \frac{\log \mu(B_{r}(x))}{\log r} \quad\text{and}\quad \underline{d}_{\mu}(x)=\liminf _{r \rightarrow 0} \frac{\log \mu(B_{r}(x))}{\log r}\end{align*} $$

are called lower and upper point-wise dimensions of $\mu $ at point x, respectively, where $B_{r}(x)=\{y\in X:d(x,y)< r\}$ . We recall two basic properties relating these quantities with the Hausdorff dimension of subsets and measures (see [21] for details):

1. (1) if $\underline d_{\mu }(x) \geqslant a$ for $\mu $ almost every $x \in \Lambda $ , then $\operatorname {dim}_{H} \mu \geqslant a$ ;

2. (2) if $\underline d_{\mu }(x)\leqslant a$ for every $x \in Z \subset \Lambda $ , then $\operatorname {dim}_{H} Z \leqslant a$ .

2.3 Measures of full Carathéodory singular dimension for repellers

In this section, we will recall the concept of Carathéodory singular dimension of subsets and invariant measures. For a non-conformal repeller of a $C^1$ map, we will show that there exists an ergodic measure of full Carathéodory singular dimension.

2.3.1 Singular valued potentials

Let $f:M\to M$ be a $C^1$ transformation of a $m_0$ -dimensional compact smooth Riemannian manifold M, and let $\Lambda $ be a compact f-invariant subset of M. Denote by $\mathcal {M}(f|_\Lambda )$ and $\mathcal {E}(f|_\Lambda )$ the set of all f-invariant measures and ergodic measures on $\Lambda $ , respectively.

If a compact f-invariant subset $\Lambda $ satisfies the following two properties:

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

   $$ \begin{align*} \|D_xf v \| \ge \kappa \|v\| \quad \text{for all } x \in \Lambda, \quad\text{and}\ v \in T_xM, \end{align*} $$
   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,

then we call $\Lambda $ a repeller for f or f is expanding on $\Lambda $ .

Given $x\in \Lambda $ and $n\ge 1$ , 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

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

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

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

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

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

Hence, the sequence of functions $ \Phi _f(t):=\{-\varphi ^t(\cdot ,f^n)\}_{n\ge 1} $ is sub-additive, which is called the sub-additive singular valued potentials.

2.3.2 Carathéodory singular dimension of sets and measures

The Carathéodory singular dimension of a repeller is introduced in [11]. Following the approach in [11], we will introduce the notions of Carathéodory singular dimension of subsets and measures.

Let $B_n(x,r):=\{x\in M: d_n(x,y)<r\}$ . 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 critical point

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

Consequently, we call the following quantity

$$ \begin{align*}{\dim_{C}Z:=\liminf_{r\to 0}\dim_{C,r}Z }\end{align*} $$

the Carathéodory singular dimension of Z. In particular, the Carathéodory singular dimension of the repeller $\Lambda $ is independent of sufficiently small $r>0$ (see [11, Theorem 4.1]).

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

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

and the following quantity

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

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

Theorem A. Let $f: M\to M$ be a $C^{1}$ transformation of an $m_0$ -dimensional compact smooth Riemannian manifold M, and $\Lambda $ a repeller of f. Then there exists an f-invariant ergodic measure $\mu $ such that

$$ \begin{align*}\dim_{C}\mu=\dim_C\Lambda=\sup\{\dim_{C}\nu: \nu\in \mathcal{M}(f|_\Lambda)\}.\end{align*} $$

2.4 Measures of full and maximal dimension for ACH sets

In this section, we first recall the concept of an average conformal hyperbolic set which is introduced in [29]. By modifying the methods in [13], we construct a Borel probability measure (not necessarily invariant) of full Hausdorff dimension for ACH sets of $C^1$ diffeomorphisms. Following Barreira and Wolf’s approach [5], we prove that there exists an ergodic measure of maximal Hausdorff dimension for ACH sets of $C^{1+\alpha }$ diffeomorphisms.

2.4.1 Definition of ACH

Let $f{\kern-0.5pt}:{\kern-0.5pt}M{\kern-0.5pt}\rightarrow{\kern-0.5pt} M$ be a $C^1$ diffeomorphism on a $m_0$ -dimensional compact Riemannian manifold. For each $x \in M$ , the following quantities

$$ \begin{align*}\|D_x f\|=\sup_{0\neq v\in T_x M} \frac{\|D_x f(v)\|}{\|v\|},\quad m(D_x f)=\inf_{0\neq v\in T_x M} \frac{\|D_x f(v)\|}{\|v\|}\end{align*} $$

are respectively called the maximal norm and minimum norm of the differentiable operator $D_{x}f:T_{x}M\rightarrow T_{f(x)}M$ , where $\|\cdot \|$ is the norm induced by the Riemannian metric on M. A compact f-invariant subset $\Lambda \subset M$ is called a locally maximal hyperbolic set if there exists an open neighborhood U such that $ \Lambda =\bigcap _{n \in \mathbb {Z}} f^{n} U$ , and a continuous splitting of the tangent bundle $T_{x} M=E_{x}^{s} \oplus E_{x}^{u}$ , and constants $0<\unicode{x3bb} <1, C>0$ such that for every $x \in \Lambda $ :

1. (1) $D_{x} f(E_{x}^{u})=E_{f(x)}^{u} , D_{x} f(E_{x}^{s})=E_{f(x)}^{s};$

2. (2) for every $n \in \mathbb {N}$ , one has $\|D_{x} f^{-n}(v)\| \leqslant C \unicode{x3bb} ^n\|v\| \text { for all } v \in E_{x}^{u}$ , and $\|D_{x} f^{n}(v)\| \leqslant C\unicode{x3bb} ^n\|v\| \text { for all } v \in E_{x}^{s}$ .

For $x \in M$ and $v\in T_{x}M$ , the Lyapunov exponent of v at x is the limit

$$ \begin{align*}\unicode{x3bb}(x,v)=\lim _{n \rightarrow \infty} \frac{1}{n} \log \|D_{x} f^{n}(v)\|\end{align*} $$

whenever the limit exists. Given an invariant measure $\mu \in \mathcal {M}(f|_\Lambda )$ , by the Oseledec multiplicative ergodic theorem [20], for $\mu $ -almost every x, every vector $v\in T_xM$ has a Lyapunov exponent, and they can be denoted by $\unicode{x3bb} _{1}(x)\geqslant \unicode{x3bb} _{2}(x)\geqslant \cdots \geqslant \unicode{x3bb} _{m_0}(x)$ . Furthermore, if $\mu $ is ergodic, since the Lyapunov exponents are f-invariant, we write the Lyapunov exponents as $\unicode{x3bb} _{1}(\mu )\geqslant \unicode{x3bb} _{2}(\mu )\geqslant \cdots \geqslant \unicode{x3bb} _{m_0}(\mu )$ . Notice that $\|D_x f\|=\|D_x f|_{E_{x}^{u}}\|, ~\|D_x f^{-1}\|=\|D_x f^{-1}|_{E_{x}^{s}}\|$ in the hyperbolic setting.

A hyperbolic set $\Lambda \subset M$ is called an average conformal hyperbolic set if for each $\mu \in \mathcal {E}(f|_\Lambda )$ , one has $\unicode{x3bb} _{1}(\mu )=\unicode{x3bb} _{2}(\mu )=\cdots =\unicode{x3bb} _{d_u}(\mu )>0$ and $\unicode{x3bb} _{d_u+1}(\mu )=\unicode{x3bb} _{d_u+2}(\mu )=\cdots =\unicode{x3bb} _{m_0}(\mu )<0$ , where $d_u=\operatorname {dim} E^u$ and $d_s=\operatorname {dim} E^s=m_0-d_u$ . In other words, it has only two Lyapunov exponents $\unicode{x3bb} _u(\nu )>0$ and $\unicode{x3bb} _s(\nu )<0$ with respect to each $\nu \in \mathcal {E}(f|_\Lambda )$ .

2.4.2 Statements of main results

Although we cannot obtain an invariant measure of full dimension even in the case of conformal hyperbolic dynamical systems (see [4, Ch. 5] for a detailed description), the following result shows that there exists a measure (not necessarily invariant) of full dimension for ACH sets of a $C^{1}$ diffeomorphism.

Theorem B. Let $f:M\rightarrow M$ be a $C^{1}$ diffeomorphism on a $m_0$ -dimensional compact Riemannian manifold, and let $\Lambda \subset M$ be a compact locally maximal hyperbolic set such that $f|_{\Lambda }$ is topologically mixing and average conformal. Then there exists a Borel probability measure $\mu $ on $\Lambda $ with support strictly inside $\Lambda $ such that

$$ \begin{align*}\operatorname{dim}_{H} \mu=\dim_{H} \Lambda.\end{align*} $$

Under the setting of the above theorem, if f is of $C^{1+\alpha }$ smoothness, then there exists an f-invariant ergodic measure of maximal dimension.

Theorem C. Let $f:M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact Riemannian manifold, and let $\Lambda \subset M$ be a compact locally maximal hyperbolic set such that $f|_{\Lambda }$ is topologically mixing and average conformal. Then there exists an f-invariant ergodic probability measure $\mu $ on $\Lambda $ such that

$$ \begin{align*}\operatorname{dim}_{H} \mu=\sup \{\operatorname{dim}_{H} \nu:\nu \in \mathcal{M}(f|_\Lambda)\}.\end{align*} $$

3 Proofs

In this section, we provide the detailed proof of the main results in this paper.

3.1 Proof of Theorem A

By the definition of Carathéodory singular dimension of subsets and measures, for every $\mu \in \mathcal {M}(f|_{\Lambda })$ , we have that

$$ \begin{align*} \dim_C \Lambda=\dim_{C,r} \Lambda\ge \dim_{C,r}\mu \end{align*} $$

for all sufficiently small $r>0$ , see [11, Theorem 4.1] for the first equality. Letting $r\to 0$ , one has $\dim _C \Lambda \ge \dim _{C}\mu $ for every $\mu \in \mathcal {M}(f|_{\Lambda })$ . Hence, we have that

$$ \begin{align*} \dim_C\Lambda\ge \sup\{\dim_{C}\mu: \mu \in \mathcal{M}(f|_\Lambda)\}. \end{align*} $$

However, for each f-invariant measure $\mu $ , let

$$ \begin{align*}P_\mu(f|_\Lambda, \Phi_f(t)):=h_\mu(f)+\mathcal{L}_*(\Phi_f(t),\mu).\end{align*} $$

It is easy to see that $P_\mu (f|_\Lambda , \Phi _f(t))=0$ has a unique root, since $P_\mu (f|_\Lambda , \Phi _f(t))$ is strictly decreasing and continuous with respect to t. If $\mu \in \mathcal {E}(f|_{\Lambda })$ , it follows from [12, Theorem A] that

$$ \begin{align*}\dim_C\mu=t_\mu,\end{align*} $$

where $t_\mu $ is the unique solution of the equation $P_\mu (f|_\Lambda , \Phi _f(t))=0$ . Let $t^*$ be the unique zero of Bowen’s equation $P(f|_\Lambda ,\Phi _f(t))=0$ . Then $\dim _C\Lambda =t^*$ (see [11, Theorem 4.1] for details).

Since f is expanding on $\Lambda $ , the map $\mu \mapsto P_\mu (f|_\Lambda , \Phi _f(t^*))$ is upper semi-continuous on $\mathcal {M}(f|_{\Lambda })$ . It follows from the variational principle of sub-additive topological pressure that there exists an f-invariant ergodic measure $\widetilde {\mu }$ so that

$$ \begin{align*} 0=P(f|_\Lambda,\Phi_f(t^*))= P_{\widetilde{\mu}}(f|_\Lambda, \Phi_f(t^*)). \end{align*} $$

Hence,

$$ \begin{align*} \dim_C\Lambda=t^*=t_{\widetilde{\mu}}=\dim_{C}\widetilde{\mu}. \end{align*} $$

This completes the proof of the theorem. $\square $

3.2 Proof of Theorem B

We first recall some facts of average conformal hyperbolic sets. For each $x \in \Lambda $ , denote

(3.1) $$ \begin{align} \phi_{u}(x)=-\log|\!\operatorname {det} D_{x} f|_{E_{x}^{u}}|^{{1}/{d_u}},\quad\phi_{s}(x)=\log|\!\operatorname {det} D_{x} f|_{E_{x}^{s}}|^{{1}/{d_s}}, \end{align} $$

where $d_u=\dim E^u_x$ , $d^s=\dim E^s_x$ , and $d_u+d_s=m_0$ . It is clear that $\phi _u$ and $\phi _s$ are continuous functions, since f is a $C^1$ diffeomorphism. Furthermore, the following properties hold:

1. (1) for any $n\in \mathbb {Z}$ , one has $m(D_{x} f^{n}|_{E_{x}^{i}}) \leqslant |\!\operatorname {det} D_{x} f^n|_{E_{x}^{i}}|^{{1}/{d_i}} \leqslant \|D_{x} f^{n}|_{E_{x}^{i}}\|$ and

   (3.2) $$ \begin{align} \lim _{n \rightarrow \pm\infty} \frac{1}{|n|}(\log \|D_{x} f^{n}|_{E_{x}^{i}} \|-\log m(D_{x} f^{n}|_{E_{x}^{i}}))=0 \end{align} $$
   uniformly on $\Lambda $ , for $i \in \{u, s\}$ . In fact, let $\psi _n(x) = \log \|D_{x} f^{n}|_{E_{x}^{u}} \|-\log m (D_{x} f^{n} |_{E_{x}^{u}})$ . Then the sequence of continuous functions $\Psi :=\{\psi _n\}_{n\ge 1}$ is sub-additive. By [19, Theorem A.3], one has
   $$ \begin{align*} \begin{aligned} \lim_{n\to\infty}\frac{1}{n}\max_{x\in\Lambda }\psi_n(x)&=\sup\{\mathcal{L}_*(\Psi,\mu):\mu\in \mathcal{E}(f|_\Lambda)\}\\ &=\sup\{\unicode{x3bb}_1(\mu)-\unicode{x3bb}_{d_u}(\mu):\mu\in \mathcal{E}(f|_\Lambda)\}\\ &=0. \end{aligned} \end{align*} $$
   The case of $i=s$ can be proven in a similar fashion. This yields the uniformly convergence in equation (3.2). See [2, Theorem 4.2] for the detailed proof of the case of average conformal repellers;

2. (2) let $t_{u}$ and $t_{s}$ denote the unique root of $P(f|_\Lambda , t \phi _{u})=0$ and $P(f|_\Lambda , t \phi _{s})=0$ , respectively. Then

   (3.3) $$ \begin{align} \dim_H \Lambda=\dim_B \Lambda =t_{u}+t_{s}, \end{align} $$
   see [29, Theorem A and Remark 7] for a detailed description.

Since f is hyperbolic on $\Lambda $ , it is expansive, so we let $0<c<1$ be an expansive constant of $f|_{\Lambda }$ . In the rest of the proof of Theorem B, we fix a small number $\delta>0$ . According to equation (3.2), there exists $N(\delta )$ such that

$$ \begin{align*} 1 \leqslant \frac{\|D_{x} f^{n}|_{E_{x}^{u}}\|}{\exp\{-\sum_{i=0}^{n-1} \phi_{u}(f^{i} x)\}} \leqslant e^{n \delta},\quad e^{-\ell \delta}\leqslant \frac{\|D_{x} f^{-\ell}|_{E_{x}^{s}}\|} {\exp\{-\sum_{i=0}^{\ell-1} \phi_{s}(f^{-i} x)\}} \leqslant 1 \end{align*} $$

for any $n,\ell \geqslant N(\delta )$ and any $x\in \Lambda $ . Fix a positive integer $L\geqslant N(\delta )$ . It follows from the uniform continuity of the map $x \mapsto \|D_{x} f^{L}\|$ that there exists $~0<\varepsilon _{0}<{c}/{4}~$ such that

$$ \begin{align*} e^{-\delta} \leqslant \frac{\|D_{z} f^{L}\|}{\|D_{y} f^{L}\|} \leqslant e^{\delta} \quad\text{and}\quad e^{-\delta} \leqslant \frac{\|D_{z} f^{-L}\|}{\|D_{y} f^{-L}\|} \leqslant e^{\delta} \end{align*} $$

for any $y,z\in \Lambda $ with $d(y,z)<\varepsilon _{0}$ .

Choose a Markov partition $\mathcal {R}=\{R_{1}, \ldots , R_{s}\}$ of $\Lambda $ such that

$$ \begin{align*}\operatorname{diam} \mathcal{R}:=\max \{\operatorname{diam} R_{i} \mid i=1, \ldots, s\}<\varepsilon_{0}\end{align*} $$

and $\#\{1 \leqslant q \leqslant s: R_{p} \cap R_{q}=\varnothing \}>0$ for every $p\in \{1,\ldots ,s\}$ (see [7]). Let $A=(a_{i j})_{1\leqslant i,j\leqslant s}$ be the structure matrix of $\mathcal {R}$ and $(\Sigma _{A}, \sigma )$ be the corresponding Markov subshift. We denote the set of all words of length n of $\Sigma _{A}$ by $\Sigma {(n)}$ and let

$$ \begin{align*} {}_{-k}[a_{-k} \ldots a_{0} \ldots a_{\ell}] _{\ell}=\{\mathbf{b}=(b_{i}) \in \Sigma_{A} \mid a_{i}=b_{i},\quad\text{for all }~i=-k,\ldots,0 ,\ldots,\ell \} \end{align*} $$

for each $\mathbf {a}=(a_{i}) \in \Sigma _{A}$ and $k, \ell \in \mathbb {N}$ . Define a coding map $h:\Sigma _{A}\rightarrow \Lambda $ by

$$ \begin{align*}h({}_{-k}[a_{-k} \ldots a_{0} \ldots a_{\ell}] _{\ell})=\bigcap_{j=-k}^{\ell} f^{-j} R_{a_{j}}\quad\text{for all }(a_{-k}\ldots a_{\ell}) \in \Sigma(k+\ell+1),\end{align*} $$

then the map h is a continuous surjection which satisfies $h \circ \sigma =f \circ h$ . Let

$$ \begin{align*}\mathcal{R}(-k, \ell)=\bigg\{\bigcap_{j=-k}^{\ell} f^{-j} R_{a_{j}}:(a_{-k} \ldots a_{\ell}) \in \Sigma(k+\ell+1)\bigg\}.\end{align*} $$

Since h is bounded finite to one (see [1]), there is an integer $e_{0}>0$ such that

(3.4) $$ \begin{align} \#\{W \in \mathcal{R}(-k, \ell)\mid x \in W\} \leqslant e_{0} \end{align} $$

for every $x \in \Lambda $ and $k, \ell \in \mathbb {N}$ .

We construct the desired Borel probability measure as follows. Since $t_u \phi _{u} \circ h$ and $t_s \phi _{s} \circ h$ are continuous functions and

$$ \begin{align*} P(\sigma, t_u \phi_{u}\circ h)=P(f|_\Lambda, t_u \phi_{u})=0,\quad P(\sigma, t_s \phi_{s}\circ h)=P(f|_\Lambda, t_s \phi_{s})=0, \end{align*} $$

there exist two weak Gibbs measures $m_+, m_-$ on $\Sigma _{A}$ in the sense that there exist two sequences of positive constants $\{A_\ell \}_{\ell \in \mathbb {N}}$ and $\{B_k\}_{k\in \mathbb {N}}$ satisfying $\lim _{\ell \rightarrow \infty } ({1}/{\ell }) \log A_{\ell }=0$ and $\lim _{k \rightarrow \infty } ({1}/{k}) \log B_{k}=0$ such that

$$ \begin{align*} \frac{1}{A_{l+1}} \leqslant \frac{m_{+}({}_{0}[a_{0} \ldots a_{l}]_{l})}{\exp \bigg\{\sum_{j=0}^{l} t_{u} \phi_{u}\circ h(\sigma^{j} \mathbf{a})\bigg\}} \leqslant A_{l+1} \end{align*} $$

and

$$ \begin{align*} \frac{1}{B_{k+1}} \leqslant \frac{m_{-}({}_{-k}[a_{-k} \ldots a_{0}]_{0})}{\exp \bigg\{\sum_{j=0}^{k} t_{s} \phi_{s}\circ h(\sigma^{-j} \mathbf{a})\bigg\}} \leqslant B_{k+1} \end{align*} $$

for every $\mathbf {a}=(a_i)\in \Sigma _{A}$ and $k,\ell \in \mathbb {N}$ (see [3, pp. 289]).

Let m be a Borel probability measure on $\Sigma _{A}$ such that

$$ \begin{align*}m({}_{-k}[a_{-k} \ldots a_{0} \ldots a_{\ell}]_{\ell})=\begin{cases} C_{1}m_{+}({}_{0}[a_{0} \ldots a_{\ell}]_{\ell}) m_{-}({}_{-k}[a_{-k} \ldots a_{0}]_{0}), & a_0=1,\\ 0, &a_0 \neq 1, \end{cases}\end{align*} $$

for every $\mathbf {a}=(a_i)\in \Sigma _{A}$ , $k,\ell \in \mathbb {N}$ , and $C_{1}=m_{+}({}_{0}[1]_{0})^{-1} m_{-}({}_{0}[1]_{0})^{-1}.$

Define a Borel probability measure $\mu $ on $\Lambda $ by

$$ \begin{align*}\mu(A)=m(h^{-1}(A))\end{align*} $$

for each Borel subset $A\subset \Lambda $ . It is clear from the definition that the support of $\mu $ is $R_{1}$ . Furthermore, if $a_{0}=1\text { and }(a_{-k} \ldots a_{0} \ldots a_{\ell }) \in \Sigma (k+\ell +1),~k,\ell \in \mathbb {N}$ , then we have that for $ x \in \bigcap _{j=-k}^{\ell } f^{-j} R_{a_{j}}$ ,

(3.5) $$ \begin{align} \frac{C_{1}}{A_{\ell+1}B_{k+1}} \leqslant \frac{\mu(\bigcap_{j=-k}^{\ell} f^{-j} R_{a_{j}})}{\exp \{\sum_{i=0}^{\ell} t_{u} \phi_{u} (f^{i} x)+\sum_{j=0}^{k} t_{s} \phi_{s} (f^{-j} x)\}} \leqslant C_{1}A_{\ell+1}B_{k+1}. \end{align} $$

To prove Theorem B, we first prove some auxiliary results.

Lemma 3.1. Let $f:M\rightarrow M$ be a $C^{1}$ diffeomorphism on an $m_0$ -dimensional compact Riemannian manifold, and let $\Lambda \subset M$ be a compact locally maximal hyperbolic set such that $f|_{\Lambda }$ is topologically mixing and average conformal. There exists $\varepsilon _0>0$ (make $\varepsilon _0$ small if necessarily) such that for any $n,m\in \mathbb {N}$ , if $ x,z\in \Lambda $ satisfy $\max _{0 \leqslant j \leqslant n-1} d(f^{j} x, f^{j} z) <\varepsilon _0$ and $ y,w\in \Lambda $ satisfy $\max _{0 \leqslant j \leqslant m-1} d(f^{-j} y, f^{-j} w) <\varepsilon _0$ , then

$$ \begin{align*} e^{-n\delta} < \frac{|\!\det D_{z} f^{n}|_{E^u_{z}}|^{{1}/{d_{u}}}}{|\!\det D_{x} f^{n}|_{E^u_{x}}|^{{1}/{d_{u}}}} < e^{n\delta} \end{align*} $$

and

$$ \begin{align*} e^{-m\delta} < \frac{|\!\det D_{w} f^{-m}|_{E^s_{w}}|^{{1}/{d_{s}}}}{|\!\det D_{y} f^{-m}|_{E^s_{y}}|^{{1}/{d_{s}}}} < e^{m\delta}. \end{align*} $$

Proof. Since $\log |\!\operatorname {det} D_xf|$ is uniformly continuous on $\Lambda $ , there exists $\varepsilon _0>0$ such that if $d(x,z)<\varepsilon _0$ , then

$$ \begin{align*}|\!\log |\! \det D_{z} f|- \log |\! \det D_{ x} f||<\delta.\end{align*} $$

Hence, if $ x,z\in \Lambda $ satisfy $\max _{0 \leqslant j \leqslant n-1} d(f^{j} x, f^{j} z) <\varepsilon _0$ , then

$$ \begin{align*} \bigg|\log \frac{|\det D_{z} f^{n}|_{E^u_{z}}|^{{1}/{d_u}}}{|\det D_{x} f^{n}|_{E^u_{x}}|^{{1}/{d_u}}}\bigg| &\le \sum_{j=0}^{n-1}\bigg|\log \frac{|\operatorname {det} D_{f^{j} z} f|_{E^u_{f^{j} z}}|^{{1}/{d_u}}}{|\det D_{f^{j} x} f|_{E^u_{f^{j} x}}|^{{1}/{d_{u}}}}\bigg|\\ &< n\delta. \end{align*} $$

The other one can be proven in a similar fashion. This completes the proof of the lemma.

Set $\delta _{0}=\inf \{ d(x, y):\ x \in R_{p},\ y \in R_{q},\ R_{p} \cap R_{q}=\emptyset ,\ 1 \leqslant p<q \leqslant s \}>0$ , $ r_{0}=\min \{{\delta _{0}}/{2}, {\varepsilon _{0}}/{2}\}>0$ , and $ C_2=\max _{1 \leqslant i \leqslant {L-1}} \max _{x\in M}\{\|D_{x} f^{i}\|\}$ .

Lemma 3.2. For $n, \ell> L\geqslant N(\delta )$ , if $x\in \Lambda $ and $0<r<r_0$ satisfy that $ C_{2} r \exp \{2n \delta -\sum _{i=0}^{n-1} \phi _{u}(f^{i} x)\}<r_0$ and $C_{2} r \exp \{2\ell \delta -\sum _{i=0}^{\ell -1} \phi _{s}(f^{-i} x)\}<r_{0}$ , then

$$ \begin{align*}B_{r}(x) \subset B_{r_{0}}^{f}(x,-\ell, n),\end{align*} $$

where $B_{r_{0}}^{f}(x,-\ell , n)=\{z \in M: d(f^{j} x, f^{j} z) < r_{0}, \text { for all } j=-\ell , \ldots , n\}.$

Proof. Take $y\in B_{r}(x)$ , we first show that $ d(f^{j} x, f^{j} y) < r_{0}$ for every $j=1, \ldots , n$ if $ C_{2} r \exp \{2n \delta -\sum _{i=0}^{n-1} \phi _{u}(f^{i} x)\}<r_0.$

Choose $0<\varepsilon _{1}<{\varepsilon _{0}}/{2}$ so small that $ C_{2}(r+\varepsilon _{1}) \exp \{2n \delta -\sum _{i=0}^{n-1} \phi _{u}(f^{i} x)\}<r_{0}$ . According to the definition of the Riemannian metric, there exists a smooth curve $\xi :[0,1]\rightarrow M$ such that

$$ \begin{align*}\xi(0)=x,~\xi(1)=y\quad\text{and}\quad\int_{0}^{1}\|\dot{\xi} (s)\| \,d s \le r+\varepsilon_{1}.\end{align*} $$

Since

$$ \begin{align*}d(x, \xi(t)) \leqslant \int_{0}^{t}\|\dot{\xi}(s)\|\, d s\leqslant r+\varepsilon_{1}<\varepsilon_{0}~(0\leqslant t \leqslant 1),\end{align*} $$

we have

$$ \begin{align*} d(f^{L}x, f^{L}\xi(t)) &\le \int_{0}^{t}\|D_{\xi(s)} f^L\| \cdot\|\dot{\xi}(s)\|\, d s\\[-2pt]&\leqslant e^\delta\|D_{x} f^L\|\int_{0}^{t}\|\dot{\xi}(s)\|\, d s\\[-2pt]&= e^\delta\|D_{x} f^L|_{E^u_{x}}\|\int_{0}^{t}\|\dot{\xi}(s)\|\, d s\\[-2pt]&\leqslant (r+\varepsilon_{1}) e^{(L+1) \delta}\cdot \prod_{i=0}^{L-1}\|\!\det D_{f^{i} x} f|_{E_{f^{i} x}^{u}}\|^{{1}/{d_u}}\\[-2pt]&= (r+\varepsilon_{1}) \exp \bigg\{(L+1) \delta-\sum_{i=0}^{L-1} \phi_{u}(f^{i} x)\bigg\}\\[-2pt]&<r_{0}(<\varepsilon_{0}). \end{align*} $$

Furthermore, we have that

$$ \begin{align*} d(f^{2L}x, f^{2L}\xi(t)) &\leqslant \int_{0}^{t}\|D_{\xi(s)} f^L\|\cdot\|D_{f^{L}\xi(s)} f^L\| \cdot\|\dot{\xi}(s)\| \,d s\\ &\leqslant e^{2\delta}\|D_{x} f^L\|\cdot\|D_{f^{L}x} f^L\|\int_{0}^{t}\|\dot{\xi}(s)\| \,d s\\ &= e^{2\delta}\|D_{x} f^L|_{E^u_{x}}\|\cdot\|D_{f^{L}x} f^L|_{E^u_{f^L x}}\|\int_{0}^{t}\|\dot{\xi}(s)\| \,d s\\ &\leqslant (r+\varepsilon_{1}) e^{2(L+1) \delta}\cdot\prod_{i=0}^{2L-1} \|\!\operatorname {det} D_{f^{i} x} f|_{E_{f^{i} x}^{u}}\|^{{1}/{d_u}}\\ &= (r+\varepsilon_{1}) \exp \bigg\{2(L+1) \delta-\sum_{i=0}^{2L-1} \phi_{u}(f^{i} x)\bigg\}. \end{align*} $$

Therefore, for every $j=1, \ldots , n $ , write $j=g_{j}L+t_{j}$ , where $g_{j}\in \mathbb {N}$ and $0\leqslant t_{j}< L$ , we have

$$ \begin{align*} d(f^{j}x, f^{j}\xi(t)) &\leqslant \int_{0}^{t}\prod_{i=0}^{g_{j}-1}\|D_{f^{iL}\xi(s)} f^L\|\cdot\|D_{f^{g_{j}L}\xi(s)} f^{t_j}\| \cdot\|\dot{\xi}(s)\| \,d s\\[-2pt]&\leqslant C_{2}e^{g_{j}\delta}\cdot\prod_{i=0}^{g_{j}-1}\|D_{f^{iL}x}f^L\|\cdot\int_{0}^{t}\|\dot{\xi}(s)\| \,d s\\[-2pt]&\leqslant C_{2}(r+\varepsilon_{1}) \exp \bigg\{g_{j}(L+1) \delta-\sum_{i=0}^{g_{j}L-1} \phi_{u}(f^{i} x)\bigg\}\\[-2pt]&\leqslant C_{2}(r+\varepsilon_{1}) \exp \bigg\{2j\delta-\sum_{i=0}^{j-1} \phi_{u}(f^{i} x)\bigg\}\\[-2pt]&\leqslant C_{2}(r+\varepsilon_{1}) \exp \bigg\{2n\delta-\sum_{i=0}^{n-1} \phi_{u}(f^{i} x)\bigg\}\\[-2pt]&<r_0~(0\leqslant t \leqslant 1). \end{align*} $$

Setting $t=1$ , we obtain $d(f^{j} x, f^{j} y) \leqslant r_{0}$ for every $1\leqslant j\leqslant n$ . Analogously, one can show that $d(f^{-j}x, f^{-j}y){\kern-1pt}<{\kern-1pt}r_0$ for $j{\kern-1pt}={\kern-1pt}1, \ldots , \ell $ if $ C_{2} r \exp \{2\ell \delta {\kern-1pt}-{\kern-1pt}\sum _{i=0}^{\ell -1} \phi _{s}(f^{-i} x)\}{\kern-1pt}<{\kern-1pt}r_{0}$ . This completes the proof of the lemma.

For each $x\in \Lambda $ and sufficiently small $0<r<r_0$ , let

$$ \begin{align*} n_{1}=\min \bigg\{n\in{\mathbb{Z}}^{+}: C_{2} r \exp \bigg\{2(n+1) \delta-\sum_{i=0}^{n} \phi_{u}(f^{i} x)\bigg\} \geqslant r_{0}\bigg\}, \end{align*} $$

$$ \begin{align*} n_{2}=\min \bigg\{n\in{\mathbb{Z}}^{+}: C_{2} r \exp \bigg\{2(n+1) \delta-\sum_{i=0}^{n} \phi_{s}(f^{-i} x)\bigg\} \geqslant r_{0}\bigg\}. \end{align*} $$

It follows from the definition of $n_1,~n_2$ that $n_1,n_2\rightarrow +\infty ~ (r\rightarrow 0)$ . Recall that $\mathcal {R}=\{R_{1}, \ldots , R_{s}\}$ is a Markov partition of $\Lambda $ and $e_0$ is defined in equation (3.4).

Lemma 3.3. For each $x\in \Lambda $ , take a sufficiently small $0<r<r_0$ so that $n_1,~n_2>L$ . Then there exist $W_{1}, \ldots , W_{m} \in \mathcal {R}(-n_{2}, n_{1})$ with $m=m(x, r_{0},-n_{2}, n_{1}) \leq s^{2} e_{0}$ such that

$$ \begin{align*} W_{k} \cap B_{r_0}^{f}(x,-n_{2}, n_{1}) \neq \varnothing \quad \text{for all } k=1, \ldots, m, \end{align*} $$

and $ B_{r}(x) \cap \Lambda \subset \bigcup _{k=1}^{m} W_{k}$ .

Proof. For $y\in \Lambda $ , let $\mathcal {R}_{y}=\{Q \in \mathcal {R} \mid R \cap Q \neq \varnothing , y \in R \in \mathcal {R}\}$ and $P_y=\bigcup _{\mathcal {R}_{y}} Q$ , then $P_{y} \subset B_{2 \varepsilon _{0}}(y) \cap \Lambda $ . From the definition of $\delta _{0}$ , if $z \in \Lambda $ and $d(y, z)<\delta _{0}$ , then for any $Q\in \mathcal {R}$ containing z, $Q \in \mathcal {R}_{y}$ .

Let

$$ \begin{align*} &\mathcal{P}(x,-n_{2}, n_{1})\\&\quad=\bigg\{W=\bigcap_{j=-n_{2}}^{n_1} f^{-j} R_{a_j} \in \mathcal{R}(-n_{2}, n_{1}) : R_{a_j} \subset P_{f^{j} x}~\text{ for all } j=-n_{2},\ldots, n_{1}\bigg\} \end{align*} $$

and $ \mathcal {P}(x, r_{0},-n_{2}, n_{1})=\{W \in \mathcal {P}(x,-n_{2}, n_{1}): B_{r_{0}}^{f}(x,-n_{2}, n_{1}) \cap W \neq \varnothing \}$ . By Lemma 3.2 and the choice of $n_1,n_2$ , we have

$$ \begin{align*} B_r(x) \cap \Lambda\subset B_{r_{0}}^{f}(x,-n_{2}, n_{1}) \cap \Lambda\subset \bigcup_{W \in \mathcal{P}(x,r_0,-n_{2}, n_{1})} W. \end{align*} $$

Let $m=\#\mathcal {P}(x, r_{0},-n_{2}, n_{1})$ and $\mathcal {P}(x, r_{0},-n_{2}, n_{1})=\{W_{1}, \ldots , W_{m}\}$ . To complete the proof of the lemma, it suffices to show that

$$ \begin{align*}m=m(x, r_{0},-n_{2}, n_{1}) \le s^{2} e_{0}.\end{align*} $$

To prove this, set $\ell _{1}=\# \mathcal {R}_{f^{n_1} x},~~ \ell _{2}=\# \mathcal {R}_{f^{-n_2} x}$ and take $\alpha _{1}, \ldots , \alpha _{\ell _1}, \beta _{1}, \ldots , \beta _{\ell _{2}} \in \{1, \ldots , s\}$ so that

$$ \begin{align*}P_{f^{n_1} x}=\bigcup_{p=1}^{\ell_1} R_{\alpha_{p}},\quad P_{f^{-n_2} x}=\bigcup_{q=1}^{\ell_2} R_{\beta_{q}}.\end{align*} $$

Then,

$$ \begin{align*}\mathcal{P}(x, r_{0},-n_{2}, n_{1})=\bigcup_{p=1}^{l_1}\ \bigcup_{q=1}^{l_2} Q_{p, q},\end{align*} $$

where $Q_{p, q}=\{W \in \mathcal {P}(x, r_{0},-n_{2}, n_{1}):W \subset f^{n_{2}} R_{\beta _{q}} \cap f^{-n_{1}} R_{\alpha _{p}}\}$ . Fix $1 \leqslant p \leqslant \ell _{1}, 1 \leqslant q \leqslant \ell _{2}$ such that $Q_{p, q} \neq \varnothing $ . Put $t=\# Q_{p, q}$ and let

$$ \begin{align*}Q_{p, q}=\{W_{p, q}^{1}, \ldots, W_{p,q}^{t}\}.\end{align*} $$

Since $(\Sigma _{A},\sigma )$ is topologically mixing, there is a $K_0\geqslant 2$ such that $A^{K}>0$ for each $K\geqslant K_0$ . We choose $(\alpha _{p} \omega _{1} \omega _{2} \ldots \omega _{K_{0}-1} \beta _{q}) \in \Sigma (K_{0}+1)$ and take

$$ \begin{align*}z_{p, q}^i\in W_{p, q}^{i} \cap\bigg(\bigcap_{k=1}^{K_{0}-1} f^{-n_{1}-k} R_{\omega_{k}}\bigg)\quad\text{such that }f^{n_{1}+n_{2}+K_{0}} z_{p, q}^{i}=z_{p, q}^{i}\end{align*} $$

for each $i=1,2, \ldots ,t$ .

Hence, for $1\le i,~j\le t$ , one has

$$ \begin{align*} d(f^{k} z_{p, q}^{i}, f^{k} z_{p,q}^{j}) &\leqslant d(f^{k} z_{p, q}^{i}, f^{k} x)+d(f^{k} x, f^{k} z_{p, q}^{j}) \\ & \leqslant 2 \varepsilon_{0}+2 \varepsilon_{0}\\ &<c \end{align*} $$

for each $k=-n_2,\ldots ,0,\ldots ,n_1$ , where c is the expansive constant of f. Moreover, for $k=1,\ldots ,K_0-1$ , we have that

$$ \begin{align*} d(f^{n_{1}+k} z_{p, q}^{i}, f^{n_1+k} z_{p, q}^{j}) \leqslant \mathrm{diam}\, R_{\omega_{k}} <c. \end{align*} $$

This implies that $d(f^{m} z_{p, q}^{i}, f^{m} z_{p, q}^{j})<c$ for every $m\in \mathbb {Z}$ . Thus, $z_{p, q}^{i}=z_{p, q}^{j}$ for each $1\leqslant i,j\leqslant t$ . Hence, we have that

$$ \begin{align*}z_{p, q}^{1} \in \bigcap_{i=1}^{t} W_{p, q}^{i}=\bigcap_{W \in Q_{p, q}} W.\end{align*} $$

Furthermore, one has $\# Q_{p, q} \le e_{0}$ since $\#\{W \in \mathcal {R}(-n_{2}, n_{1}): z_{p, q}^{1} \in W\} \leqslant e_{0}$ . Hence, we have

$$ \begin{align*} m &=\# \mathcal{P}(x, r_{0},-n_{2}, n_{1}) \\ &=\sum_{p=1}^{l_{1}} \sum_{q=1}^{l_2} \# Q_{p, q} \\ & \leqslant s^{2} e_0. \end{align*} $$

This completes the proof of the lemma.

Using the previous results, we proceed with the proof of Theorem B.

Proof of Theorem B

For every $x\in \text {supp}\mu ~(=R_1)$ and sufficiently small $0<r<r_0$ , choose $n_1,~n_2>L\geqslant N(\delta )$ and $W_1,\ldots ,W_m\in \mathcal {R}(-n_2,n_1)$ with $m\leqslant s^2 e_0$ , as in Lemma 3.3. For each $k=1, \ldots , m$ , pick

$$ \begin{align*}y_{k} \in W_{k} \cap B_{r_0}^{f}(x,-n_{2}, n_{1}).\end{align*} $$

By Lemma 3.1, we have

$$ \begin{align*} \exp \bigg\{\sum_{i=0}^{n_{1}} \phi_{u}(f^{i} y_{k})\bigg\} \leq \exp \bigg\{(n_1+1)\delta+\sum_{i=0}^{n_1} \phi_{u}(f^{i} x)\bigg\} \end{align*} $$

and

$$ \begin{align*} \exp \bigg\{\sum_{j=0}^{n_2} \phi_{s}(f^{-j} y_{k})\bigg\} \leqslant \exp \bigg\{(n_2+1)\delta+\sum_{j=0}^{n_{2}} \phi_{s}(f^{-j} x)\bigg\}. \end{align*} $$

Furthermore, for each $k=1, \ldots , m$ , it follows from equation (3.5) that

$$ \begin{align*} \mu(W_{k}) &\leqslant C_{1}A_{n_1+1}B_{n_2+1} \exp\bigg\{\sum_{i=0}^{n_{1}} t_{u} \phi_{u}(f^{i} y_{k})+\sum_{j=0}^{n_{2}} t_s \phi_{s}(f^{-j} y_{k})\bigg\}\\[-3pt]&\leq C_{1}A_{n_1+1}B_{n_2+1} \exp \bigg\{(t_u (n_1+1)+t_s (n_2+1))\delta+\sum_{i=0}^{n_{1}}t_u \phi_{u}(f^{i} x)\\[-3pt]& \quad+\sum_{j=0}^{n_{2}}t_{s}\phi_{s}(f^{-j} x)\bigg\}\\[-3pt]&\leq C_{1}A_{n_1+1}B_{n_2+1} \exp \{3(t_{u}(n_{1}+1) +t_{s}(n_{2}+1)) \delta\}\\[-3pt]&\quad\cdot\exp \bigg\{-2t_{u}(n_{1}+1) \delta+\sum_{i=0}^{n_{1}} t_u \phi_{u}(f^{i} x)-2t_{s}(n_{2}+1) \delta+\sum_{j=0}^{n_{2}} t_{s} \phi_{s}(f^{-j} x)\bigg\}\\[-3pt]&\leqslant C_{1}A_{n_1+1}B_{n_2+1}\bigg(\frac{C_{2} r}{r_{0}}\bigg)^{\dim_{H} \Lambda}\cdot \exp \{3(t_{u}(n_{1}+1) +t_{s}(n_{2}+1)) \delta\}, \end{align*} $$

where we use the fact that $\dim _H\Lambda =t_u+t_s$ . Hence, we have

(3.6) $$ \begin{align} \begin{split} \mu&(B_{r}(x) \cap \Lambda) \\ &\leqslant \sum_{k=1}^{m} \mu(W_{k})\\ &\leqslant CA_{n_1+1}B_{n_2+1}r^{\dim_H \Lambda} \cdot \exp \{3(t_{u}(n_{1}+1) +t_{s}(n_{2}+1)) \delta\}, \end{split} \end{align} $$

where $C=s^{2} e_{0}C_{1} C_{2}^{\operatorname {dim} _H \Lambda }r_{0}^{-\operatorname {dim} _H \Lambda }.$

By the definition of $n_1,~n_2$ , we have

(3.7) $$ \begin{align} &2n_{1} \delta-\sum_{i=0}^{n_1-1} \phi_u(f^{i} x)<\log r_{0}-\log C_{2}-\log r, \end{align} $$

(3.8) $$ \begin{align} &2n_{2} \delta-\sum_{i=0}^{n_2-1} \phi_{s}(f^{-i} x)<\log r_{0}-\log C_{2}-\log r. \end{align} $$

Let $ M:=\max _{x \in \Lambda }\{\phi _{u}(x), \phi _s (x)\}<0$ . Thus, by equation (3.7), we conclude

$$ \begin{align*} \liminf _{r \rightarrow 0} \frac{2n_{1} \delta-n_{1}M}{\log r} \geqslant \liminf _{r \rightarrow 0}\frac{\log r_{0}-\log C_{2}-\log r}{\log r}, \end{align*} $$

that is,

$$ \begin{align*} \liminf _{r\rightarrow 0} \frac{n_{1}}{\log r} \geqslant\frac{-1}{2\delta-M}. \end{align*} $$

Similarly, by equatoin (3.8), one has

$$ \begin{align*} \liminf _{r \rightarrow 0} \frac{n_{2}}{\log r} \geqslant\frac{-1}{2\delta-M}. \end{align*} $$

Therefore, by equation (3.6), we have

$$ \begin{align*} \liminf_{r \rightarrow 0} \frac{\log \mu(B_{r}(x))}{\log r}\geqslant \operatorname{dim}_{H} \Lambda-\frac{3({t_u}+{t_s})\delta}{2\delta-M} \end{align*} $$

for every $x\in \text {supp}\mu $ . The arbitrariness of $\delta $ implies

$$ \begin{align*}\underline{d}_\mu (x)\geqslant\operatorname{dim}_{H} \Lambda\end{align*} $$

for every $x\in \text {supp}\mu $ . Thus, we have that

$$ \begin{align*}\operatorname{dim}_{H} \mu \geqslant\operatorname{dim}_{H} \Lambda.\end{align*} $$

However, the reverse inequality $\dim _{H} \mu \leqslant \dim _{H} \Lambda $ clearly follows from the definitions. This completes the proof of Theorem B.

3.3 Proof of Theorem C

Assume that $f:M\rightarrow M$ is a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact Riemannian manifold, and let $\Lambda \subset M$ be a compact locally maximal hyperbolic set such that $f|_{\Lambda }$ is topologically mixing and average conformal. Recall that $\mathcal {M}(f|_\Lambda )$ and $\mathcal {E}(f|_\Lambda )$ denote the set of f-invariant measures and ergodic measures on $\Lambda $ , respectively.

Recall that the topological pressure $P(\phi )$ of a continuous function $\phi :\Lambda \mapsto \mathbb {R}$ (with respect to $f|_{\Lambda }$ ) satisfies the following variational principle:

(3.9) $$ \begin{align} P(\phi)=\sup _{\mu \in \mathcal{M}(f|_\Lambda)}\bigg\{h_{\mu}(f)+\int \phi ~d \mu\bigg\}. \end{align} $$

Remark that $P(0)=h_{\text {top}}(f)$ is the topological entropy of $f|_{\Lambda }$ . A measure $\mu \in \mathcal {M}(f|_\Lambda )$ which attains the supermum in equation (3.9) is called an equilibrium measure of $\phi $ , and two functions $\phi , \psi :\Lambda \rightarrow \mathbb {R}$ are said to be cohomologous if $\phi -\psi =\eta -\eta \circ f$ for some continuous function $\eta : \Lambda \rightarrow \mathbb {R}$ . Denote by $C^{\alpha }(\Lambda )$ the space of Hölder continuous functions $\varphi :\Lambda \rightarrow \mathbb {R}$ with Hölder exponent $\alpha $ . We list several properties of the topological pressure in the following (see [23] for details):

1. (1) the map $\phi \mapsto P(\phi )$ is analytic in $C^{\alpha }(\Lambda )$ ;

2. (2) each function $\phi \in C^{\alpha }(\Lambda )$ has a unique equilibrium measure $\nu _{\phi } \in \mathcal {E}(f|_\Lambda )$ ;

3. (3) for each $\phi ,\psi \in C^{\alpha }(\Lambda )$ , we have $\nu _{\phi }=\nu _{\psi }$ if and only if $\phi -\psi $ is cohomologous to a constant;

4. (4) for each $\phi ,\psi \in C^{\alpha }(\Lambda )$ and $t\in \mathbb {R}$ , we have

   $$ \begin{align*}\frac{d}{d t} P(\phi+t \psi) \geqslant 0,\end{align*} $$
   with equality if and only if $\psi $ is cohomologous to a constant.

Note that the functions $\phi _u$ and $\phi _s$ defined in equation (3.1) are $\alpha $ -Hölder continuous in this case. For each $\nu \in \mathcal {M}(f|_\Lambda )$ , put

$$ \begin{align*}\unicode{x3bb}_{u}(\nu)=-\int \phi_{u}~ d \nu,~~\unicode{x3bb}_{s}(\nu)=\int \phi_{s}~ d \nu \quad\text{and}\quad d(\nu)=h_{\nu}(f)\bigg(\frac{1}{\unicode{x3bb}_{u}(\nu)}-\frac{1}{\unicode{x3bb}_{s}(\nu)}\bigg),\end{align*} $$

then $\unicode{x3bb} _{u}(\nu )>0,~ \unicode{x3bb} _{s}(\nu )<0$ . Furthermore, if $\nu \in \mathcal {E}(f|_\Lambda )$ , then

(3.10) $$ \begin{align} \dim_{H} \nu=d(\nu). \end{align} $$

See [28] for the detailed proofs, which can be viewed as an extension of Young’s results in [31] to the case of an average conformal hyperbolic setting. If $\mu \in \mathcal {M}(f|_\Lambda )$ , Fang, Cao, and Zhao [16, Theorem 4.4] proved that

$$ \begin{align*}\dim_{H} \mu=\text{ess~sup} \{\dim_{H} \nu: \nu\in \mathcal{E}(f|_\Lambda)\},\end{align*} $$

with the essential supremum taken with respect to the ergodic decomposition $\tau $ of $\mu $ . See [6, Theorem 2] for the case of hyperbolic surface diffeomorphisms. Their approach extends without change to general conformal hyperbolic diffeomorphisms (see [4, Theorem 13.2.4]). Consequently, to prove Theorem C, it suffices to show that there exists $\mu \in \mathcal {E}(f|_\Lambda )$ so that

$$ \begin{align*}\dim_{H}\mu=\sup\{\operatorname{dim}_{H} \nu:\nu \in \mathcal{E}(f|_\Lambda)\}.\end{align*} $$

Next, one can show the desired result by following mutatis mutandis Barreira and Wolf’s proof [5] (see also [4, Ch. 5]). We outline some key steps for the reader’s convenience.

Consider the following bivariate function:

$$ \begin{align*}Q:\mathbb{R}^{2}\rightarrow\mathbb{R},\quad Q(p,q)=P(p \phi_{u}+q \phi_{s}).\end{align*} $$

Since $\phi _{u}$ , $\phi _{s}\in C^{\alpha }(\Lambda )$ , $p \phi _{u}+q \phi _{s}$ has a unique equilibrium measure $\nu _{p,q} \in \mathcal {E}(f|_\Lambda )$ for each $(p, q) \in \mathbb {R}^{2}$ . Let

$$ \begin{align*}\unicode{x3bb}_{u}(p,q)=\unicode{x3bb}_{u}(\nu_{p, q}), \quad\unicode{x3bb}_{s}(p,q)=\unicode{x3bb}_{s}(\nu_{p,q}),\quad h(p,q)=h_{\nu_{p, q}}(f)\end{align*} $$

and $Q(p, q)=h(p, q)-p \unicode{x3bb} _{u}(p, q)+q \unicode{x3bb} _{s}(p, q)$ . By properties (1)–(4) of the topological pressure, one can show $\unicode{x3bb} _{u},\unicode{x3bb} _{s}$ and h as functions in $\mathbb {R}^{2}$ are real-analytic. Furthermore, let

$$ \begin{align*}d_{u}(p,q)=\frac{h(p,q)}{\unicode{x3bb}_{u}(p,q)},\quad d_{s}(p,q)=-\frac{h(p,q)}{\unicode{x3bb}_{s}(p,q)}.\end{align*} $$

Then $d_s$ and $d_u$ are also real-analytic.

Since the maps $\nu \mapsto \unicode{x3bb} _{u} (\nu )$ and $\nu \mapsto \unicode{x3bb} _{s} (\nu )$ are continuous on the compact space $\mathcal {M}(f|_\Lambda )$ , put

(3.11) $$ \begin{align} \unicode{x3bb}_{u}^{\min }=\min_{\mu\in\mathcal{M}(f|_\Lambda)} \unicode{x3bb}_{u}(\mu),\quad\unicode{x3bb}_{u}^{\max }=\max_{\mu\in\mathcal{M}(f|_\Lambda)} \unicode{x3bb}_{u}(\mu),\nonumber\\ \unicode{x3bb}_{s}^{\min }=\min_{\mu\in\mathcal{M}(f|_\Lambda)} \unicode{x3bb}_{s}(\mu),\quad\unicode{x3bb}_{s}^{\max }=\max_{\mu\in\mathcal{M}(f|_\Lambda)} \unicode{x3bb}_{s}(\mu). \end{align} $$

Set

$$ \begin{align*}I_{u}=(\unicode{x3bb}_{u}^{\min },~\unicode{x3bb}_{u}^{\max }),\quad I_{s}=(\unicode{x3bb}_{s}^{\min },~\unicode{x3bb}_{s}^{\max }).\end{align*} $$

Note that $I_{u}\neq \varnothing $ (respectively $I_{s}\neq \varnothing $ ) if and only if $\phi _{u}$ (respectively $\phi _{s}$ ) is not cohomologous to a constant.

Let $\{\nu _{n}\}_{n\geqslant 1}$ be a sequence of measures in $\mathcal {E}(f|_\Lambda )$ such that

$$ \begin{align*}\lim _{n \rightarrow \infty} \dim_{H} \nu_{n}=\sup \{\dim_{H} \nu: \nu \in \mathcal{E}(f|_\Lambda)\}.\end{align*} $$

Without loss of generality, assume that $\{\nu _{n}\}_{n\geqslant 1}$ converges to some measure $m \in \mathcal {M}(f|_\Lambda )$ . By equation (3.10) and the upper semi-continuity of the entropy map $\nu \mapsto h_{\nu }(f)$ , one has

$$ \begin{align*} \limsup_{n\to\infty}\dim_{H} \nu_{n} =\limsup_{n\to\infty} d(\nu_n)\le d(m). \end{align*} $$

To prove the desired result, it suffices to show that there exists $\mu \in \mathcal {E}(f|_\Lambda )$ such that

(3.12) $$ \begin{align} \dim_{H}\mu=d(m). \end{align} $$

We also note that when m is ergodic, it follows from equation (3.10) that $\dim _{H}m=d(m)$ , this completes the proof. However, m may be non-ergodic.

As in [4, Lemmas 5.24, 5.25, and 5.26], one can show the following properties:

1. (i) if $\unicode{x3bb} _{s}(m) \in I_{s}$ , then there exists ${p \in [0, {h_{m}(f)}/{\unicode{x3bb} _{u}(m)}]}$ such that $\unicode{x3bb} _{u}(p, \gamma _{s}(p))=\unicode{x3bb} _{u}(m);$

2. (ii) assume that neither $\phi _{u}$ nor $\phi _{s}$ are cohomologous to a constant, then $\unicode{x3bb} _u(m)\in I_u \text { if and only if }\unicode{x3bb} _s(m)\in I_s$ ;

3. (iii) if $\unicode{x3bb} _{u}(p, q)=\unicode{x3bb} _{u}(m)\text { and }\unicode{x3bb} _{s}(p, q)=\unicode{x3bb} _{s}(m)$ for some $p, q\in \mathbb {R}$ , then $m=\nu _{p,q}$ .

Item (ii) implies that it is sufficient to consider the following four cases:

1. (I) $\unicode{x3bb} _u(m)\in I_u \text { and }\unicode{x3bb} _s(m)\in I_s;$

2. (II) $\unicode{x3bb} _s(m)\in I_s$ and $\phi _{u}$ is cohomologous to a constant;

3. (III) $\unicode{x3bb} _u(m)\in I_u$ and $\phi _{s}$ is cohomologous to a constant;

4. (IV) $\unicode{x3bb} _u(m)\notin I_u $ and $\unicode{x3bb} _s(m)\notin I_s.$

For case (I), as in [5, Lemma 4], one can prove that there exists $(p,q)\in \mathbb {R}^2$ so that $m=\nu _{p,q}$ . For cases (II) and (III), following the proof of [5, Lemmas 5 and 6], one can also show that $m=\nu _{p,q}$ for some $(p,q)\in \mathbb {R}^2$ . For the last case, one can show that: (1) $\unicode{x3bb} _{u}(m)=\unicode{x3bb} _{u}^{\min } \text { and } \unicode{x3bb} _{s}(m)=\unicode{x3bb} _{s}^{\max }$ ; (2) there exists $\nu \in \mathcal {E}(f|_\Lambda )$ such that

$$ \begin{align*}\unicode{x3bb}_{u}(\nu)=\unicode{x3bb}_{u}(m),~~ \unicode{x3bb}_{s}(\nu)=\unicode{x3bb}_{s}(m) \quad\text{and}\quad h_{\nu}(f)=h_{m}(f).\end{align*} $$

This completes the proof of equation (3.12).