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 $(f,\Phi )$ .

Cao, Feng, and Huang [Reference Cao, Feng and Huang9] 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 [Reference Walters27] 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 [Reference Pesin21] 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 [Reference Cao, Pesin and Zhao11]. Following the approach in [Reference Cao, Pesin and Zhao11], 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 [Reference Cao, Pesin and Zhao11, 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 [Reference Wang, Wang, Cao and Zhao29]. By modifying the methods in [Reference Chung13], 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 [Reference Barreira and Wolf5], 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 [Reference Oseledec20], 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 [Reference Barreira4, 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

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

3.1 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 [Reference Cao, Pesin and Zhao11, 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 [Reference Cao, Wang and Zhao12, 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 [Reference Cao, Pesin and Zhao11, 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 [Reference Morris19, 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 [Reference Ban, Cao and Hu2, 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 [Reference Wang, Wang, Cao and Zhao29, 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 [Reference Bowen7]). 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 [Reference Aoki1]), 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 [Reference Barreira3, 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 [Reference Ruelle23] 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 [Reference Wang and Cao28] for the detailed proofs, which can be viewed as an extension of Young’s results in [Reference Young31] to the case of an average conformal hyperbolic setting. If $\mu \in \mathcal {M}(f|_\Lambda )$ , Fang, Cao, and Zhao [Reference Fang, Cao and Zhao16, 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 [Reference Barreira and Wolf6, Theorem 2] for the case of hyperbolic surface diffeomorphisms. Their approach extends without change to general conformal hyperbolic diffeomorphisms (see [Reference Barreira4, 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 [Reference Barreira and Wolf5] (see also [Reference Barreira4, 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 [Reference Barreira4, 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 [Reference Barreira and Wolf5, 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 [Reference Barreira and Wolf5, 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).