2.1 Hyperbolic set

Let f be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional smooth compact Riemannian manifold M. We say an f-invariant compact subset $\Lambda \subset M$ is a hyperbolic set if for any $x\in \Lambda $ , the tangent space admits a decomposition ${T_{x}M=E^{s}(x)\oplus E^{u}(x)}$ such that the following properties hold:

1. (1) the splitting is $Df$ -invariant, that is, for every $x\in \Lambda $ , $D_{x}f E^{\sigma }(x)=E^{\sigma }(f(x))$ for $\sigma =s,u$ ;

2. (2) the stable subspace $E^{s}(x)$ is uniformly contracting and the unstable subspace $E^{u}(x)$ is uniformly expanding in the sense that there are constants $C\geq 1$ and $0<\chi <1$ such that for every $n\geq 0$ and $v^{\sigma }\in E^{\sigma }(x)$ ( $\sigma =s$ or u), we have

   $$ \begin{align*}\|D_{x}f^{n}v^{s}\|\leq C\chi^{n}\|v^{s}\|\quad \text{and} \quad \|D_{x}f^{-n}v^{u}\|\leq C\chi^{n}\|v^{u}\|.\end{align*} $$

Recall that a hyperbolic set $\Lambda $ is locally maximal if there exists an open neighbourhood U of $\Lambda $ such that $\Lambda =\bigcap _{n\in \mathbb {Z}}f^{n}(U)$ , and a diffeomorphism f is called topologically transitive on $\Lambda $ if for every two non-empty (relative) open subsets $U,V\subset \Lambda $ , there exists $n>0$ such that $f^n(U)\cap V\neq \emptyset $ . Given a point $x\in \Lambda $ , for each small $\beta>0$ , the local stable and unstable manifolds at the point x are defined as follows:

$$ \begin{align*} W^{s}_{\mathrm{loc}}(x, f) = \{y\in M:d(f^{n}(x),f^{n}(y))\leq \beta \text{ for all } n\geq 0\}, \end{align*} $$

and

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

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

$$ \begin{align*} W^{s}(x, f) = \bigcup_{n\geq 0}f^{-n}(W^{s}_{\mathrm{loc}}(f^{n}(x), f)), \quad W^{u}(x, f) = \bigcup_{n\geq 0}f^{n}(W^{u}_{\mathrm{loc}}(f^{-n}(x), f)). \end{align*} $$

Let $d^s/d^u$ be the metric induced by the Riemannian structure on the stable/unstable manifold $W^s/W^u$ .

2.2 Dimension

Let X be a compact Riemannian manifold with a Riemannian metric. Given a subset Z of X, for $s\geq 0$ and $\delta>0$ , define

$$ \begin{align*} \mathcal{H}_{\delta}^{s}(Z) := \inf \bigg\{\!\sum_{i}|U_i|^s: \ Z\subset \bigcup_{i}U_i,~|U_i|\leq \delta \text{ for all } i\bigg\}, \end{align*} $$

where $|\cdot |$ denotes the diameter of a subset. The quantity

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

is called the s-dimensional Hausdorff measure of Z. It is easy to show that there is a jump-up value

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

which is called the Hausdorff dimension of Z.

Given a Borel probability measure $\mu $ on X, the Hausdorff dimension of the measure  $\mu $ is defined as

$$ \begin{align*}\dim_{H}\mu=\inf\{\dim_{H}Y:Y\subset X ,~\mu(Y)=1\}.\end{align*} $$

The lower and upper pointwise dimension of $\mu $ at point $x\in X$ are defined respectively by

$$ \begin{align*} \underline{d}_\mu(x)=\liminf_{r\to 0}\frac{\log\mu(B(x,r))}{\log r}\quad \text{and}\quad \overline{d}_\mu(x)=\limsup_{r\to 0}\frac{\log\mu(B(x,r))}{\log r}, \end{align*} $$

where $B(x,r)$ denotes the ball of radius r centred at x. If $\underline {d}_\mu (x)=\overline {d}_\mu (x)$ , then we denote the common value by $d_\mu (x)$ . In particular, Barreira and Wolf [Reference Barreira and Wolf5] proved that

(5) $$ \begin{align} \dim_H\mu=\text{ess sup}\{\underline{d}_\mu(x): x\in X\}, \end{align} $$

where the essential supremum is taken with respect to $\mu $ . The following well-known result gives the relation between the Hausdorff dimension and the lower pointwise dimension.

Let $f:X\rightarrow X$ be a $C^{1+\alpha }$ diffeomorphism on an $m_{0}$ -dimensional compact Riemannian manifold X, and let $\mu $ be a hyperbolic ergodic measure on X. Let $\Gamma $ be the set of points which are regular in the sense of Oseledets [Reference Oseledets26]. A measurable partition $\xi ^u$ / $\xi ^s$ of X is said to be subordinate to the unstable/stable manifold if for $\mu $ -almost every x, $\xi ^u(x)\subset W^u(x,f)$ / $\xi ^s(x)\subset W^s(x, f)$ and contains an open neighbourhood of x in $W^u(x,f)$ / $W^s(x, f)$ . Let $\{\mu _x^u\}$ and $\{\mu _x^s\}$ be the collections of conditional measures associated with $\xi ^u$ and $\xi ^s$ , respectively. For every $x\in \Gamma $ , Ledrappier and Young [Reference Ledrappier and Young22] proved the existence of the following limits:

(6) $$ \begin{align} d_\mu^u(x):=\lim_{r\to0}\frac{\log \mu_x^u(B^u(x,r))}{\log r} \quad \text{and} \quad d_\mu^s(x):=\lim_{r\to0}\frac{\log \mu_x^s(B^s(x,r))}{\log r}, \end{align} $$

which are called the stable and unstable dimension of the measure $\mu $ , respectively. Here $B^\sigma (x,r):=\{y\in W^\sigma (x,f): d^\sigma (x,y)<r\}$ with $\sigma \in \{u,s\}$ . Since we consider the limit $r\to 0$ in equation (6), the definition of $d_\mu ^u(x)$ will remain unchanged if we consider the global metric d in the dynamical ball $B^\sigma (x,r)$ instead.

2.3 Pressure

Let $(M,f)$ be a topological dynamical system (TDS for short), that is, $f:M\rightarrow M$ is a continuous map on a compact metric space M equipped with the metric d. Denote by $\mathcal {M}_{\mathrm {inv}}(f|_M)$ and $\mathcal {M}_\mathrm{erg}(f|_M)$ the set of all f-invariant and ergodic Borel probability measures on M, respectively. Given $n\in \mathbb {N}$ and $x,y\in M$ , let

$$ \begin{align*} d_{n}(x,y)=\max\{d(f^{k}(x),f^{k}(y)):0\leq k<n\}. \end{align*} $$

Given $\varepsilon>0$ , denote by $B_{n}(x,\varepsilon )=\{y:d_{n}(x,y)<\varepsilon \}$ the Bowen’s ball of radius $\varepsilon $ centred at x of length n. A subset $E\subset M$ is called $(n,\varepsilon )$ -separated if $d_n(x,y)>\varepsilon $ for any two distinct points $x,y\in E$ . A sequence of continuous functions $\Psi =\{\psi _n\}_{n\geq 1}$ on M is called sub-additive if

$$ \begin{align*} \psi_{m+n}\leq\psi_n+\psi_m\circ f^n\quad \text{for all } m,n\geq1. \end{align*} $$

Similarly, one calls a sequence of continuous functions $\Phi =\{\phi _n\}_{n\geq 1}$ on M super-additive if $-\Phi =\{-\phi _n\}_{n\geq 1}$ is sub-additive.

Let $\Psi =\{\psi _n\}_{n\geq 1}$ be a sub-additive sequence of continuous potentials on M, set

$$ \begin{align*} P_n(f,\Psi, \varepsilon)=\sup\bigg\{\!\sum_{x\in E}e^{\psi_n(x)}: E \text{ is an } (n,\varepsilon) \text{-separated subset of } M\bigg\}. \end{align*} $$

The quantity

$$ \begin{align*} P(f,\Psi):=\lim_{\varepsilon\to0}\limsup_{n\to\infty}\frac1n\log P_n(f,\Psi,\varepsilon) \end{align*} $$

is called the sub-additive topological pressure of $\Psi $ .

The sub-additive topological pressure satisfies the following variational principle, see [Reference Cao, Feng and Huang6] for more details.

Theorem 2.1. Let $\Psi =\{\psi _n\}_{n\geq 1}$ be a sub-additive sequence of continuous potentials on M. Then

$$ \begin{align*} P(f,\Psi)=\sup\{h_{\mu}(f)+\mathcal{F}_*(\Psi,\mu)| ~\mu\in\mathcal{M}_{\mathrm{inv}}(f|_M), \mathcal{F}_*(\Psi,\mu)\neq-\infty\}, \end{align*} $$

where $h_\mu (f)$ is the measure theoretic entropy of f with respect to the measure $\mu $ and ${\mathcal {F}_*(\Psi ,\mu )=\lim _{n\to \infty }(1/n)\int \psi _n\,d\mu }.$

Next we recall the super-additive topological pressure introduced in [Reference Cao, Pesin and Zhao8] by the variational relation for topological pressure, although it is unknown whether the variational principle holds for super-additive topological pressure defined via separated sets. Given a sequence of super-additive continuous potentials $\Phi =\{\phi _{n}\}_{n\geq 1}$ on M, the super-additive topological pressure of $\Phi $ is defined as

$$ \begin{align*} P(f,\Phi):= \sup \{h_{\mu}(f)+\mathcal{F}_{\ast}(\Phi,\mu):\mu\in \mathcal{M}_{\mathrm{inv}}(f|_M)\}, \end{align*} $$

where

$$ \begin{align*}\mathcal{F}_{\ast}(\Phi,\mu)=\lim_{n\rightarrow \infty}\frac{1}{n}\int \phi_{n}\,d\mu=\sup_{n\in \mathbb{N}} \frac{1}{n}\int \phi_{n}\,d\mu.\end{align*} $$

The second equality is due to the standard sub-additive argument. The following result gives the relation between the sub-additive (super-additive) topological pressure and the topological pressure for additive potentials.

The first statement is proved in [Reference Ban, Cao and Hu3], where the sub-additive topological pressure is defined via separated sets, so one requires that the entropy map be upper semi-continuous. The second statement is proved in [Reference Cao, Pesin and Zhao8], and one does not need any additional condition since the super-additive topological pressure is defined via the variational relations.

Following the approach described in [Reference Pesin29], we recall the topological pressure on an arbitrary subset of unstable manifolds. Let $f: M\to M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional smooth compact Riemannian manifold M and let $\Lambda \subset M$ be a hyperbolic set. Let $\Psi =\{\psi _n\}_{n\geq 1}$ be a sub-additive sequence of continuous functions on $\Lambda $ . For every $x\in \Lambda $ , denote $Z=\Lambda \cap W^u_{\mathrm {loc}}(x,f)$ . Given $s\in \mathbb {R}$ , set

(7) $$ \begin{align} m(Z,\Psi,s,\delta):=\lim_{N\to\infty} \inf \bigg\{\!\sum_i\exp\Big(-sn_i + \sup_{y\in B_{n_i}^u(x_i,\delta)}\psi_{n_i}(y)\Big)\bigg\}, \end{align} $$

where the infimum is taken over all collections $\{B^u_{n_i}(x_i,\delta )\}$ with $x_i\in \Lambda $ , $n_i\geq N$ that cover Z, and

$$ \begin{align*}B_{n_i}^u(x_i,\delta):=\{y\in W^u(x,f): d^u(f^j(x_i), f^j(y))<\delta\ \text{for } j=0,1,\ldots,n_i-1\}.\end{align*} $$

It is easy to show that there is a jump-up value

$$ \begin{align*}P_Z(f,\Psi,\delta): = \inf\{s: m(Z,\Psi, s,\delta)=0\} = \sup\{s: m(Z,\Psi,s,\delta)=+\infty\}. \end{align*} $$

The quantity

$$ \begin{align*} P_Z(f,\Psi):=\lim_{\delta\to0} P_Z(f,\Psi,\delta) \end{align*} $$

is called the topological pressure of $\Psi $ on the subset Z. It is not difficult to show that $P_\Lambda (f, \Psi )=P(f|_\Lambda , \Psi )$ (see [Reference Cao, Feng and Huang6, Proposition 4.4]).

Let $\mu $ be an f-invariant Borel probability measure on M. Given a sub-additive potential $\Phi =\{\phi _{n}\}_{n \geq 1}$ on M, for $0<\delta < 1, n \geq 1$ and $\varepsilon>0$ , a subset $F\subset M$ is called an $(n,\varepsilon ,\delta )$ -spanning set if the union $\bigcup _{x\in F}B_n(x,\varepsilon )$ has $\mu $ -measure more than or equal to $1-\delta $ . Put

$$ \begin{align*} P_{\mu}(f, \Phi, n, \varepsilon, \delta):=\inf \bigg\{\!\sum_{x \in F} \exp \Big(\sup _{y \in B_{n}(x, \varepsilon)} \phi_{n}(y)\Big) : F \text{ is an } (n, \varepsilon, \delta)\text{-spanning set}\bigg\} \end{align*} $$

and let further that

$$ \begin{align*} P_{\mu}(f, \Phi, \varepsilon, \delta)&:=\limsup _{n \rightarrow \infty} \frac{1}{n} \log P_{\mu}(f, \Phi, n, \varepsilon, \delta), \\ P_{\mu}(f, \Phi, \delta)&:=\liminf_{\varepsilon \rightarrow 0} P_{\mu}(f, \Phi, \varepsilon, \delta), \\ P_{\mu}(f, \Phi)&:=\lim_{\delta \rightarrow 0} P_{\mu}(f, \Phi, \delta), \end{align*} $$

and we call $P_{\mu }(f, \Phi )$ the sub-additive measure-theoretic pressure of $(f,\Phi )$ with respect to $\mu $ . If one considers a super-additive potential $\Phi =\{\phi _n\}_{n\ge 1}$ on M, replacing $\sup _{y \in B_{n}(x, \varepsilon )} \phi _{n}(y)$ by $\phi _{n}(x)$ in $P_{\mu }(f, \Phi , n, \varepsilon , \delta )$ , then the corresponding quantity $P_{\mu }(f, \Phi )$ is called the super-additive measure theoretic pressure of $(f,\Phi )$ with respect to  $\mu $ .

In the following, we recall some properties of sub-additive/super-additive measure- theoretic pressure which are proved in [Reference Cao, Hu and Zhao7].

Theorem 2.2. [Reference Cao, Hu and Zhao7, Theorem A]

Let $(M,f)$ be a TDS and $\Phi =\{\phi _n\}_{n\geq 1}$ a sub-additive potential on M. For every $\mu \in \mathcal {M}_\mathrm{erg}(f|_M)$ with $\mathcal {F}_*(\Phi , \mu )\neq -\infty $ , we have that

$$ \begin{align*} P_\mu(f,\Phi)=h_\mu(f)+\mathcal{F}_*(\Phi, \mu). \end{align*} $$

Theorem 2.3. [Reference Cao, Hu and Zhao7, Proposition 3.2]

Let $(M,f)$ be a TDS and $\Phi =\{\phi _n\}_{n\geq 1}$ a super-additive potential on M. For every $\mu \in \mathcal {M}_\mathrm{erg}(f|_M)$ , we have that

$$ \begin{align*} P_\mu(f,\Phi)=h_\mu(f)+\mathcal{F}_*(\Phi, \mu). \end{align*} $$

Remark 2.3. In Theorem 2.2, to avoid the indeterminate form $\infty -\infty $ , the condition $\mathcal {F}_*(\Phi , \mu )\neq -\infty $ is necessary. However, we do not need this condition in Theorem 2.3. If $\Phi =\{\phi _n\}_{n\geq 1}$ is an additive potential on M, that is, $\phi _n(x)=S_n\phi (x)$ for some continuous function $\phi $ , then we have

$$ \begin{align*} P_\mu(f,\phi)=h_\mu(f)+\int\phi\,d\mu\quad\text{for all } \mu\in \mathcal{M}_\mathrm{erg}(f|_M). \end{align*} $$

The above formula is also proven in [Reference He, Lv and Zhou16].

2.4 Singular dimension

Let $f: M\to M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and $\Lambda \subset M$ a hyperbolic set. Consider the sub-additive singular valued potential $\Psi _f(t)=\{-\psi ^t(\cdot , f^n)\}_{n\geq 1}$ given by equation (4). Fix $x\in \Lambda $ and let $Z=\Lambda \cap W_{\mathrm {loc}}^u(x,f)$ . Following the approach described in [Reference Cao, Pesin and Zhao8], we introduce the Carathéodory singular dimension of Z. Put

(8) $$ \begin{align} m(Z, \Psi_f(t),\delta):=\lim_{N\to\infty}\inf\bigg\{\!\sum_i\exp \Big[\sup_{y\in B_{n_i}^u(x_i,\delta)} -\psi^t(y, f^{n_i})\Big]\bigg\}, \end{align} $$

where the infimum is taken over all collections $\{B_{n_i}^u(x_i,\delta )\}$ with $x_i\in \Lambda $ , $n_i\geq N$ that cover Z. It is easy to see that there is a jump-up value

(9) $$ \begin{align} \nonumber \dim_{C,\delta}^{\Psi_f} Z := &\inf\{t: m(Z, \Psi_f(t), \delta)=0 \} \\ = &\sup\{t: m(Z, \Psi_f(t), \delta)=+\infty \}. \end{align} $$

The quantity

(10) $$ \begin{align} \dim_C^{\Psi_f} Z := \lim_{\delta\to0} \dim_{C,\delta}^{\Psi_f} Z \end{align} $$

is called the Carathéodory singular dimension of Z with respect to the sub-additive singular valued potential $\Psi _f$ .

Consider the super-additive singular valued potential $\Phi _f(t)=\{-\phi ^t(\cdot , f^n)\}_{n\geq 1}$ given by equation (3), replacing $ {\sup _{y\in B_{n_i}^u(x_i,\delta )} -\psi ^t(y, f^{n_i})}$ by $-\phi ^t(x_i, f^{n_i})$ in equation (8), one can define $m(Z, \Phi _f(t),\delta )$ and $\dim _{C,\delta }^{\Phi _f} Z$ in a similar fashion as equations (8) and (9). The corresponding quantity $\dim _C^{\Phi _f} Z$ as in equation (10) is called the Carathéodory singular dimension of Z with respect to the super-additive singular valued potential $\Phi _f$ .

2.5 Approximation of hyperbolic measures by hyperbolic sets with dominated splitting

First we recall the definition of the dominated splitting. Let $f:M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_{0}$ -dimensional compact smooth Riemannian manifold M. Suppose $\Lambda \subset M$ is a compact f-invariant set. We say $\Lambda $ admits a dominated splitting if there is a continuous invariant splitting $T_{\Lambda }M=E\oplus F$ and constants $C>0,\unicode{x3bb} \in (0,1)$ such that for each $x\in \Lambda $ , $n\in \mathbb {N}$ , $0\neq u\in E(x)$ and $0\neq v\in F(x)$ , it holds that

$$ \begin{align*} \frac{\|D_{x}f^{n}(u)\|}{\|u\|}\leq C\unicode{x3bb}^{n}\frac{\|D_{x}f^{n}(v)\|}{\|v\|}. \end{align*} $$

We say F dominates E and write it as $E\preceq F$ . Furthermore, given $0<\ell \leq m_{0}$ , we say a continuous invariant splitting $T_{\Lambda }M=E_{1}\oplus \cdots \oplus E_{\ell }$ dominates if there are numbers $\chi _{1}<\chi _{2}<\cdots <\chi _{\ell }$ , constants $C>0$ and $0<\varepsilon <\min _{1\leq i\leq \ell -1}\{({\chi _{i+1}-\chi _{i}})/{100}\}$ such that for every $x\in \Lambda $ , $n\in \mathbb {N}$ and $1\leq j\leq \ell $ and each unit vector $u\in E_{j}(x)$ , it holds that

$$ \begin{align*} C^{-1}\exp[n(\chi_{j}-\varepsilon)]\leq \|D_{x}f^{n}(u)\|\leq C \exp[n(\chi_{j}+\varepsilon)]. \end{align*} $$

In particular, it is clear that $E_{1}\preceq \cdots \preceq E_{\ell }$ . We shall use the notion $\{\chi _{j}\}$ -dominated when we want to stress the dependence on the numbers $\{\chi _{j}\}$ .

Refining Katok’s approximation theory in non-uniformly hyperbolic dynamical systems [Reference Katok18], Avila, Crovisier and Wilkinson [Reference Avila, Crovisier and Wilkinson2] obtained the following approximation result.

In the second statement, the original result does not show that $h_{\text {top}}(f|_{\Lambda _{\varepsilon }})\le h_{\mu }(f)+\varepsilon $ . However, only a slight modification can give the upper bound of the topological entropy of f on the horseshoe.

3.1 Proof of Theorem A

(i) Since $\mu $ is a hyperbolic ergodic SRB measure for a $C^{1+\alpha }$ diffeomorphism f and has a one-dimensional stable manifold, by [Reference Wang, Qu and Cao34, Lemma 15 and 25], one has

for $\mu $ -a.e. x. Barreira, Pesin and Schmeling [Reference Barreira, Pesin and Schmeling4] proved that $d_\mu (x)=d_\mu ^u(x)+d_\mu ^s(x)$ for $\mu $ -a.e. x. As a consequence, one has that

for $\mu $ -a.e. x. Hence, one has

If $\unicode{x3bb} _1(\mu )+\unicode{x3bb} _2(\mu )+\cdots +\unicode{x3bb} _{m_0}(\mu )<0$ , then one can show that

since $\mu $ is an SRB measure and has a one-dimensional stable manifold. Therefore, we have that

$$ \begin{align*}\dim_H\mu=\dim_L\mu.\end{align*} $$

If $\unicode{x3bb} _1(\mu )+\unicode{x3bb} _2(\mu )+\cdots +\unicode{x3bb} _{m_0}(\mu )\geq 0$ , it follows from the definition of Lyapunov dimension that $\dim _L\mu =m_0$ . Since $\mu $ is an SRB measure for f and has a one-dimensional stable manifold, one has that

This together with the fact that

implies that $({h_\mu (f)}/{-\unicode{x3bb} _{m_0}(\mu )})=1$ . This yields that $\dim _H\mu =\dim _L\mu $ .

By [Reference Wang, Qu and Cao34, Theorem B], there exists a sequence of hyperbolic sets $\Lambda _{n}$ such that

$$ \begin{align*} \dim_H \Lambda_{n}\rightarrow \dim_{L}\mu~( n\to\infty). \end{align*} $$

(ii) Since $\mu $ is a hyperbolic ergodic SRB measure for a $C^{1+\alpha }$ diffeomorphism f, by [Reference Wang, Qu and Cao34, Lemma 15], one has that

$$ \begin{align*} d_\mu^u(x)=u\quad \mu\text{-a.e. }x. \end{align*} $$

Considering $f^{-1}$ instead of f, since $h_\mu (f)=-\unicode{x3bb} _{u+1}(\mu )-\unicode{x3bb} _{u+2}(\mu )-\cdots -\unicode{x3bb} _{m_0}(\mu )$ , by [Reference Ledrappier and Young21, Theorem A], we have that the measure $\mu $ has absolutely continuous conditional measures on stable manifolds of f. Using the same arguments as the proof of [Reference Wang, Qu and Cao34, Lemma 15], we have that

$$ \begin{align*}d_\mu^s(x)=s\quad \mu\text{-a.e. }x.\end{align*} $$

Hence, $d_\mu (x)=u+s=m_0$ for $\mu $ -a.e. x, which implies that $\dim _H\mu =m_0$ . Since $\mu $ is an SRB measure for f and $h_\mu (f)=-\unicode{x3bb} _{u+1}(\mu )-\unicode{x3bb} _{u+2}(\mu )-\cdots -\unicode{x3bb} _{m_0}(\mu )$ , one can conclude that

then $\dim _L\mu =m_0$ by the definition of Lyapunov dimension. This proves that

$$ \begin{align*}\dim_L\mu=\dim_H\mu.\end{align*} $$

Finally, for each $\varepsilon>0$ , there exists a hyperbolic set $\Lambda _\varepsilon $ satisfying properties (a)–(d) in Theorem 2.4. Fix a positive integer $n\ge 1$ . Let $t_n$ be the unique root of Bowen’s equation $P(f^{2^n}|\Lambda _\varepsilon ,-\psi ^t(\cdot , f^{2^n}))=0$ and let $\mu _n^u$ be the unique equilibrium state for the topological pressure $P(f^{2^n}|\Lambda _\varepsilon ,-\psi ^{t_n}(\cdot , f^{2^n}))$ . Similarly, let $t_n^{\prime }$ be the root of Bowen’s equation $P(f^{2^n}|\Lambda _\varepsilon , \phi ^t(\cdot , f^{2^n}))=0$ and let $\mu _n^s$ be the unique equilibrium state for the topological pressure $P(f^{2^n}|\Lambda _\varepsilon , \phi ^{t_n}(\cdot , f^{2^n}))$ . As in the proof of [Reference Wang, Qu and Cao34, Theorem B], the following properties hold:

1. (e) ${\lim _{\varepsilon \to 0}\lim _{n\to \infty }t_n=u}$ and ${\lim _{\varepsilon \to 0}\lim _{n\to \infty }t_n'=s}$ ;

2. (f) there is a Markov partition $\mathcal {P}=\{P_1,P_2,\ldots ,P_\ell \}$ of $\Lambda _\varepsilon $ . For every $i\in \{1,2,\ldots ,\ell \}$ , there is a family of conditional measures $\{\mu _{n,x}^u\}_{x\in P_i}$ ( $\{\mu _{n,x}^s\}_{x\in P_i}$ ) of $\mu _n^u$ ( $\mu _n^s$ ) on the local unstable (stable) sets $W^u_{P_i}$ ( $W^s_{P_i}$ ) such that for every $x\in P_i$ , there is small $r_0>0$ such that for every $r\in (0, r_0)$ ,

   $$ \begin{align*}r^{u+\varepsilon} \leq \mu_{n,x}^u(B^u(x,r)) \leq r^{t_n-\varepsilon}\end{align*} $$
   and
   $$ \begin{align*}r^{s+\varepsilon} \leq \mu_{n,x}^s(B^s(x,r)) \leq r^{t_n'-\varepsilon},\end{align*} $$
   where $W_{P_i}^u(x,f):=W_{loc}^u(x,f)\cap P_i$ and $W_{P_i}^s(x,f):=W_{loc}^s(x,f)\cap P_i$ for every $x\in P_i$ .

Define a measure $\hat {\mu }_n$ on $P_i$ as follows:

$$ \begin{align*} \hat{\mu}_n(B(x,r)) = \mu_{n,x}^u(B^u(x,r)) \cdot \mu_{n,x}^s(B^s(x,r)) \end{align*} $$

for every $x\in P_i$ and each sufficiently small $r>0$ . This yields that

$$ \begin{align*} t_n+t_n'-2\varepsilon \leq \underline{d}_{\hat{\mu}_n}(x) \leq \overline{d}_{\hat{\mu}_n}(x) \leq m_0+2\varepsilon \end{align*} $$

for every $x\in P_i$ . By Proposition 2.1 and the fact that $\Lambda _\varepsilon =\bigcup _{i=1}^\ell P_i$ , we have that

$$ \begin{align*} \lim_{\varepsilon\to0}\dim_H\Lambda_\varepsilon = m_0 = \dim_L\mu. \end{align*} $$

This completes the proof of Theorem A.

3.2 Proof of Theorem B

Let $\Gamma $ be the set of points which are regular in the sense of Oseledets [Reference Oseledets26] with respect to the measure $\mu $ . For every $x\in \Gamma $ , denote its Lyapunov exponents by

To prove Theorem B, we need a coarse upper bound for the unstable and stable pointwise dimension $d_\mu ^u(x)$ , $d_\mu ^s(x)$ of an ergodic f-invariant hyperbolic probability measure $\mu $ for almost every x. We now provide the following useful lemma, which estimates the Hausdorff measure of the image of a small ball along unstable/stable direction under f.

Lemma 3.1. Fix $t\in [0,u]$ , then for any $b_0>2\sqrt {u}$ and $C_0>2^tu^{{t}/{2}}$ , there is $\rho _0>0$ such that for all $x\in \Gamma $ , if $B^u(x,\rho )\subset B(x,\rho _0)\cap W^u(x,f)$ for some $0<\rho <\rho _0$ , then we have

$$ \begin{align*}\mathcal{H}_{b\rho}^t(B^u(x,\rho)) \leq C \mathcal{H}_\rho^t(f(B^u(x,\rho))),\end{align*} $$

where $b=b_0\exp \{-\log \alpha _{u-[t]}(x,f)\} \text { and } C=C_0\exp \{-\psi ^t(x,f)\}$ .

Proof. For simplicity, we just prove the lemma on the assumption that M is the Euclid space $\mathbb {R}^{m_0}$ . For the general case, one can use local charts to prove it.

Given a small positive number $\varepsilon $ with ${e^\varepsilon }/({1-\varepsilon })<2$ , since $f: M\to M$ is a $C^{1+\alpha }$ diffeomorphism on M, there exists $\rho _0>0$ such that for every $y,z\in B(x,\rho _0)\cap W^u(x,f)$ , the following properties hold:

1. (a) $\|y-z- (D_yf)^{-1}(f(y)-f(z)) \| \leq \varepsilon \|y-z\|$ ;

2. (b) $|\log \alpha _i(y,f) - \log \alpha _i(z,f)|\leq \varepsilon $ for $i=1,2,\ldots ,u$ .

See [Reference Jordan and Pollicott17, Lemma 4] for the detailed proof of the above properties. Fix $0<\rho <\rho _0$ . Let $A:= B^u(x,\rho )$ and $a=\mathcal {H}^t_\rho (f(A))$ . Assume that a is finite, otherwise the conclusion is clear. For every $\eta>0$ , there are points $\{z_j\}\subset f(B(x,\rho _0)\cap W^u(x,f))$ such that

$$ \begin{align*} f(A) \subset \bigcup_{j}B^u(z_j, r_j) \end{align*} $$

with $r_j\leq \rho $ for each j and

$$ \begin{align*} \sum_{j}r_j^t<a+\eta. \end{align*} $$

Let $B_j'=\{y\in A: f(y)\in B^u(z_j,r_j)\}$ , then $A\subset \bigcup _j B_j'$ . By property (a), we conclude that $B_j'$ is contained in an ellipse with principal axes

$$ \begin{align*} \frac{1}{1-\varepsilon} r_j \cdot \alpha_1(y_j, f)^{-1}, \frac{1}{1-\varepsilon} r_j \cdot \alpha_2(y_j, f)^{-1}, \ldots, \frac{1}{1-\varepsilon} r_j \cdot \alpha_u(y_j, f)^{-1}, \end{align*} $$

where $y_j\in B^u(x,\rho )$ and $f(y_j)=z_j$ . This together with property (b) yield that $B_j'$ is contained in an ellipse with principal axes

$$ \begin{align*}\frac{e^\varepsilon}{1-\varepsilon} r_j \cdot \alpha_1(x, f)^{-1}, \frac{e^\varepsilon}{1-\varepsilon} r_j \cdot \alpha_2(x, f)^{-1}, \ldots, \frac{e^\varepsilon}{1-\varepsilon} r_j \cdot \alpha_u(x, f)^{-1}. \end{align*} $$

Hence, $B_j'$ is covered by

$$ \begin{align*}\frac{\exp\{-\sum_{j=u-[t]+1}^u \log\alpha_j(x,f) \} }{ \exp\{-[t]\log\alpha_{u-[t]}(x,f) \}}\end{align*} $$

balls with radius $({e^\varepsilon }/({1-\varepsilon })) \sqrt {u} r_j\cdot \exp \{-\log \alpha _{u-[t]}(x,f)\}$ . In fact, the radius

$$ \begin{align*} &\hspace{-6pt}\frac{e^\varepsilon}{1-\varepsilon} \sqrt{u} r_j \cdot \exp\{-\log\alpha_{u-[t]}(x,f)\}\\ &\leq \ 2\sqrt{u}\exp\{-\log\alpha_{u-[t]}(x,f)\} \cdot \rho\\ &\leq \ b\rho. \end{align*} $$

Therefore,

$$ \begin{align*} \mathcal{H}^t_{b\rho}(B_j')&\leq \exp\bigg\{-\sum_{j=u-[t]+1}^u \log\alpha_j(x,f) + [t]\log\alpha_{u-[t]}(x,f) \bigg\}\\ &\quad \cdot \bigg(\frac{e^\varepsilon}{1-\varepsilon} \sqrt{u}\bigg)^t r_j^t \cdot \exp\{-t\log\alpha_{u-[t]}(x,f)\}\\ & \leq (2\sqrt{u})^t \cdot \exp\{-\psi^t(x,f)\} \cdot r_j^t. \end{align*} $$

Summing up over all j, we have that

$$ \begin{align*} \mathcal{H}^t_{b\rho}(A)&\leq \sum_{j}\mathcal{H}_{b\rho}^t(B_j')\\ &\leq 2^t (\sqrt{u})^t \exp\{-\psi^t(x,f)\} \cdot \sum_j r_j^t\\ &\leq 2^t (\sqrt{u})^t \exp\{-\psi^t(x,f)\} \cdot (a+\eta). \end{align*} $$

The choice of $C_0$ and the arbitrariness of $\eta>0$ implies the desired result.

The following result relates the zero of measure-theoretic pressure with the upper bound of the unstable pointwise dimension of $\mu $ .

Proof. Fix a small number $\varepsilon>0$ such that $-\unicode{x3bb} _u(\mu )+2\varepsilon <0$ and choose $t>t_{u,1}^*$ such that

$$ \begin{align*}h_\mu(f)-\int\psi^t(x,f)\,d\mu = -3\varepsilon. \end{align*} $$

Claim. There exists an integer $N_1$ (depending only on $\varepsilon $ ) such that, for $\mu $ -a.e. x and every $N\ge N_1$ , the Birkhoff averages

$$ \begin{align*} \frac{1}{kN}\sum_{j=0}^{k-1}\log \alpha_{u}(f^{jN}x, f^N) \end{align*} $$

converge towards a number bigger than $\unicode{x3bb} _u(\mu )-\varepsilon $ , as k goes to $+\infty $ .

Proof of the Claim

We give the proof of the Claim by modifying slightly the arguments in the proof of [Reference Abdenur, Bonatti and Crovisier1, Lemma 8.4].

Since ${\lim _{n\to \infty } ({1}/{n}) \log \alpha _u(x,f^n) = \lim _{n\to \infty } (1/n) \int \log \alpha _u(x,f^n)\,d\mu = \unicode{x3bb} _u(\mu )}$ for $\mu $ -a.e. x, there exists a positive integer L such that

(11)

The measure $\mu $ may be not ergodic for $f^L$ , one can decompose it as

$$ \begin{align*}\mu=\frac{1}{m}(\mu_1+\mu_2+\cdots+\mu_m),\end{align*} $$

where $m\in \mathbb {N}^+$ divides L and each $\mu _i$ is an ergodic $f^L$ -invariant measure such that $f_*\mu _i=\mu _{i+1}$ for each $i(\text {mod}\,\, m)$ . Let $A_1\cup A_2\cup \cdots \cup A_m$ be a measurable partition of $(M,\mu )$ such that $f(A_i)=A_{i+1}$ for each $i(\text {mod}\,\, m)$ and $\mu _i(A_i)=1$ . By equation (11), there exists $j_0\in \{1,2,\ldots , m\}$ such that

For every $N\ge 1$ and $\mu $ -a.e. x, one decomposes the orbit $\{f^i(x)\}_{i=0}^{N-1}$ as $(x,\ldots, f^{j-1}(x))$ , $(f^j(x),\ldots ,f^{j+(r-1)L-1}(x))$ and $(f^{j+(r-1)L}(x),\ldots , f^{N-1}(x))$ , where $j{\kern-1.2pt}<{\kern-1.2pt}L$ , $j+rL{\kern-1.2pt}\ge{\kern-1.2pt} N$ and the points $\{f^{j+sL}(x)\}_{s=0}^{r}$ belong to $A_{j_0}$ . Using the super-additivity of $\{\log \alpha _u(x,f^n) \}_{n\ge 1}$ , we have that

$$ \begin{align*} \log \alpha_u(x,f^N)&\ge \log \alpha_u(x,f^j) + \sum_{s=0}^{r-2} \log \alpha_u(f^{j+sL}x,f^L)\\ &\quad +\log \alpha_u(f^{j+(r-1)L}x,f^{N-j-L(r-1)}). \end{align*} $$

Hence, one has

$$ \begin{align*} \log \alpha_u(x,f^N)\ge 2C_f + \sum_{s=0}^{r-2} \log \alpha_u(f^{j+sL}x,f^L), \end{align*} $$

where ${C_f=\max _{0\le i<L}\max _{x\in M}|{\log}\, \alpha _u(x,f^i)|}$ with the convention that $|{\log}\, \alpha _u (x,f^0)|=0$ . Since

and

$$ \begin{align*} \lim_{k\to +\infty}\frac{1}{kN}\sum_{j=0}^{k-1}\log \alpha_{u}(f^{jN}x, f^N)\ge \frac{2C_f}{N}+\lim_{k\to +\infty} \frac{1}{kL}\sum_{\ell=0}^{k-1}\log \alpha_u(f^{j+\ell L}x,f^L), \end{align*} $$

and there exists an integer $N_1$ (depending on $\varepsilon $ ) so that $|{2C_f}/{N}|< {\varepsilon }/{2}$ for every $N>N_1$ , for $\mu $ -a.e. x and every $N>N_1$ , we have that

Take $b_0>2\sqrt {u}$ and $C_0>2^tu^{{t}/{2}}$ , choose $N>N_1$ large enough such that

(12)

By the above Claim and Birkhoff ergodic theorem, for $\mu $ -a.e. $x\in M$ , we have that

and

$$ \begin{align*} \lim_{n\to\infty}\frac{1}{nN}\sum_{j=0}^{nN-1} \psi^t(f^jx,f)=\int\psi^t(x,f)\,d\mu. \end{align*} $$

Let $\rho _0$ be as in Lemma 3.1. Fix $\delta \in (0,\rho _0)$ . Ledrappier and Young [Reference Ledrappier and Young22] proved that

$$ \begin{align*} \limsup_{n\to\infty} \frac{-\log\mu_x^u(B^u(x,n,\delta/2))}{n} &\leq \lim_{\delta\to0}\limsup_{n\to\infty} \frac{-\log\mu_x^u(B^u(x,n,{\delta}/{2}))}{n}\\ &=h_\mu(f)\, \mu\text{-a.e.}\, x, \end{align*} $$

where $B^u(x,n, \delta /2):=\{y\in W^u(x,f): d^u(f^jx,f^jy)< \delta /2 \text { for } 0\leq j<n\}$ . Hence, one can find sets $A_n\subset \Gamma $ with $\mu (A_n)\to 1\, (n\to \infty )$ , for every $x\in A_n$ where the following properties hold:

1. (a) ${\exp [-nN(h_\mu (f)+\varepsilon )] \leq \mu _x^u(B^u(x,nN, {\delta }/{2}))}$ ;

2. (b) ${nN(-\int \psi ^t(x,f)-\varepsilon ) \leq -\sum _{j=0}^{nN-1}\psi ^t(f^{j}x,f) \leq nN(-\int \psi ^t(x,f)+\varepsilon )}$ ;

3. (c) ${\sum _{j=0}^{n-1}\log \alpha _{u-[t]}(f^{jN}x, f^N) \geq nN(\unicode{x3bb} _u(\mu )-2\varepsilon )}$ .

Take a point $x\in A_n$ . Let E be a maximal $(nN, \delta )$ -separated subset of $A_n\cap \xi ^u(x)$ , then

$$ \begin{align*} A_n\cap\xi^u(x) \subset \bigcup_{x_j\in E} B^u(x_j, nN, \delta). \end{align*} $$

Furthermore, by property (a), the number of balls $B^u(x_j,nN, \delta /2)$ is less than or equal to $\exp \{nN[h_\mu (f)+\varepsilon ]\}$ . Let

$$ \begin{align*}b_k(x)=(b_0)^k \exp \bigg[-\sum_{j=n-k}^{n-1}\log\alpha_{u-[t]}(f^{jN}x, f^N)\bigg]\end{align*} $$

for $k=1,2,\ldots , n$ and $\beta _n=\{b_0\exp [(-\unicode{x3bb} _u(\mu )+2\varepsilon )N]\}^n \cdot \rho $ , where $0<\rho <\rho _0$ . By property (c), we have

For each $x_j\in E$ , using Lemma 3.1 n times, we conclude that

$$ \begin{align*} \mathcal{H}^t_{\beta_n}(B^u(x_j, nN, \delta)) &\leq \mathcal{H}^t_{b_n(x_j)\rho}(B^u(x_j,nN,\delta))\\ &\leq C_0 \exp\{-\psi^t(x_j, f^N)\} \cdot \mathcal{H}^t_{b_{n-1}(x_j)\rho} (f^N(B^u(x_j,nN, \delta)))\\ &\leq C_0 \exp\{-\psi^t(x_j, f^N)\} \cdot \mathcal{H}^t_{b_{n-1}(x_j)\rho} (B^u(f^N(x_j),(n-1)N, \delta))\\ &\leq (C_0)^2 \exp\{-\psi^t(x_j, f^N)\} \cdot \exp\{-\psi^t(f^N(x_j), f^N)\} \\ &\quad\cdot \mathcal{H}^t_{b_{n-2}(x_j)\rho} (B^u(f^{2N}(x_j),(n-2)N, \delta))\\ &\leq \cdots\\&\leq (C_0)^n \exp\bigg\{ -\sum_{j=0}^{n-1}\psi^t(f^{jN}x_j, f^N) \bigg\} \cdot \mathcal{H}^t_\rho(B^u(f^{nN}x_j, \delta))\\ &\leq (C_0)^n C_1 \cdot \exp\bigg\{ -\sum_{j=0}^{n-1}\psi^t(f^{jN}x_j, f^N) \bigg\}, \end{align*} $$

where $C_1=\sup _{y\in M}\mathcal {H}^t_\rho (B(y,\delta ))$ . By property (b) and the sub-additivity of $\{-\psi ^t(\cdot ,f^n)\}_{n\geq 1}$ , we have that

$$ \begin{align*} \mathcal{H}_{\beta_n}^t(A_n\cap\xi^u(x)) &\leq \sum_{x_j\in E}\mathcal{H}^t_{\beta_n}(B^u(x_j, nN, \delta))\\ &\leq \sum_{x_j\in E} (C_0)^n C_1 \cdot \exp\bigg\{ -\sum_{i=0}^{n-1}\psi^t(f^{jN}x_j, f^N) \bigg\}\\ &\leq \sum_{x_j\in E} (C_0)^n C_1 \cdot \exp\bigg\{ -\sum_{i=0}^{nN-1}\psi^t(f^{i}x_j, f) \bigg\}\\ &\leq (C_0)^n C_1 \cdot \exp[nN(h_\mu(f)+\varepsilon)] \cdot \exp\bigg[nN\bigg({-}\int\psi^t(x,f)\,d\mu + \varepsilon\bigg)\kern-1pt\bigg] \\ &= (C_0)^n C_1 \cdot \exp\bigg[nN\bigg(h_\mu(f)- \int\psi^t(x,f)\,d\mu +2\varepsilon\bigg) \bigg]\\ &= (C_0)^n C_1\cdot e^{-nN\varepsilon}\\ &=(C_0 e^{-N\varepsilon})^n C_1. \end{align*} $$

Since N satisfies $C_0e^{-N\varepsilon }<1$ , we have that

$$ \begin{align*} \lim_{n\to\infty} \mathcal{H}^t_{\beta_n}(A_n\cap\xi^u(x))=0. \end{align*} $$

Since ${\lim _{n\to \infty }\beta _n=0}$ and ${\lim _{n\to \infty }\mu _x^u(A_n\cap \xi ^u(x))= 1}$ for $\mu $ -a.e. x, by [Reference Jordan and Pollicott17, Lemma 6], we obtain that

$$ \begin{align*}\dim_H\mu_x^u\leq t\end{align*} $$

for $\mu $ -a.e. x. Combining with equation (5) and the choice of t yield that $d_\mu ^u(x)\leq t_{u,1}^*$ for $\mu $ -a.e. x.

Now we are ready to present the proof of Theorem B.

Proof of Theorem B

For each $n>1$ , the measure $\mu $ is f-invariant ergodic, but it may be not ergodic for $f^n$ although $\mu $ is still $f^n$ -invariant. In either case, one can find an $f^n$ -invariant ergodic probability measure $\nu $ such that

$$ \begin{align*}\mu=\frac1m[\nu + f_*\nu + \cdots + f^{m-1}_*\nu], \end{align*} $$

where $m\in \mathbb {N}\setminus \{0\}$ divides n. Let

$$ \begin{align*} \widetilde{P}_\mu(f^n, -\psi^n(\cdot, f^n)):=h_\mu(f^n)-\int \psi^n(x, f^n)\,d\mu, \end{align*} $$

then one can show that

$$ \begin{align*} \widetilde{P}_\mu(f^n, -\psi^n(\cdot, f^n))&= \frac1m \sum_{i=0}^{m-1}\bigg( h_{f_*^i\nu}(f^n)-\int \psi^n(x, f^n) \,df_*^i\nu \bigg)\\ &=\frac1m \sum_{i=0}^{m-1} P_{f_*^i\nu}(f^n, -\psi^n(\cdot, f^n)). \end{align*} $$

Hence, there exists $j_0\in \{0,1,\ldots , m-1\}$ such that

$$ \begin{align*} \widetilde{P}_\mu(f^n, -\psi^t(\cdot, f^n)) \geq P_{f_*^{j_0}\nu}(f^n, -\psi^t(\cdot, f^n)). \end{align*} $$

Since $f_*^{j_0}\nu $ is hyperbolic and $f^n$ -invariant ergodic, by Lemma 3.2, there is a set $\tilde {A}$ with $\nu \circ f^{-j_0}(\tilde {A})=1$ such that for each $x\in \tilde {A}$ ,

$$ \begin{align*}d_{f_*^{j_0}\nu}^u (x) \leq t_{u,n}^*, \end{align*} $$

where $t_{u,n}^*$ is the unique root of the equation $P_\mu (f^n,-\psi ^t(\cdot , f^n))=0$ . Note that $d_\mu ^u(x), d_{f_*^{j_0}\nu }^u(x)$ are constants almost everywhere (see [Reference Ledrappier and Young22]) and $d_\mu ^u(x)\leq d_{f_*^{j_0}\nu }^u(x) \leq t_{u,n}^*$ for each $x\in \tilde {A}$ with $\mu (\tilde {A})\geq 1/m$ . Consequently, we have that

$$ \begin{align*}d_\mu^u(x)\leq t_{u,n}^*\end{align*} $$

for $\mu $ -a.e. x.

By the sub-additive of $\{-\psi ^t(\cdot , f^n)\}_{n\geq 1}$ , we obtain

$$ \begin{align*}\frac{1}{2^{k+1}} \bigg[h_\mu(f^{2^{k+1}}) - \int \psi^t(x,f^{2^{k+1}})\, d\mu\bigg] \leq \frac{1}{2^k} \bigg[ h_\mu(f^{2^k}) -\int \psi^t(x,f^{2^k})\, d\mu \bigg]. \end{align*} $$

Hence,

$$ \begin{align*}\frac{1}{2^{k+1}} \widetilde{P}_\mu(f^{2^{k+1}}, -\psi^t(\cdot, f^{2^{k+1}})) \leq \frac{1}{2^k} \widetilde{P}_\mu(f^{2^{k}}, -\psi^t(\cdot, f^{2^{k}})). \end{align*} $$

This yields that $t_{u,2^{k+1}}^*\leq t_{u,2^k}^*$ for every $k\ge 1$ . Let ${t_u^*:=\lim _{k\to \infty } t_{u, 2^k}^*}$ , then one has that

$$ \begin{align*}d_\mu^u(x)\leq t_u^*\quad \mu\text{-a.e.}\, x. \end{align*} $$

Since $P_\mu (f, \{-\psi ^t(\cdot , f^n)\})$ is continuous and strictly decreasing with respect to t, there exists at most one solution of the equation. To complete the proof of Theorem B, it suffices to show that $P_\mu (f, \{-\psi ^{t^*}(\cdot , f^n)\})=0$ .

Since $t_{u,2^k}^*\ge t_u^{*}$ for every $k\ge 1$ , by Theorem 2.2, one has that

$$ \begin{align*} 0\le \lim_{k\to\infty}\frac{1}{2^k} \widetilde{P}_\mu(f^{2^k}, -\psi^{t_u^{*}}(\cdot, f^{2^k}))= P_\mu(f, \{-\psi^{t_u^{*}}(\cdot, f^n)\}). \end{align*} $$

However, for each small number $\varepsilon>0$ , there exists K so that $t_{u,2^k}^*\le t_u^{*}+\varepsilon $ for every ${k\ge K}$ . Hence, we have that

$$ \begin{align*} P_\mu(f, \{-\psi^{t_u^*+\varepsilon}(\cdot, f^n)\}) &= h_\mu(f) - \lim_{n\to\infty} \frac1n\int\psi^{t_u^*+\varepsilon}(x,f^n)\, d\mu\\ &= \lim_{k\to\infty} \frac{1}{2^k}\widetilde{P}_\mu(f^{2^k}, -\psi^{t_u^*+\varepsilon}(\cdot, f^{2^k}))\le 0. \end{align*} $$

The previous arguments imply that $P_\mu (f, \{-\psi ^{t^*}(\cdot , f^n)\})=0$ . One can prove in a similar fashion that $d_\mu ^s(x)\leq t_s^*$ for $\mu $ -a.e. x. This completes the proof of Theorem B.

3.3 Proof of Theorem C

For each $\varepsilon>0$ , there exists a hyperbolic set $\Lambda _\varepsilon $ satisfying properties (a)–(d) in Theorem 2.4. The following lemma shows that the zero of the super-additive topological pressure of $\Phi _f(t)$ provides a lower bound of the Carathéodory singular dimension of the hyperbolic set on the local unstable leaf with respect to the super-additive singular valued potential $\Phi _f(t)$ .

Lemma 3.3. Let $f: M\to M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and let $\Lambda \subset M$ be a hyperbolic set. Assume that $f|_{\Lambda }$ is topologically transitive, then for every $x\in \Lambda $ ,

$$ \begin{align*}\dim_C^{\Phi_f} (\Lambda\cap W^u_{\mathrm{loc}}(x,f)) \geq t_*, \end{align*} $$

where $t_*$ is the unique root of the equation $P(f|_\Lambda , \Phi _f(t))=0$ .

Proof. For every $x\in \Lambda $ , we denote $Z=\Lambda \cap W^u_{\mathrm {loc}}(x, f)$ and $P(t)=P(f|_\Lambda , \Phi _f(t))$ . Since the function $P(t)$ is strictly decreasing in t, then for each $t<t_*$ , we have that $P(t)>0$ . Fix such a number t and take $\varepsilon>0$ with $P(t)-\varepsilon>0$ . By Proposition 2.2, one has that

$$ \begin{align*} P(t)=\lim_{n\to\infty}\frac1nP(f^n|_\Lambda, -\phi^t(\cdot, f^n)), \end{align*} $$

then there exists $N_1\in \mathbb {N}$ such that for every $n\geq N_1$ , we obtain

$$ \begin{align*} P(f^n|_\Lambda, -\phi^t(\cdot, f^n))> n( P(t) - \varepsilon ) >0. \end{align*} $$

Fix an integer $L\geq N_1$ , by [Reference Climenhaga, Pesin and Zelerowicz11, Proposition 5.4], one has that

$$ \begin{align*} P(f^L|_\Lambda, -\phi^t(\cdot, f^L)) = P_{Z}(f^L|_\Lambda, -\phi^t(\cdot, f^L)). \end{align*} $$

Hence, there is $\delta _1>0$ such that

$$ \begin{align*}P_{Z} (f^L|_\Lambda, -\phi^t(\cdot, f^L), \delta)> (P(t)-\varepsilon)L \end{align*} $$

for every $0<\delta <\delta _1$ . Consequently, fixing such a $\delta>0$ , one has that

$$ \begin{align*}m(Z, -\phi^t(\cdot, f^L), (P(t)-\varepsilon)L, \delta)=+\infty. \end{align*} $$

Hence, for each $K>0$ , there exists $S\in \mathbb {N}$ such that for each $N\ge S$ , we have that

(13) $$ \begin{align} \nonumber K & \leq \inf \sum_i\exp [-(P(t)-\varepsilon)Lm_i -S_{m_i}\phi^t(x_i, f^L)] \\ &\le e^{-NL(P(t)-\varepsilon)} \inf \sum_i\exp[ -S_{m_i}\phi^t(x_i, f^L)], \end{align} $$

where the infimum is taken over all collections $\{B^u_{m_i}(x_i, \delta , f^L)\}$ with $x_i\in \Lambda $ , $m_i\geq N$ which cover Z, $-S_{m_i}\phi ^t(x_i, f^L)\kern1.2pt{=}\kern1.2pt{-}\phi ^t(x_i, f^L)\kern1.2pt{-}\kern1.2pt\phi ^t(f^Lx_i,f^L)-\cdots -\phi ^t(f^{(m_i-1)L}x_i, f^L)$ and

$$ \begin{align*}\displaystyle{B^u_{m_i}(x_i, \delta, f^L):= \left\{y\in W^u(x_i,f): \max_{0\le j <m_i}d^u(f^{jL}(y), f^{jL}(x_i))<\delta \right\}}.\end{align*} $$

Fixing such an N and taking an integer $R\ge NL$ , let the collection of balls $\{B_{n_i}^u(x_i,\delta )\}$ with $x_i\in \Lambda $ , $n_i\ge R$ be a cover of Z. One can write $n_i=m_iL+s_i$ with $0\le s_i<L$ and $m_i\geq N$ for each i. Since $B_{n_i}^u(x_i,\delta )\subset B_{m_i}^u(x_i,\delta , f^L)$ for each i, the collection of balls $B_{m_i}^u(x_i,\delta , f^L)$ is also a cover of Z with $x_i\in \Lambda $ , $m_i\ge N$ . By the super-additivity of $\{-\phi ^t(\cdot , f^n)\}_{n\ge 1}$ , one has

$$ \begin{align*} \sum_{i}\exp [-\phi^t(x_i,f^{n_i})]&\ge \sum_{i}\exp [-S_{m_i}\phi^t(x_i,f^{L})-\phi^t(f^{m_iL}y, f^{s_i})]\\ &\ge C \sum_{i}\exp [-S_{m_i}\phi^t(x_i,f^{L})], \end{align*} $$

where ${C=\min _{0\le s<L}\min _{x\in M}\exp [-\phi ^t(x, f^{s})]}$ . This together with equation (13) yield that

$$ \begin{align*} \sum_{i}\exp [-\phi^t(x_i,f^{n_i})]\ge CK e^{NL(P(t)-\varepsilon)}. \end{align*} $$

Since the cover of Z is taken arbitrarily, one can conclude that

$$ \begin{align*} \inf\sum_{i}\exp [-\phi^t(x_i,f^{n_i})] \geq CK e^{NL(P(t)-\varepsilon)}, \end{align*} $$

where the infimum is taken over all collections $\{B^u_{n_i}(x_i, \delta )\}$ with $x_i\in \Lambda $ , $n_i\geq NL$ which cover Z. Letting $N\to \infty $ , we obtain

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

for every $t<t_*$ . This implies that

$$ \begin{align*}\dim_{C}^{\Phi_f}Z\geq t_*.\\[-37pt] \end{align*} $$

Proof of Theorem C(i)

By Lemma 3.3, for every $x\in \Lambda _\varepsilon $ , we obtain

$$ \begin{align*}\dim_C^{\Phi_f}(\Lambda_\varepsilon\cap W^u_{\mathrm{loc}}(x,f)) \geq t_{\varepsilon*}, \end{align*} $$

where $t_{\varepsilon *}$ is the unique root of the equation $P(f|_{\Lambda _\varepsilon }, \Phi _f(t))=0$ . By the variational principle of topological entropy, take $\nu \in \mathcal {M}_{\mathrm {inv}}(f|_{\Lambda _\varepsilon })$ such that $h_{\text {top}}(f|_{\Lambda _\varepsilon })=h_\nu (f|_{\Lambda _\varepsilon })$ . By properties (b) and (d) in Theorem 2.4, it holds that

(14)

where $\unicode{x3bb} _1(\nu )\geq \unicode{x3bb} _2(\nu ) \geq \cdots \geq \unicode{x3bb} _{m_0}(\nu )$ are the Lyapunov exponents of $\nu $ . However, let $\tau \in \mathcal {M}_{\mathrm {inv}}(f|_{\Lambda _\varepsilon })$ be an equilibrium state of $P(f|_{\Lambda _\varepsilon }, \Phi _f(t_{\varepsilon *}))$ , then one has that

This together with equation (14) yield that

$$ \begin{align*} -(u+1)\varepsilon\le P_{\mu}(f, \Phi_f(t_{\varepsilon*}))\le (u+1)\varepsilon. \end{align*} $$

Hence, we have that

$$ \begin{align*} \lim_{\varepsilon\to 0}P_{\mu}(f, \Phi_f(t_{\varepsilon*}))=0. \end{align*} $$

This implies that ${\lim _{\varepsilon \to 0}t_{\varepsilon *}=t_{u*}}$ , where $t_{u*}$ is the unique root of $P_\mu (f, \Phi _f(t))=0$ . Consequently, we have that

(15) $$ \begin{align} \liminf_{\varepsilon\to0}\dim_C^{\Phi_f}(\Lambda_\varepsilon\cap W^u_{\mathrm{loc}}(x,f)) \geq t_{u*}. \end{align} $$

As a counterpart of Lemma 3.3, we have the following result.

Lemma 3.4. Let $f: M\to M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and let $\Lambda \subset M$ be a hyperbolic set. Assume that $f|_{\Lambda }$ is topologically transitive. Then for every $x\in \Lambda $ ,

$$ \begin{align*}\dim_C^{\Psi_f} (\Lambda\cap W^u_{\mathrm{loc}}(x,f)) \leq t^*, \end{align*} $$

where $t^*$ is the unique root of the equation $P(f|_\Lambda , \Psi _f(t))=0$ .

Proof. Denote $P(t)=P(f|_\Lambda , \Psi _f(t))$ . For each $t>t_*$ ,

$$ \begin{align*}0>P(t)=\lim_{n\to\infty}\frac1nP(f^n, -\psi^t(\cdot, f^n)). \end{align*} $$

Fix such a number t and take $\varepsilon>0$ with $P(t)+\varepsilon <0$ . Then there exists $N_1\in \mathbb {N}$ such that for every $n\geq N_1$ , we obtain

$$ \begin{align*}P(f^n, -\psi^t(\cdot, f^n)) < n( P(t) + \varepsilon ) <0. \end{align*} $$

Fix an integer $L\geq N_1$ such that

$$ \begin{align*}P(f^L, -\psi^t(\cdot, f^L)) < L ( P(t) + \varepsilon ) <0. \end{align*} $$

For each $x\in \Lambda $ , set $Z=\Lambda \cap W^u_{\mathrm {loc}}(x,f)$ . By [Reference Climenhaga, Pesin and Zelerowicz11, Proposition 5.4], one has that

$$ \begin{align*} P(f^L, -\psi^t(\cdot, f^L)) = P_{Z}(f^L, -\psi^t(\cdot, f^L)). \end{align*} $$

Thus, there is $\delta _1>0$ such that for every $0<\delta <\delta _1$ , one has

$$ \begin{align*}P_{Z} (f^L, -\psi^t(\cdot, f^L), \delta) < (P(t)+\varepsilon)L. \end{align*} $$

Hence, one has that

$$ \begin{align*} m(Z, -\psi^t(\cdot, f^L), (P(t)+\varepsilon)L, \delta)=0. \end{align*} $$

For each $\xi>0$ , there exists $N\in \mathbb {N}$ and a cover $\{B^u_{n_i}(x_i, \delta , f^L)\}$ of Z with $x_i\in \Lambda $ , $n_i\geq N$ such that

$$ \begin{align*} \xi &\geq \sum_i\exp\Big[-(P(t)+\varepsilon)Ln_i + \sup_{y\in B^u_{n_i}(x_i, \delta, f^L)}-S_{n_i}\psi^t(y, f^L)\Big]. \\ &\geq e^{-NL(P(t)+\varepsilon)} \sum_i \exp \Big[\sup_{y\in B^u_{n_i}(x_i, \delta, f^L)}-S_{n_i}\psi^t(y, f^L)\Big]. \end{align*} $$

Note that $d^u(f^Lx, f^Ly)<\delta $ implies $d^u(f^ix, f^iy)<\delta $ for $i=0, 1,\ldots , L-1$ , since f is expanding along the unstable manifold. This implies that $ B^u_{(n_i-1)L+1}(x_i,\delta ) = B^u_{n_i}(x_i, \delta , f^L) $ for every i. Since

$$ \begin{align*} -S_{n_i}\psi^t(y, f^L)&=-\psi^t(y, f^L)-\psi^t(f^Ly,f^L)-\cdots -\psi^t(f^{(n_i-1)L}y, f^L)\\ &\geq -\psi^t(y,f^{(n_i-1)L}) +C_1\\ &= -\psi^t(y,f^{(n_i-1)L}) - \psi^t(f^{(n_i-1)L}y,f)+\psi^t(f^{(n_i-1)L}y,f)+C_1\\ &\geq -\psi^t(y,f^{(n_i-1)L+1})+ C_1+C_2, \end{align*} $$

where $C_1=\min _{x\in M}\{-\psi ^t(x,f^L)\}$ and $C_2=\min _{x\in M}\psi ^t(x,f)$ , we have that

$$ \begin{align*} \xi &\geq e^{-NL(P(t)+\varepsilon)}e^{C_1+C_2} \sum_i \exp\Big[\sup_{y\in B^u_{(n_i-1)L+1}(x_i,\delta) }-\psi^t(y,f^{(n_i-1)L+1}) \Big] \\ &\geq e^{-NL(P(t)+\varepsilon)}e^{C_1+C_2} \inf \sum_i\exp \Big[\sup_{y\in B^u_{m_i}(x_i,\delta)} -\psi^t(y, f^{m_i})\Big] \end{align*} $$

and

$$ \begin{align*} \inf \sum_i\exp \Big[\sup_{y\in B^u_{m_i}(x_i,\delta)} -\psi^t(y, f^{m_i})\Big]\leq \xi e^{NL(P(t)+\varepsilon)}e^{-C_1-C_2}, \end{align*} $$

where the infimum is taken over all collections $\{B^u_{m_i}(x_i, \delta )\}$ with $x_i\in \Lambda $ , $m_i\geq (N-1)L$ which cover Z. Letting $N\to \infty $ , we obtain

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

for every $t>t_*$ . This yields that

$$ \begin{align*} \dim_{C}^{\Psi_f}Z\leq t_*.\\[-42pt] \end{align*} $$

Remark 3.1. Let $f: M\to M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact Riemannian manifold M and $\Lambda \subset M$ be a hyperbolic set. Assume that $f|_{\Lambda }$ is topologically transitive. Then for every $x\in \Lambda $ ,

$$ \begin{align*}\dim_C^{\Phi_f} (\Lambda\cap W^u_{\mathrm{loc}}(x,f)) = t^{\Phi_f}_u,\quad \dim_C^{\Psi_f} (\Lambda\cap W^u_{\mathrm{loc}}(x,f)) = t^{\Psi_f}_u, \end{align*} $$

where $t^{\Phi _f}_u$ , $t^{\Psi _f}_u$ are the unique roots of the equations

$$ \begin{align*}P_{\Lambda\cap W^u(x,f)}(f, \Phi_f(t))=0, \quad P_{\Lambda\cap W^u(x,f)}(f, \Psi_f(t))=0,\end{align*} $$

respectively. The proof is a slight modification of Lemmas 3.3 and 3.4. See [Reference Cao, Wang and Zhao9] for more details about the Carathéodory singular dimension of each subset of a repeller. However, we do not know whether $P_{\Lambda \cap W^u(x,f)}(f, \Phi _f(t))=P_\Lambda (f, \Phi _f(t))$ and $P_{\Lambda \cap W^u(x,f)}(f, \Psi _f(t))=P_\Lambda (f, \Psi _f(t))$ hold.

Proof of Theorem C(ii)

By Lemma 3.4, we have that

$$ \begin{align*}\dim_C^{\Psi_f}(\Lambda_\varepsilon\cap W^u_{\mathrm{loc}}(x,f)) \leq t_{\varepsilon}^*, \end{align*} $$

where $t_{\varepsilon }^*$ is the unique root of the equation $P(f|_{\Lambda _\varepsilon }, \Psi _f(t))=0$ . Take $\nu \in \mathcal {M}_{\mathrm {inv}}(f|_{\Lambda _\varepsilon })$ such that $h_\nu (f|_{\Lambda _\varepsilon })=h_{\text {top}}(f|_{\Lambda _\varepsilon })$ , by properties (b) and (d) in Theorem 2.4, it holds that

(16)

where $\unicode{x3bb} _1(\nu )\geq \unicode{x3bb} _2(\nu ) \geq \cdots \geq \unicode{x3bb} _{m_0}(\nu )$ are the Lyapunov exponents of $\nu $ . Similarly, let ${\tau \in \mathcal {M}_{\mathrm {inv}}(f|_{\Lambda _\varepsilon })}$ be an equilibrium state of $P(f|_{\Lambda _\varepsilon }, \Psi _f(t_{\varepsilon }^*))$ , then one has that

This together with equation (16) yield that

$$ \begin{align*} -(u+1)\varepsilon\le P_{\mu}(f, \Psi_f(t_{\varepsilon}^*))\le (u+1)\varepsilon. \end{align*} $$

Hence, we have that

$$ \begin{align*} \lim_{\varepsilon\to 0}P_{\mu}(f, \Psi_f(t_{\varepsilon}^*))=0. \end{align*} $$

This implies that ${\lim _{\varepsilon \to 0}t_{\varepsilon }^*=t_{u}^*}$ , where $t_{u}^*$ is the unique root of $P_\mu (f, \Psi _f(t))=0$ . Consequently, we have that

$$ \begin{align*}\limsup_{\varepsilon\to0}\dim_C^{\Psi_f}(\Lambda_\varepsilon\cap W^u_{\mathrm{loc}}(x,f)) \leq t_{u}^*.\\[-44pt] \end{align*} $$

Finally, to complete the proof of Theorem C, assume that $\mu $ is an SRB measure from now on, then $h_\mu (f)=\unicode{x3bb} _1(\mu )+\unicode{x3bb} _2(\mu )+\cdots +\unicode{x3bb} _u(\mu )$ . Thus, $P_\mu (f, \Phi _f(u))=0$ . Since the Carathéodory singular dimension with respect to $\Psi _f$ is always less than u, by property (i) of Theorem C, we have that

$$ \begin{align*} \lim_{\varepsilon\to0}\dim_C^{\Phi_f}(\Lambda_\varepsilon\cap W^u_{\mathrm{loc}}(x,f)) =u. \end{align*} $$

If $\unicode{x3bb} _1(\mu )+\unicode{x3bb} _2(\mu )+\cdots +\unicode{x3bb} _{m_0}(\mu )\geq 0$ , then $P_\mu (f, \Xi _f(m_0-u))\ge 0$ since $\mu $ is an SRB measure for f. Consider $f^{-1}$ , by Margulis–Ruelle inequality, we have that

which implies that $P_\mu (f, \Xi _f(m_0-u))\le 0$ . Hence, we have that

$$ \begin{align*} P_\mu(f, \Xi_f(m_0-u))= 0. \end{align*} $$

Thus, we have that $t_s^*=m_0-u$ . By the definition of Lyapunov dimension, we have that $\dim _L\mu =m_0=u+t_s^*$ .

Now, we assume that $\unicode{x3bb} _1(\mu )+\unicode{x3bb} _2(\mu )+\cdots +\unicode{x3bb} _{m_0}(\mu )< 0$ and let $\ell $ be the largest integer such that $\unicode{x3bb} _1(\mu )+\unicode{x3bb} _2(\mu )+\cdots +\unicode{x3bb} _{\ell }(\mu )\geq 0$ . By a standard computation, one can show that

Combining with

one has

$$ \begin{align*}\dim_L\mu=u+t_s^*.\end{align*} $$

This completes the proof of Theorem C.