2.1.1 Topological pressure defined via separated sets

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

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

Definition 2.1. We call the quantity

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

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

Let $\mathcal {M}_f(X)$ denote the space of all f-invariant measures on X. For $\mu \in \mathcal {M}_f(X)$ , let $h_\mu (f)$ denote the metric entropy of f with respect to $\mu $ (see Walters’ book [Reference Walters39] for details of metric entropy), and let

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

The existence of the above limit follows from a sub-additive argument. In [Reference Cao, Feng and Huang9], the authors proved the following variational principle.

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

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

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

Although it is unknown whether the variational principle holds for super-additive topological pressure, Cao, Pesin, and Zhao gave an alternative definition via variational principle in [Reference Cao, Pesin and Zhao11]. Given a sequence of super-additive continuous potentials $\Psi =\{\psi _n\}_{n\geq 1}$ on a compact dynamical system $(X, f)$ , the super-additive topological pressure of $\Psi $ is defined as

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

where

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

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

2.1.2 Measure theoretic pressure

We first follow the approach in [Reference Pesin33] to give the definitions of topological pressures on arbitrary subsets. Given a sub-additive potential $\Phi =\{\phi _n\}_{n\ge 1}$ on X, a subset $Z\subset X$ , and $\alpha \in \mathbb {R}$ , let

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

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

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

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

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

Definition 2.4. We call the quantity

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

the topological pressure of $(f, \Phi )$ on the set Z (see [Reference Feng and Huang14] for the weighted version of this quantity).

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

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

We set

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

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

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

respectively.

Definition 2.5. We call the quantities

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

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

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

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

and then we call the quantity

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

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

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

We call the quantities

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

the lower and upper measure theoretic pressures of $(f,\Phi )$ with respect to $\mu $ , respectively. It is proved in [Reference Cao, Hu and Zhao10, Theorem A] that

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

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

Remark 2.6. In fact, one can show that

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

where $\mathcal {P}$ denotes either P or $\underline {CP}$ or $\overline {CP}$ , see [Reference Zhao46] for a proof.

3.1.1 Definitions of repeller and Lyapunov dimension

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

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

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

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

Given an f-invariant ergodic measure $\mu $ supported on the repeller $\Lambda $ , let $\unicode{x3bb} _1(\mu )\ge \unicode{x3bb} _2(\mu )\ge \cdots \ge \unicode{x3bb} _{m_0}(\mu )$ and $h_{\mu }(f)$ denote the Lyapunov exponents and the measure theoretic entropy of $(f,\mu )$ , respectively. We refer the reader to [Reference Barreira and Pesin6, Reference Walters39] for detailed descriptions of Lyapunov exponents and the measure theoretic entropy. We further define the Lyapunov dimension of $\mu $ as follows:

(3.1)

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

The original definition of Lyapunov dimension in [Reference Alexander and Yorke1, Reference Kaplan, Yorke, Peitgen and Walther19, Reference Ledrappier22] is defined only for hyperbolic systems as follows: assume that $\nu $ is an ergodic measure of a smooth diffeomorphism f with Lyapunov exponents $\unicode{x3bb} _1\ge \cdots \ge \unicode{x3bb} _u>0\ge \unicode{x3bb} _{u+1}\ge \cdots \ge \unicode{x3bb} _{m_0}$ , then the the Lyapunov dimension is

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

and $\ell =\max \{i:-\unicode{x3bb} _{u+1}-\cdots -\unicode{x3bb} _i\ge h_\nu (f)\}$ . Hence, the definition in equation (3.1) is a reasonable substitute. For a $C^1$ expanding map f, Feng and Simon [Reference Feng and Simon15] defined the Lyapunov dimension of an ergodic measure as the zero of the measure theoretic pressure $P_\mu (f,\Phi _f(t))=0$ (see equation (3.4)). In this paper, we will prove that the unique solution of the equation $P_\mu (f,\Phi _f(t))=0$ is indeed our definition of Lyapunov dimension (see Theorem A). Furthermore, this paper shows that the Lyapunov dimension of an ergodic measure defined in equation (3.1) is equal to its Carathéodory singular dimension (see Proposition 3.2), so the Carathéodory singular dimension (see §3.1.3 for the detailed definition) can be regarded as a geometric explanation of the Lyapunov dimension.

3.1.2 Singular valued potentials

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

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

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

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

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

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

It follows that the sequence of functions

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

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

3.1.3 Carathéodory singular dimension

We recall the definition of Carathéodory singular dimension of a repeller which is introduced in [Reference Cao, Pesin and Zhao11].

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

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

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

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

The quantity

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

is called the Carathéodory singular dimension of Z. Particularly, the Carathéodory singular dimension of the repeller $\Lambda $ is independent of the parameter r for small values of $r>0$ (see [Reference Cao, Pesin and Zhao11, Theorem 4.1]).

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

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

and the quantity

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

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

3.1.4 Approximation of Carathéodory singular dimension of repellers

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

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

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

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

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

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

Some comments on the previous theorem are in order. First, the map of higher smoothness $C^{1+\alpha }$ is crucial as it allows us to use some powerful results of Pesin theory. Second, if f is a local diffeomorphism preserving an ergodic expanding measure $\mu $ of positive entropy, that is, $(f,\mu )$ has only positive Lyapunov exponent, in this case, one can also obtain an approximation result as in [Reference Cao, Pesin and Zhao11] so that we can obtain the second statement in the previous theorem in this setting. In [Reference Sánchez-Salas37], for a $C^2$ interval map f with finitely many non-degenerate critical points, the author proved that the Hausdorff dimension of an expanding measure $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of repellers.

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

3.2.1 Lyapunov exponents and holonomy maps

Let $f: M\to M$ be a diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M. By Oseledec’s multiplicative ergodic theorem (see [Reference Oseledec30]), there exists a total measure set $\mathcal {O}\subset M$ such that for each $x \in \mathcal {O}$ and each invariant measure $\mu $ , there exist positive integers $d_1(x),d_2(x),\ldots , d_{p(x)}(x)$ , numbers $\unicode{x3bb} _1(x)>\unicode{x3bb} _2(x)>\cdots >\unicode{x3bb} _{p(x)}(x)$ , and a splitting

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

which satisfy that:

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

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

   

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3.2.2 Approximation of Lyapunov exponents and entropy

For a $C^{1+\alpha }$ diffeomorphism $f:M\rightarrow M$ , Katok [Reference Katok20] showed that an f-invariant ergodic hyperbolic measure (a measure has no zero Lyapunov exponents) with positive metric entropy can be approximated by horseshoes. However, Katok’s result does not explicitly mention a control of the Oseledets splitting over the horseshoes. Recently, Avila, Crovisier, and Wilkinson [Reference Avila, Crovisier and Wilkinson2] showed that there is a dominated splitting over the horseshoes, with approximately the same Lyapunov exponents on each sub-bundle of the splitting.

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

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

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

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

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

then we say that

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

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

For the reader’s convenience, we recall Avila, Crovisier, and Wilkinson’s approximation results in the following, see [Reference Avila, Crovisier and Wilkinson2] for more details.

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

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

3.2.3 Statements of results

Let $f: M\to M$ be a $C^{1+\alpha }$ diffeomorphism of a compact Riemannian manifold M, and let $\mu $ be a hyperbolic ergodic f-invariant probability measure with positive entropy. Suppose that $(f,\mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ . Ledrappier, Young [Reference Ledrappier and Young23] and Barreira, Pesin, Schmeling [Reference Barreira, Pesin and Schmeling7] proved that

(3.7)

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

for $i=u, s$ and every $x\in \Lambda $ . Burns and Wilkinson [Reference Burns and Wilkinson8] prove that the holonomy maps of the stable and unstable foliations for $(f, \Lambda _\varepsilon )$ are Lipschitz continuous. Consequently, one can show that

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

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

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

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

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

In [Reference Wang, Cao and Zou41], the authors relaxed the smoothness of Theorem 3.3 to $C^1$ under the additional condition that Oseledec’s splitting $E^u\oplus E^s$ of $(f,\mu )$ is dominated. In this setting, one does not have Lipschitz continuity of the holonomy map in general. However, using Palis and Viana’s method [Reference Palis, Viana, Bamón, Labarca and Palis31], one can show that for every $\gamma \in (0, 1)$ , there is some $D_\gamma>0$ such that the holonomy maps of the stable and unstable foliations for the hyperbolic dynamical system $(f, \Lambda _\varepsilon )$ (see Lemma 4.2) are $(D_\gamma , \gamma )$ -Hölder continuous. Since $\gamma $ is arbitrary, using the ideas in [Reference Wang, Wang, Cao and Zhao43], one can prove the following theorem.

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

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

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