2.1 Cocycles and Lyapunov exponents for cocycles

Let f be a $C^r (r>1)$ diffeomorphism of a compact Riemannian manifold M, and $L(\mathfrak {X})$ be the space of bounded linear operators on a Banach space $\mathfrak {X}$ . Assume $A:M \to L(\mathfrak {X})$ is a Hölder continuous map. The cocycle over f generated by A is a map $\mathcal {A}:M\times \mathbb {N}\to L(\mathfrak {X})$ defined by $\mathcal {A}(x,0)=Id,$ and $\mathcal {A}(x,n)=A(f^{n-1}x)\ldots A(fx)A(x)$ . We also denote $\mathcal {A}(x,-n):=A(f^{-n}x)^{-1}\ldots A(f^{-2}x)^{-1}A(f^{-1}x)^{-1}$ for $n>0.$ Note that $\mathcal {A}(x,-n)$ is not always defined. To be more flexible, we also denote $\mathcal {A}_x^n:=\mathcal {A}(x,n).$

Let $\mathcal {M}_f(M)$ be the set of f-invariant probability measures. Then for a fixed dense subset $\{\varphi _j\}_{j= 1}^\infty $ of the unit sphere of $C(M)$ , it induces a metric on $\mathcal {M}_f(M)$ :

$$ \begin{align*}D(\mu,\nu)= \sum\limits_{j=1}^{\infty}\frac{|\int \varphi_j\,d\mu-\int \varphi_j\,d\nu|}{2^j}\quad \text{for all } \mu,\nu\in \mathcal{M}_f(M).\end{align*} $$

We now study some properties of Lyapunov exponents. The following version of multiplicative ergodic theorem was established by Froyland, LLoyd, and Quas [Reference Froyland, Lloyd and Quas10], based on the work of [Reference Lian and Lu18, Reference Oseledec24, Reference Thieullen28]. To state the multiplicative ergodic theorem, we introduce some definitions.

Denote by $B_1$ the unit ball of $\mathfrak {X}$ . Then for any $T\in L(\mathfrak {X}),$ we define the Hausdorff measure of non-compactness of T by

$$ \begin{align*}\|T\|_\kappa :=\inf\{\varepsilon>0: T(B_1)~\text{can be covered by a finite number of} ~\varepsilon\text{-balls} \}.\end{align*} $$

Then by the definition, we have $\|T\|_\kappa \leq \|T\|$ , and $\|\cdot \|_\kappa $ is sub-multiplicative, that is, $\|T_2T_1\|_\kappa \leq \|T_2\|_\kappa \cdot \|T_1\|_\kappa $ for any $T_1,T_2\in L(\mathfrak {X}).$ Let $\mu $ be an ergodic f-invariant measure on $M$ . Then by the sub-additive ergodic theorem, the limits

$$ \begin{align*} \unicode{x3bb}(\mathcal{A},\mu):=\lim\limits_{n\to +\infty}\frac{1}{n}\int \log \|\mathcal{A}_x^n\|\,d\mu,\\ \kappa(\mathcal{A},\mu):=\lim\limits_{n\to +\infty}\frac{1}{n}\int \log \|\mathcal{A}_x^n\|_\kappa \,d\mu \end{align*} $$

exist. We say that the cocycle $\mathcal {A}$ is quasi-compact with respect to $\mu $ , if $\unicode{x3bb} (\mathcal {A},\mu )>\kappa (\mathcal {A},\mu )$ .

For a given f-invariant set $\Lambda ,$ a splitting $\mathfrak {X}=E_1(x)\oplus \cdots \oplus E_i(x)\oplus F_i(x)$ on $\Lambda $ is called an $\mathcal {A}$ -invariant splitting if $E_j(x)=A(x)^{-1}E_j(fx)$ and $F_i(x)= A(x)^{-1}F_i(fx)$ for every $x\in \Lambda , j=1,\ldots ,i.$

Given two topological spaces Y, Z and a Borel measure $\mu $ on Y, a map $g:Y\to Z$ is called $\mu $ -continuous if there exists a sequence of pairwise disjoint compact subsets $Y_n\subset Y$ , such that $\mu (\bigcup _{n\geq 1}Y_n)=1$ and $g|_{Y_n}$ is continuous for every $n\geq 1$ .

We now state a version of multiplicative ergodic theorem used in this paper. To simplify the statement, we write $\mathbb {N}_k:=\{1,2,\ldots ,k \}$ for $1\leq k <+\infty $ and $\mathbb {N}_{+\infty }:=\mathbb {N}_+$ .

The numbers $\unicode{x3bb} _{1}>\unicode{x3bb} _{2}>\cdots $ indexed in $\mathbb {N}_{k_0}$ are called the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu ,$ and the decomposition ${\mathfrak {X}}=E_1(x)\oplus \cdots \oplus E_i(x)\oplus F_i(x)$ is called the Oseledets decomposition. Denote $d_j:=\dim (E_j).$

2.2 Main result

Recall that an ergodic f-invariant measure is called a hyperbolic measure if it has no zero Lyapunov exponents for $Df$ . We now state the main result of this paper.

The main difficulty in the proof of Theorem 2.2 is to obtain the $\mathcal {A}$ -invariant splitting in property (iv) of Theorem 2.2. If $\mathfrak {X}$ is finite dimensional, there is a classical argument to obtain the linear subspaces $E_i(x)$ (see [Reference Barreira and Pesin3, Theorem 6.1.2] for instance). However, this argument relies on the compactness of the unit sphere of $\mathfrak {X}$ , which is invalid in the infinite dimensional case. To overcome this difficulty, we obtain the invariant splitting by showing that $E_i(x)$ is actually a limit of a certain Cauchy sequence in the Grassmannian of closed subspaces of $\mathfrak {X}$ .

2.3 Application: continuity of sub-additive topological pressure

For a given continuous map $f:M\to M$ , a sequence of continuous functions $\Phi =\{\varphi _n\}_{n\geq 1}$ on M is called sub-additive if

$$ \begin{align*}\varphi_{m+n}\leq \varphi_n+\varphi_m\circ f^n\quad \text{for all } m,n\geq 1.\end{align*} $$

Recall that a subset $E\subset M$ is called (n, $\epsilon $ )-separated if for any $x,y\in E$ , there exists ${0\leq i<n}$ such that $d(f^ix,f^iy)>\epsilon .$ Now for a given sub-additive potential ${\Phi =\{\varphi _n\}_{n\geq 1}}$ , the sub-additive topological pressure of $\Phi $ is defined by

$$ \begin{align*} P(f,\Phi):=\lim\limits_{\epsilon\to 0}\limsup\limits_{n\to\infty}\frac1n\log P_n(\Phi,\epsilon), \end{align*} $$

where

$$ \begin{align*} P_n(\Phi,\epsilon)=\sup\bigg\{\sum\limits_{x \in E} e^{\varphi_n(x)}: E \text{ is an } (n,\epsilon)\text{-}\text{separated subset of }M\bigg\}. \end{align*} $$

The variational principle for sub-additive topological pressure was proved by Cao, Feng, and Huang [Reference Cao, Feng and Huang5].

Theorem 2.3. [Reference Cao, Feng and Huang5]

Let $f:M\to M$ be a continuous map on a compact metric space M, and $\Phi =\{\varphi _n\}_{n\ge 1}$ be a sub-additive potential on M. Then,

$$ \begin{align*} P(f,\Phi)=\sup\{h_\mu(f)+\mathcal{F}_*(\Phi,\mu): \mu\in \mathcal{M}(M,f), \mathcal{F}_*(\Phi,\mu)\ne -\infty\}, \end{align*} $$

where $h_{\mu }(f)$ is the metric entropy of f with respect to $\mu $ (for details see [Reference Walters29]) and

$$ \begin{align*} \mathcal{F}_*(\Phi,\mu)=\lim_{n\to\infty}\frac1n\int\varphi_n\, d\mu=\inf_{n\ge 1}\frac1n\int\varphi_n\,d\mu. \end{align*} $$

Let $A:M\to L(\mathfrak {X})$ be a continuous map. We shall consider the singular value functions of A. The concept of singular values in a general Banach space is not well defined, instead, many different generalizations exist (for details, see [Reference Quas, Thieullen and Zarrabi26, §A.3]). We choose the number

$$ \begin{align*}\sigma_k(T)=\sup\limits_{\dim(V)=k}\inf\limits_{v\in V\setminus\{0\}}\frac{\|Tv\|}{\|v\|} \end{align*} $$

to be the singular value of index k of an operator T. Let

$$ \begin{align*}\psi^s(x,n):= \sum_{i=1}^{[s]}\log\sigma_i(\mathcal{A}_x^n)+(s-[s])\log \sigma_{[s]+1}(\mathcal{A}_x^n), \end{align*} $$

and denote $P(A,s):=P(f,\Psi )$ as the topological pressure of the singular value potential $\Psi =\{\psi ^s(\cdot ,n)\}_{n\geq 1}$ .

Remark. Though $\Psi $ is not sub-additive, it is equivalent to a sub-additive potential. Indeed, we define a k-dimensional volume of vectors $(v_1,v_2,\ldots ,v_k)$ by

$$ \begin{align*}\mathrm{Vol}_k(v_1,v_2,\ldots,v_k)=\bigg(\prod_{i=1}^{k-1} {\mathrm{dist}} (v_i,\mathrm{span}(v_{i+1},\ldots,v_k))\bigg)\|v_k\|, \end{align*} $$

where ${\mathrm{dist}} (v_i,\mathrm{span}(v_{i+1},\ldots ,v_k))=\inf \{\|v_i-v\|:v\in \mathrm{span}(v_{i+1},\ldots ,v_k)\}.$ For ${T\in L(\mathfrak {X}),}$ define

$$ \begin{align*}V_k(T)=\sup\limits_{\substack{\|v_i\|=1 \\ 1\leq i\leq k}}\mathrm{Vol}_k(Tv_1,Tv_2,\ldots,Tv_k).\end{align*} $$

Then by [Reference Lian and Lu18, Lemma 4.7], $V_k$ is sub-multiplicative. Moreover, by [Reference González-Tokman and Quas13], there exists a constant $C\geq 1$ which is independent of $T\in L(\mathfrak {X})$ such that

(2.2) $$ \begin{align} C^{-1}\cdot V_k(T)\leq \prod_{i=1}^k\sigma_i(T)\leq C\cdot V_k(T). \end{align} $$

Let

(2.3) $$ \begin{align} \phi^s(x,n)=(s-[s])\log V_{[s]+1}(\mathcal{A}_x^n)+([s]+1-s)\log V_{[s]}(\mathcal{A}_x^n). \end{align} $$

Then it gives that $\Phi =\{\phi ^s(\cdot ,n)\}_{n\geq 1}$ is sub-additive and $|\phi ^s(x,n)-\psi ^s(x,n)|\leq \log C,$ which implies $P(A,s)=P(f,\Psi )=P(f,\Phi ).$

We shall consider a $C^{r}(r>1)$ diffeomorphism $f:M\to M$ which satisfies the assumptions:

1. (A1) the entropy map $\mu \mapsto h_\mu (f)$ is upper semi-continuous;

2. (A2) every f-invariant measure is hyperbolic, that is, the Lyapunov exponents of f are non-zero for every f-invariant measure.

Paradigms of diffeomorphisms which satisfy assumptions (A1) and (A2) but not are uniformly hyperbolic are given, for example, by a one-parameter family of $C^k$ maps constructed by Rios [Reference Rios27], and a family of $C^2$ Hénon-like maps constructed by Cao, Luzzatto, and Rios [Reference Cao, Luzzatto and Rios6].

Theorem 2.4. Let $f:M\to M$ be a $C^{r}(r>1)$ diffeomorphism satisfying the hypothesis of assumptions (A1) and (A2), and let $A:M\to L(\mathfrak {X})$ be an $\alpha $ -Hölder continuous map such that $A(x)$ is a compact operator for every $x\in M$ . Then for any $s<k_0$ , $A\mapsto P(A,s)$ is continuous at A, that is, for any $\epsilon>0$ , there exist $\delta>0$ such that

$$ \begin{align*} |P(B,s)-P(A,s)|<\epsilon \end{align*} $$

for every $\alpha $ -Hölder continuous map $B: M\to L(\mathfrak {X})$ with $\|A-B\|_0<\delta $ , where $k_0$ is given by Theorem 2.1 and $\|A-B\|_0:=\max _{x\in M}\|A(x)-B(x)\|$ .

If the Banach space $\mathfrak {X}$ is finite dimensional, then we have the following corollary.

Corollary 2.5. Let $f:M\to M$ be a $C^{r}(r>1)$ diffeomorphism satisfying the hypothesis of assumptions (A1) and (A2), and let $A:M\to M_d(\mathbb {R})$ be an $\alpha $ -Hölder continuous map. Then $A\mapsto P(A,s)$ is continuous at A, that is, for any $\epsilon>0$ , there exist $\delta>0$ such that

$$ \begin{align*} |P(B,s)-P(A,s)|<\epsilon \end{align*} $$

for every $\alpha $ -Hölder continuous map $B: M\to M_d(\mathbb {R})$ with $\|A-B\|_0<\delta $ , where $\|A-B\|_0:=\max _{x\in M}\|A(x)-B(x)\|$ .

In [Reference Cao, Pesin and Zhao7, Theorem 7.1], Cao, Pesin, and Zhao proved a similar result for f being a sub-shift of finite type. This corollary relaxes the condition of f to be a totally non-uniformly hyperbolic diffeomorphism, that is, f satisfies the hypothesis of assumptions (A1) and (A2).

3.1 Regular neighborhoods

Let f be a $C^r \, (r>1) \, $ diffeomorphism of a compact Riemannian d-dimensional manifold M, preserving an ergodic hyperbolic measure $\mu $ . Then by Oseledets multiplicative ergodic theorem [Reference Oseledec24], there exists an f-invariant set $\mathcal {R}^{Df}$ with $\mu $ -full measure, a number $\chi>0$ , and a $Df$ -invariant decomposition $TM=E^u\oplus E^s$ on $\mathcal {R}^{Df}$ such that for any $x\in \mathcal {R}^{Df}$ , we have

$$ \begin{align*}\lim\limits_{n\rightarrow \pm \infty} \frac{1}{n}\log\|D_xf^n(u)\|> \chi\quad \text{for all } u\in E^u(x)\setminus \{0\},\end{align*} $$

$$ \begin{align*}\lim\limits_{n\rightarrow \pm \infty} \frac{1}{n}\log\|D_xf^n(v)\| < -\chi\quad \text{for all } v\in E^s(x)\setminus \{0\}.\end{align*} $$

Denote $d_u=\dim (E^u), d_s=\dim (E^s)$ , and denote by $B(0,r)$ the standard Euclidean r-ball in $\mathbb {R}^d$ centered at $0.$ We now introduce some properties of regular neighborhoods, see [Reference Pesin25] for the proofs.

The set $\mathcal {N}(x):=\Psi _x(B(0,l^\prime (x)^{-1}))$ is called a regular neighborhood of x. Let $r(x)$ be the radius of maximal ball contained in $\mathcal {N}(x)$ . Then Theorem 3.1 implies $r(x)\geq l^\prime (x)^{-2}.$ By Lusin’s theorem, for any $\delta>0,$ there exists a compact subset $\mathcal {R}^{Df}_{\delta }\subset \mathcal {R}^{Df}$ with $\mu (\mathcal {R}^{Df}_{\delta })>1-\delta $ such that $x\mapsto \Psi _x$ , $l^\prime (x)$ and the Oseledets splitting $T_xM=E_x^u\oplus E_x^s$ vary continuously on $\mathcal {R}^{Df}_\delta $ .

3.2 $(\rho ,\beta ,\gamma )$ -rectangles

Let $M,f,\mu $ be as above and let $d_u=\dim (E^u),d_s=\dim (E^s)$ . Then $d_u+d_s=d.$ Let $I=[-1,1]$ , so then we say $R(x)\subset M$ is a rectangle in M if there exists a $C^1$ embedding $\Phi _x:I^{d}\to M$ such that $\Phi _x(I^{d})=R(x)$ and $\Phi _x(0)=x$ . A set $\widetilde {H}$ is called an admissible u-rectangle in  $R(x)$ , if there exist $0<\unicode{x3bb} <1, C^1$ maps $\phi _1,\phi _2:I^{d_ u}\to I^{d_s}$ satisfying $\|\phi _1(u)\|\geq \|\phi _2(u)\|$ for $u\in I^{d_u}$ and $\|D\phi _i\|\leq \unicode{x3bb} $ for $i=1,2,$ such that $\widetilde {H}=\Phi _x(H)$ , where

$$ \begin{align*}H=\{(u,v)\in I^{d_u}\times I^{d_s}: v=t\phi_1(u)+(1-t)\phi_2(u), 0\leq t\leq1 \}.\end{align*} $$

Similarly, define an admissible s-rectangle in  $R(x)$ .

The following lemma is a simplified statement of [Reference Katok and Hasselblatt17, Theorem S.4.16].

3.3 Lyapunov norm for cocycles

We now establish some preliminary results for cocycles. Let f be a $C^r$ diffeomorphism of a compact Riemannian manifold M with $r>1$ , and $\mu $ be an ergodic hyperbolic measure for f with $h_{\mu }(f)>0$ . Let $A:M\to L(\mathfrak {X})$ be an $\alpha $ -Hölder continuous map such that $\unicode{x3bb} (\mathcal {A},\mu )>\kappa (\mathcal {A},\mu )$ . Then by the multiplicative ergodic theorem stated in §2.1, there are Lyapunov exponents $\unicode{x3bb} _1>\unicode{x3bb} _2>\cdots $ indexed in $\mathbb {N}_{k_0}$ for some $k_0\leq +\infty .$

Fix any $i\in \mathbb {N}_{k_0}$ , $\varepsilon>0$ , and $x\in \mathcal {R}^{\mathcal {A}}$ . We define the Lyapunov norm $\|\cdot \|_x=\|\cdot \|_{x,i,\varepsilon }$ on ${\mathfrak {X}}$ as follows.

For any $u=u_1+\cdots +u_{i+1}\in \mathfrak {X},$ where $u_j\in E_j(x)~ \text { for all } j=1,\ldots ,i, $ and $ u_{i+1}\in F_i(x)$ , we define

(3.1) $$ \begin{align} \|u\|_x:=\sum_{j=1}^{i+1} \|u_j\|_x, \end{align} $$

where

and

(3.2)

Then the following lemma holds.

Proof. (i) We will prove the inequality

the others can be proved analogously. By the definition, we have

(ii) For any $u=u_1+\cdots +u_{i+1}\in {\mathfrak {X}}$ , by the definition,

$$ \begin{align*} \|u\|\leq \|u_1\|+\cdots+\|u_{i+1}\|\leq \|u_1\|_x+\cdots+\|u_{i+1}\|_x=\|u\|_x.\end{align*} $$

This estimates the lower bound. To estimate the upper bound, we define

Then,

(3.7) $$ \begin{align} \|u\|_x &\leq \sum_{j=1}^{i}\sum_{n=-\infty}^{+\infty}M_{j}(x)e^{-({1}/{2})\varepsilon |n|}\cdot \|u_j\| + \sum_{n=0}^{+\infty}M_{i+1}(x)e^{-({1}/{2})\varepsilon n}\cdot\|u_{i+1}\| \nonumber\\ & \leq c_0 \cdot \bigg(\sum_{j=1}^{i}M_{j}(x)\cdot\|\pi^{j}_{E}(x)-\pi^{j-1}_{E}(x)\|\cdot\|u\|+M_{i+1}(x)\cdot\|\pi^{i}_{F}(x)\|\cdot\|u\|\bigg)\nonumber\\ & =: M(x)\cdot\|u\|, \end{align} $$

where $c_0=\sum _{n=-\infty }^{+\infty }e^{-(1/2)\varepsilon |n|}$ and $\pi ^{0}_{E}(x)=0$ .

Claim. $M(x)$ is tempered on an f-invariant subset of $\mathcal {R}^{\mathcal {A}}$ with $\mu $ -full measure, that is,

$$ \begin{align*}\lim\limits_{n\to\pm\infty}\frac{1}{n}\log {M}(f^nx)=0\quad \text{for}~ \mu\text{-almost every (a.e.)} ~x\in \mathcal{R}^{\mathcal{A}}. \end{align*} $$

Proof of the claim

Since $\|\pi ^j_{E}(x)\|,\|\pi ^j_{F}(x)\|$ are tempered by equation (2.1), it is enough to prove $M_{j}(x)$ is tempered for any $ 1\leq j\leq i+1$ . Since

where $c_1=\max \nolimits _{x\in M}{\|\mathcal {A}_x\|}/{e^{\unicode{x3bb} _{i+1}-(1/2)\varepsilon }}$ , we obtain

$$ \begin{align*}(\log M_{i+1}(x)-\log M_{i+1}(fx))^+ \leq \log \max\bigg\{c_1,\frac{1}{ M_{i+1}(fx)} \bigg\}\leq\log \max\{c_1,1\},\end{align*} $$

and then we conclude $M_{i+1 }(x)$ is tempered on a subset of full measure by [Reference Mañé22, Lemma III.8]. The result for $ 1\leq j\leq i$ is obtained similarly.

Let

$$ \begin{align*}K(x):= \sum_{n\in\mathbb{Z}} {M}(f^nx)e^{-\varepsilon|n|} \quad \text{for all } x\in\mathcal{R}^{\mathcal{A}}, \end{align*} $$

and then by [Reference Barreira and Pesin3, Lemma 3.5.7], $K(x)$ satisfies equations (3.5) and (3.6). This completes the proof of the lemma.

Fix any $i\in \mathbb {N}_{k_0}$ . Then by Lusin’s theorem, for any $\delta>0,$ there exists a compact subset $\mathcal {R}^{\mathcal {A}}_{\delta }\subset \mathcal {R}^{\mathcal {A}}$ such that $\mu (\mathcal {R}^{\mathcal {A}}_{\delta })>1-\delta $ and $K(x)$ is continuous on $\mathcal {R}^{\mathcal {A}}_\delta $ . Denote

(3.8) $$ \begin{align} l=l_\delta=\sup\limits\{K(x):x\in\mathcal{R}^{\mathcal{A}}_{\delta}\}. \end{align} $$

Since the Oseledets decomposition is $\mu $ -continuous, we may assume the Oseledets decomposition is continuous on $\mathcal {R}^{\mathcal {A}}_{\delta }$ .

Let $\widetilde {\mathfrak {X}}$ be the collection of norms on $\mathfrak {X}$ which are equivalent to $\|\cdot \|$ . Then $\widetilde {{\mathfrak {X}}}$ is a metric space with respect to the metric

$$ \begin{align*}\widetilde{D}(\varphi,\psi)=\sup\limits_{u\in {\mathfrak{X}}\setminus\{0\}}\frac{|\varphi(u)-\psi(u)|}{\|u\|}\quad \text{for all } \varphi,\psi\in\widetilde{{\mathfrak{X}}}. \end{align*} $$

We claim that the function $x\mapsto \|\cdot \|_x$ is continuous on $\mathcal {R}^{\mathcal {A}}_{\delta }$ , which will be used in the proof of equations (3.17), (3.18) in Lemma 3.5 and equation (4.9) in Proposition 4.1. Indeed, for any $x\in \mathcal {R}^{\mathcal {A}}_{\delta }, u=u_1+\cdots +u_{i+1}\in {\mathfrak {X}},$ where $u_j\in E_j(x), \text {for all } j\,{=}\,1,\ldots ,i, $ and $ u_{i+1}\in F_i(x)$ , we define

and then by equation (3.7),

$$ \begin{align*} |\|u\|_x-\|u\|_{k,x}| & \leq \sum_{|n|>k}e^{-({1}/{2})\varepsilon |n|}\cdot M(x)\cdot\|u\| \\ & \leq \frac{2e^{-({\varepsilon}/{2})k}}{e^{({\varepsilon}/{2})}-1}\cdot K(x)\cdot\|u\| \\ & \leq \frac{2l\cdot e^{-({\varepsilon}/{2})k}}{ (e^{({\varepsilon}/{2})}-1)}\cdot\|u\|, \end{align*} $$

that is,

$$ \begin{align*}\widetilde{D}(\|\cdot\|_x,~\|\cdot\|_{k,x})\leq \frac{2l\cdot e^{-({\varepsilon}/{2})k}}{ (e^{{\varepsilon}/{2}}-1)}, \end{align*} $$

which implies $\|\cdot \|_{k,x}\to \|\cdot \|_x$ uniformly for $x\in \mathcal {R}^{\mathcal {A}}_{\delta }$ as $k\to +\infty .$ Since $x\mapsto \|\cdot \|_{k,x}$ is continuous, we have $x\mapsto \|\cdot \|_x$ is continuous on $\mathcal {R}^{\mathcal {A}}_{\delta }$ . This proves the claim.

3.4 Invariant cones for Banach cocycles

We now estimate the growth of vectors in certain invariant cones for cocycles. Let f, $\mu $ , and A be as in §3.3. Fix any $i\in \mathbb {N}_{k_0}$ , and let $\varepsilon _i:=\min \{\tfrac 12\chi \alpha ,{\chi }/{4},{\chi (r-1)}/{2r},({\unicode{x3bb} _1-\unicode{x3bb} _{2}})/{8},\ldots ,({\unicode{x3bb} _{i}-\unicode{x3bb} _{i+1}})/{8} \}$ , where $\chi $ is given in §3.1. For any $0<\varepsilon < \varepsilon _i, x\in \mathcal {R}^{\mathcal {A}},$ let $\|\cdot \|_x=\|\cdot \|_{x,i,\varepsilon }$ be the Lyapunov norm on $\mathfrak {X}$ defined in equation (3.1). For any $1\leq j\leq i$ , we have the Oseledets decomposition $\mathfrak {X}=H_j(x)\oplus F_j(x)$ , where $H_j(x)=E_1(x)\oplus \cdots \oplus E_j(x)$ . For any $\theta>0$ , $u\in \mathfrak {X},$ let $u=u_H+u_F$ , where $u_H\in H_j(x), u_F\in F_j(x)$ . We consider two cones:

$$ \begin{align*}U_j(x,\theta):=\{u\in \mathfrak{X}:\|u_F\|_x\leq \theta \|u_H\|_x \}; \end{align*} $$

$$ \begin{align*}V_j(x,\theta):=\{u\in \mathfrak{X}:\|u_H\|_x\leq \theta \|u_F\|_x \}. \end{align*} $$

For any $\delta>0,$ recall that a sequence $(x_n)_{n\in \mathbb {Z}}$ is a $\rho $ -pseudo-orbit of $f^m$ in $\mathcal {R}^{\mathcal {A}}_\delta $ for some $ m \in \mathbb {N}$ , if $x_n,f^m(x_n)\in \mathcal {R}^{\mathcal {A}}_\delta $ and $d(f^mx_n,x_{n+1})\leq \rho $ for any $n\in \mathbb {Z}$ . Let $l=l_\delta $ be as in equation (3.8). The following lemma is related to a horseshoe $\Lambda ^*$ which is given in §4.1. The definition of $\Lambda ^*$ using Markov rectangles. In the definition of $\Lambda ^*$ , the integer m is fixed once for all, and any point $y\in \Lambda ^*$ is determined by a sequence $(x_n)_{n\in \mathbb {Z}}\in F_m$ , where $F_m\subset \mathcal {R}^{\mathcal {A}}_\delta $ has finite cardinality. More precisely, for any $y\in \Lambda ^*$ , there exists $\{x_n\}_{n\in \mathbb {Z}}\subset F_m$ such that

$$ \begin{align*}d(f^kx_{n},f^{k}(f^{nm}y))\leq \rho\cdot e^{-({\chi}/{2})\min\{k,m-k\}} \quad\text{for all } n\in \mathbb{Z},k=0,\ldots,m.\end{align*} $$

The sequence $\{x_n\}_{n\in \mathbb {Z}}\subset F_m$ is actually a $\rho $ -pseudo-orbit of $f^m$ in $\mathcal {R}^{\mathcal {A}}_\delta $ , and the orbit of $y\in \Lambda ^*$ shadows $\{x_n\}_{n\in \mathbb {Z}}$ .

Proof. (i) For the simplicity of notation, we prove $\mathcal {A}_y^mU_j(x_{0},\theta _0)\subset U_j(x_{1},\eta \theta _0)$ and $ e^{(\unicode{x3bb} _{j}-4\varepsilon )m}\|u\|\leq \|\mathcal {A}_y^m(u)\|\leq e^{(\unicode{x3bb} _1+4\varepsilon )m}\|u\| \text { for all } u\in U_j(x_{0},\theta _0), 1\leq j\leq i.$

For any fixed $0<\varepsilon <\varepsilon _i, \delta>0$ , denote $\theta _0:=e^{\unicode{x3bb} _{i+1}-\unicode{x3bb} _1}(e^\varepsilon -1)<e^\varepsilon -1$ and ${{\eta }_0:=\max \{e^{\unicode{x3bb} _{j+1}-\unicode{x3bb} _j+4\varepsilon }: 1\leq j\leq i\}}$ . We have the following claim.

Claim. There exist $\widetilde {\rho }_0>0,$ such that for any $x_{0},f^mx_{0}\in \mathcal {R}^{\mathcal {A}}_\delta $ , $y\in M$ with ${d(f^kx_{0},f^{k}y)\leq \rho \cdot e^{-({\chi }/{2})\min \{k,m-k\}}}$ for $k=0,\ldots ,m$ and $0<\rho <\widetilde {\rho }_0$ , we have

(3.9) $$ \begin{align} A(f^{k}y)U_j(f^{k}x_{0},\theta_0)\subset U_j(f^{k+1}x_{0},{\eta}_0\theta_0) \quad \text{for all } ~0\leq k\leq m-1. \end{align} $$

Proof of the claim

For any $0\leq k\leq m-1, ~u=u_H+u_F\in U_j(f^{k}x_{0},\theta _0)$ , by equations (3.3) and (3.4), we have

(3.10)

(3.11)

Let $w\kern1.3pt{=}\kern1.3pt(A(f^{k}y)\kern1.3pt{-}\kern1.3pt A(f^kx_{0}))u\kern1.3pt{=}\kern1.3ptw_H+w_F, $ where $w_H\kern1.3pt{\in}\kern1.3pt H_j(f^{k+1}x_{0}), w_F\kern1.3pt{\in}\kern1.3pt F_j(f^{k+1}x_{0})$ . Then

(3.12) $$ \begin{align} A(f^{k}y)u=w+A(f^kx_{0})u. \end{align} $$

Since $A:M\to L(\mathfrak {X})$ is $\alpha $ -Hölder continuous, there exists a constant $c_0>0$ such that

$$ \begin{align*}\|A(z_1)-A(z_2)\|\le c_0d(z_1,z_2)^\alpha\quad \text{for all } z_1,z_2\in M. \end{align*} $$

Since $x_0,f^m(x_0)\in \mathcal {R}^{\mathcal {A}}_\delta $ and $K(x)$ is continuous on $\mathcal {R}^{\mathcal {A}}_\delta $ , by equations (3.5) and (3.8),

$$ \begin{align*} K(f^{k+1}x_{0}) & \le \min\{e^{\varepsilon(k+1)}K(x_0),e^{\varepsilon(m-k-1)}K(f^mx_0) \}\\ & \le le^{\varepsilon\min\{k+1,m-k-1 \}}. \end{align*} $$

Then by equation (3.6),

(3.13) $$ \begin{align} \|w_H\|_{f^{k+1}x_{0}}\leq \|w\|_{f^{k+1}x_{0}} & \leq K(f^{k+1}x_{0})\|A(f^{k}y)-A(f^kx_{0})\|\cdot\|u\| \nonumber\\ & \leq le^{\varepsilon\min\{k+1,m-k-1 \}}\cdot c_0\rho^\alpha e^{-({\chi}/{2})\alpha\min\{k,m-k \}}\|u\|\nonumber\\ & \leq c_0le^\varepsilon\rho^\alpha e^{(\varepsilon-({\chi}/{2})\alpha)\min\{k,m-k \}}\|u\|_{f^{k}x_{0}}\nonumber\\ & \leq (1+\theta_0)c_0le^\varepsilon\rho^\alpha \|u_H\|_{f^{k}x_{0}}, \end{align} $$

as $\varepsilon -({\chi }/{2})\alpha <0.$ Similarly,

(3.14) $$ \begin{align} \|w_F\|_{f^{k+1}x_{0}}\leq (1+\theta_0)c_0le^\varepsilon\rho^\alpha\|u_H\|_{f^{k}x_{0}}. \end{align} $$

Let

$$ \begin{align*}A(f^{k}y)u=(A(f^{k}y)u)_H+(A(f^{k}y)u)_F,\end{align*} $$

where $(A(f^{k}y)u)_H\in H_j(f^{k+1}x_{0}), (A(f^{k}y)u)_F\in F_j(f^{k+1}x_{0})$ . Then by equations (3.10), (3.12), and (3.13),

(3.15)

if $\rho $ is small enough. Similarly, by equations (3.11), (3.12), and (3.14),

(3.16)

if $\rho $ is small enough. Thus,

$$ \begin{align*}\|(A(f^{k}y)u)_F\|_{f^{k+1}x_{0}}\leq {\eta}_0\theta_0 \|(A(f^{k}y)u)_H\|_{f^{k+1}x_{0}},\end{align*} $$

that is, $A(f^{k}y)U_j(f^{k}x_{0},\theta _0)\subset U_j(f^{k+1}x_{0},{\eta }_0\theta _0)$ .

The claim implies

$$ \begin{align*} \mathcal{A}_y^mU_j(x_{0},\theta_0)\subset U_j(f^mx_{0},{\eta}_0\theta_0)\quad\text{for all } 1\leq j \leq i. \end{align*} $$

Since, by §3.4, the Lyapunov norm and the Oseledets decomposition are uniformly continuous on the compact set $\mathcal {R}^{\mathcal {A}}_\delta $ , there exists ${\eta }_0<\eta <1$ such that

(3.17) $$ \begin{align} U_j(f^mx_{0},\eta_0\theta_0)\subset U_j(x_{1},\eta\theta_0) \quad \text{for all } d(f^mx_{0},x_{1})\leq \rho, \end{align} $$

(3.18) $$ \begin{align} V_j(x_{1},\theta_0)\subset V_j\bigg(f^mx_{0},\frac{\eta}{\eta_0}\theta_0\bigg) \quad \text{for all } d(f^mx_{0},x_{1})\leq \rho, \end{align} $$

if $\rho $ is small enough. Hence, for any $1\leq j \leq i,$ we have

(3.19) $$ \begin{align} \mathcal{A}_y^mU_j(x_{0},\theta_0)\subset U_j(x_{1},\eta\theta_0). \end{align} $$

Moreover, for any $0\leq k\leq m-1$ and $u\in U_j(f^{k}x_{0},\theta _0)$ , it follows from equation (3.15) that

Therefore, for any $1\leq j\leq i$ and $u\in U_j(x_0,\theta _0)$ , by equations (3.9) and (3.5), we conclude

This is the first time that the estimate on l is used. Similar to equation (3.15), we can also get

Thus, we obtain by using equation (3.16) that

Hence, for any $u\in U_j(x_0,\theta _0)$ , by equations (3.9) and (3.5), we conclude

This proves the conclusion (i).

(ii) For simplicity, we only prove $\mathcal {A}_{f^my}^{-m}V_j(x_{1},\theta _0)\subset V_j(x_{0},\eta \theta _0)$ and $\|\mathcal {A}_y^m(v)\|\leq e^{(\unicode{x3bb} _{j+1}+4\varepsilon )m}\|v\| \text { for all } v\in \mathcal {A}_{f^my}^{-m}V_j(x_{1},\theta _0), 1\leq j\leq i.$

Let $\theta _1=({\eta }/{\eta _0})\theta _0$ , where $\eta \in (\eta _0,1)$ is given by equation (3.17). We have the following claim.

Claim. There exists $\bar {\rho }_0>0,$ such that for any $x_{0},f^mx_{0}\in \mathcal {R}^{\mathcal {A}}_\delta $ , $y\in M$ with $d(f^kx_{0},f^{k}y)\leq \rho \cdot e^{-({\chi }/{2})\min \{k,m-k\}}$ for $k=0,\ldots ,m$ and $0<\rho <\bar {\rho }_0$ , we have

(3.20) $$ \begin{align} {A}({f^{k-1}y})^{-1}V_j(f^{k}x_{0},\theta_1)\subset V_j(f^{k-1}x_{0},{\eta}_0\theta_1) \quad \text{for all } ~1\leq k\leq m. \end{align} $$

Proof of the claim

For any $1\le k\le m, v\in {A}({f^{k-1}y})^{-1}V_j(f^{k}x_{0},\theta _1),$ one has $A(f^{k-1}y)v\in V_j(f^{k}x_{0},\theta _1).$ Let

$$ \begin{align*}A(f^{k-1}y)v=(A(f^{k-1}y)v)_H+(A(f^{k-1}y)v)_F\in H_j(f^kx_0)\oplus F_j(f^kx_0), \end{align*} $$

and

$$ \begin{align*}v=v_H+v_F\in H_j(f^{k-1}x_0)\oplus F_j(f^{k-1}x_0).\end{align*} $$

Denote $w:=(A(f^{k-1}y)-A(f^{k-1}x_0) )v.$ Since $A(f^{k-1}y)v=A(f^{k-1}x_0)v+w,$ one has

$$ \begin{align*}(A(f^{k-1}y)v)_H=A(f^{k-1}x_0)v_H+w_H,~(A(f^{k-1}y)v)_F=A(f^{k-1}x_0)v_F+w_F. \end{align*} $$

Similar to the proof of equations (3.10), (3.11), (3.13), and (3.14), we can obtain

and

$$ \begin{align*} \begin{split} \|w_H\|_{f^{k}x_{0}}\leq \|w\|_{f^{k}x_{0}} & \leq K(f^{k}x_{0})\|A(f^{k-1}y)-A(f^{k-1}x_{0})\|\cdot\|v\| \\ & \leq le^{\varepsilon\min\{k,m-k \}}\cdot c_0\rho^\alpha e^{-({\chi}/{2})\alpha\min\{k-1,m-k+1 \}}\|v\|\\ & \leq c_0le^\varepsilon\rho^\alpha e^{(\varepsilon-({\chi}/{2})\alpha)\min\{k-1,m-k+1 \}}\|v\|_{f^{k-1}x_{0}}\\ & \leq c_0le^\varepsilon\rho^\alpha \|v\|_{f^{k-1}x_{0}}, \end{split} \end{align*} $$

$$ \begin{align*}\|w_F\|_{f^{k}x_{0}}\leq \|w\|_{f^{k}x_{0}}\le c_0le^\varepsilon\rho^\alpha \|v\|_{f^{k-1}x_{0}}. \end{align*} $$

Therefore,

(3.21)

and

(3.22)

Note that $A(f^{k-1}y)v\in V_j(f^{k}x_{0},\theta _1).$ It follows that

$$ \begin{align*} \|(A(f^{k-1}y)v)_H\|_{f^{k}x_{0}}\le \theta_1 \|(A(f^{k-1}y)v)_F\|_{f^{k}x_{0}}. \end{align*} $$

Hence, by equations (3.21) and (3.22),

if $\rho $ is small enough. Thus,

that is, $v\in V_j(f^{k-1}x_{0},{\eta }_0\theta _1)$ .

This claim implies

$$ \begin{align*} \mathcal{A}_{f^my}^{-m}V_j(f^mx_{0},\theta_1)\subset V_j(x_{0},{\eta}_0\theta_1)=V_j(x_{0},{\eta}\theta_0)~\text{ for all } 1\leq j \leq i. \end{align*} $$

By equation (3.18), we obtain

$$ \begin{align*} \mathcal{A}_{f^my}^{-m}V_j(x_{1},\theta_0)\subset V_j(x_{0},{\eta}\theta_0)\quad\text{for all } 1\leq j \leq i. \end{align*} $$

Now, for any $v\in \mathcal {A}_{f^my}^{-m} V_j(x_{1},\theta _0)$ , one has

$$ \begin{align*}w:=\mathcal{A}_y^k(v)\in \mathcal{A}_{f^my}^{-m+k} V_j(x_{1},\theta_0)\subset \mathcal{A}_{f^my}^{-m+k} V_j(f^mx_{0},\theta_1)\subset V_j(f^kx_{0},\theta_0).\end{align*} $$

Let $w=w_H+w_F,$ where $w_H\in H_j(f^kx_{0}),w_F\in F_j(f^kx_{0})$ . Then,

It follows that

Similar to equation (3.13), we can obtain $\|A(f^ky)w-A(f^kx_0)w\|_{f^{k+1}x_0}\leq c_0le^{\varepsilon }\rho ^\alpha \|w\|_{f^{k}x_0}$ . Therefore,

if $\rho $ is small enough. It implies that for any $v\in \mathcal {A}_{f^my}^{-m} V_j(x_{1},\theta _0)$ ,

This completes the proof of the lemma.

4.1 Construction of hyperbolic horseshoes

The aim of this subsection is to construct a hyperbolic horseshoe $\Lambda $ satisfying the properties listed in Theorem 2.2. This construction resembles Theorem S.5.9 in [Reference Katok and Hasselblatt17].

We begin with producing a separated set with sufficiently large cardinality. For $n\geq 1,$ denote by $d_n(x,y)=\max _{0\leq k\leq n-1}d(f^kx,f^ky)$ the dynamical distance on M, and denote by $B_n(x,\rho )=\{y\in M: d_n(x,y)\leq \rho \}$ the $d_n$ -balls of radius $\rho $ . Let $N_\mu (n,\rho ,\bar {\delta })$ be the minimal numbers of $d_n$ -balls of radius $\rho $ whose union has measure at least $\bar {\delta }.$ Then by [Reference Katok16, Theorem 1.1], for any $\bar {\delta }>0,$

$$ \begin{align*}h_\mu(f)=\lim\limits_{\rho\to 0}\liminf\limits_{n\to+\infty}\frac{1}{n}\log N_\mu(n,\rho,\bar{\delta}).\end{align*} $$

Given any $0<\delta <1/2,$ let $\Lambda _\delta =\mathcal {R}^{Df}_\delta \cap \mathcal {R}^{\mathcal {A}}_\delta \cap \text {~supp}(\mu ).$ Then, $\mu (\Lambda _\delta )>1-2\delta >0.$ Take $\bar {\delta }=({1}/{2})\mu (\Lambda _\delta ),$ and then for any given $i\in \mathbb {N}_{k_0}, 0<\varepsilon <\varepsilon _i$ , there exist $0<\rho _1<\varepsilon /2, N>1,$ such that for any $0<\rho <\rho _1,n\geq N,$ one has

(4.1) $$ \begin{align} N_\mu(n,\rho,\bar{\delta})\geq e^{(h_\mu(f)-\varepsilon)n}. \end{align} $$

Fix a dense subset $\{\varphi _j\}_{j= 1}^\infty $ of the unit sphere of $C(M)$ . We consider the induced metric on $\mathcal {M}_f(M)$ :

$$ \begin{align*}D(\mu_1,\mu_2)= \sum\limits_{j=1}^{\infty}\frac{|\int \varphi_j\,d\mu_1-\int \varphi_j\,d\mu_2|}{2^j}\quad \text{for all } \mu_1,\mu_2\in \mathcal{M}_f(M).\end{align*} $$

Take J large enough such that ${1}/{2^{J}}<{\varepsilon }/{8}$ , and take $\rho <\min \{\rho _0,\rho _1/2\}$ small enough (where $\rho _0$ is given by Lemma 3.5) such that

(4.2) $$ \begin{align} |\varphi_j(x)-\varphi_j(y)|\leq \varepsilon/4\quad \text{for all } d(x,y)\leq \rho,~ j=1,\ldots,J. \end{align} $$

Since $\Lambda _\delta \subset \mathcal {R}^{Df}_\delta $ , by Lemma 3.3, there exists $0<\beta <\tfrac {1}{3}\rho $ and finite $(\rho , \beta ,{\chi }/{2})$ -rectangles $R(q_1),\ldots ,R(q_t)$ , such that $\bigcup _{j=1}^tB(q_j,\beta )\supset \Lambda _\delta $ . We consider a partition $\mathcal {P}=\{P_1,\ldots ,P_t \}$ of $\Lambda _\delta $ , where

$$ \begin{align*}P_1=B(q_1,\beta)\cap\Lambda_\delta\quad\text{and}\quad P_k=B(q_k,\beta)\cap\Lambda_\delta\setminus \bigg(\bigcup_{j=1}^{k-1}P_j\bigg) \quad\text{for all } 2\leq k\leq t.\end{align*} $$

Let

$$ \begin{align*} \Lambda_{\delta,n} := \bigg\{ & x\in \Lambda_\delta: \text{there exists~} k\in[n, (1+\varepsilon)n]\text{~such that~} f^k(x)\in \mathcal{P}(x),\\&\text{~the collection~}\{f^k(x)\}_{1\leq k\leq n} \text{~is~} \varepsilon/2\text{-dense in supp}(\mu), \text{ and~}\\ & \bigg|\frac{1}{m}\sum\limits_{p=0}^{m-1}\varphi_j(f^px)-\int \varphi_j \,d\mu\bigg|\leq \frac{\varepsilon}{4} \text{ for all } m\geq n, 1\leq j\leq J \bigg\}. \end{align*} $$

Proof of the claim

Let

$$ \begin{align*}A_n =\{ x\in \Lambda_\delta: \text{there exists~} k\in[n, (1+\varepsilon)n]\text{~such that~} f^k(x)\in \mathcal{P}(x)\},\end{align*} $$

and $A_{n,j} =\{ x\in P_j: \text {there exists~} k\in [n, (1+\varepsilon )n]\text {~such that~} f^k(x)\in P_j\}.$ Then, ${A_n=\bigcup _{j=1}^tA_{n,j}}$ . Considering $P_1$ , we may assume $\mu (P_1)>0$ . For any $\tau>0,$ let

$$ \begin{align*}A_{n,1}^{\tau}=\{x\in P_1: \mu(P_1)-\tau\leq\frac{1}{m}\sum_{j=0}^{m-1}\chi_{P_1}(f^jx)\leq \mu(P_1)+\tau \text{ for all } m\geq n \}. \end{align*} $$

Then the Birkhoff ergodic theorem gives $\mu (\bigcup _{n\geq 1}A_{n,1}^{\tau })=\mu (P_1)$ . Since $A_{1,1}^{\tau }\subset A_{2,1}^{\tau }\subset \cdots ,$ we have

$$ \begin{align*}\lim\limits_{n\to\infty}\mu(A_{n,1}^{\tau})=\mu(P_1). \end{align*} $$

Take $\tau <{\varepsilon }/({2+\varepsilon })\mu (P_1)$ . Then for any $x\in A_{n,1}^{\tau }$ , by the definition of $ A_{n,1}^{\tau }$ ,

$$ \begin{align*} \text{card}\{k\in [n,n+n\varepsilon]:f^k(x)\in P_1 \} & \geq (\mu(P_1)-\tau)(n+n\varepsilon)-(\mu(P_1)+\tau)n\\ & = n(\mu(P_1)\varepsilon-2\tau-\tau\varepsilon)\\ & \geq 1, \end{align*} $$

if n is taken large enough. Thus, $x\in A_{n,1}$ , that is, $A_{n,1}^{\tau }\subset A_{n,1}$ . Therefore, $\lim \nolimits _{n\to \infty } \mu (A_{n,1})=\mu (P_1).$ Reproduce the proof above for every $P_j$ . Then we conclude

(4.3) $$ \begin{align} \lim\limits_{n\to\infty}\mu(A_n)=\mu(\Lambda_{\delta}). \end{align} $$

Since, by [Reference Walters29, Theorem 1.7], $\{x\in M:\overline {\mathcal {O}^+(x)}=\text {supp}(\mu ) \}$ has $\mu $ -full measure, letting

$$ \begin{align*}B_n=\{x\in \Lambda_\delta:\{f^k(x)\}_{1\leq k\leq n} ~\text{is} ~\varepsilon/2\text{-dense in supp}(\mu)\},\end{align*} $$

we have $\mu (B_n)\to \mu ( \Lambda _\delta )$ as $n\to +\infty .$ Let

$$ \begin{align*}C_n=\bigg\{x\in \Lambda_\delta:\bigg|\frac{1}{m}\sum\limits_{k=0}^{m-1}\varphi_j(f^kx)-\int \varphi_j \,d\mu\bigg|\leq \frac{\varepsilon}{4} \text{ for all } m\geq n, 1\leq j\leq J \bigg \}.\end{align*} $$

Then by Birkhoff ergodic theorem, $\mu (\bigcup _{n\geq 1}C_n)=\mu (\Lambda _{\delta })$ . Since $C_1\subset C_2\subset \cdots ,$ one has

$$ \begin{align*}\lim\limits_{n\to\infty}\mu(C_n)=\mu(\Lambda_{\delta}). \end{align*} $$

Together with equation (4.3),

$$ \begin{align*}\lim\limits_{n\to\infty}\mu(\Lambda_{\delta,n})=\lim\limits_{n\to\infty}\mu(A_n\cap B_n\cap C_n) =\mu(\Lambda_{\delta}).\end{align*} $$

This proves the claim.

Choose $n>\max \{ {2\log l}/{\varepsilon }, N, {1}/{\varepsilon }\log t,{1}/{4 \varepsilon } \}$ large enough such that $\mu (\Lambda _{\delta ,n})>\tfrac {1}{2}\mu (\Lambda _{\delta })=\bar {\delta },$ and $n\varepsilon +1<e^{\varepsilon n}$ , where the constant $ ({2\log l}/{\varepsilon })$ comes from Lemma 3.5. Denote by E an $(n,2\rho )$ -separated set of $\Lambda _{\delta ,n}$ of maximum cardinality. Then $\bigcup _{x\in E}B_n(x,2\rho )\supset \Lambda _{\delta ,n}.$ By equation (4.1),

$$ \begin{align*} \text{card}(E)\geq N_\mu(n,2\rho,\mu(\Lambda_{\delta,n}))\geq N_\mu(n, 2\rho,\bar{\delta})\geq e^{(h_\mu(f)-\varepsilon)n}. \end{align*} $$

For $ k\in [n, n(1+\varepsilon )]$ , let $F_k=\{x\in E:f^k(x)\in \mathcal {P}(x)\}$ and take $m\in [n, n(1+\varepsilon )]$ satisfying $\text {card}(F_m)=\max \{\text {card}(F_k):n\leq k\leq n(1+\varepsilon ) \}.$ Then,

$$ \begin{align*}\text{card}(F_m)\geq \frac{1}{n\varepsilon+1}\text{card}(E)\geq \frac{1}{n\varepsilon+1}e^{(h_\mu(f)-\varepsilon)n}\geq e^{(h_\mu(f)-2\varepsilon)n}.\end{align*} $$

Choose $P\in \mathcal {P}$ satisfying $\text {card}(F_m\cap P)=\max \{\text {card}(F_m\cap P_k):1\leq k\leq t\}.$ Then,

$$ \begin{align*}\text{card}(F_m\cap P)\geq \frac{1}{t}\text{card}(F_m)\geq \frac{1}{t}e^{(h_\mu(f)-2\varepsilon)n}.\end{align*} $$

Without loss of generality, we may assume

(4.4) $$ \begin{align} \frac{1}{t}e^{(h_\mu(f)-2\varepsilon)n}\leq \text{card}(F_m\cap P)\leq e^{(h_\mu(f)+2\varepsilon)n}. \end{align} $$

Otherwise, since $e^{(h_\mu (f)+2\varepsilon )n}-({1}/{t})e^{(h_\mu (f)-2\varepsilon )n}\ge e^{(h_\mu (f)-2\varepsilon )n}(e^{4\varepsilon n}-1)>1$ , we can choose a subset of $F_m\cap P$ such that equation (4.4) holds.

By the definition of the partition $\mathcal {P}$ , there exists $q\in \{q_1,\ldots ,q_t\}$ , such that ${P\subset B(q,\beta )\cap \Lambda _\delta }$ . Thus, for any $x\in F_m\cap P,$ since $x,f^m(x)\in B(q,\beta )\cap \Lambda _\delta ,$ by Definition 3.2, the connected component $C(x,R(q)\cap f^{-m}R(q))$ of $R(q)\cap f^{-m}R(q)$ containing x is an admissible s-rectangle in $R(q)$ , and $f^mC(x,R(q)\cap f^{-m}R(q))$ is an admissible u-rectangle in $R(q)$ .

We claim that if $x_1,x_2\in F_m\cap P$ with $x_1\neq x_2,$ then $C(x_1,R(q)\cap f^{-m}R(q))\cap C(x_2,R(q)\cap f^{-m}R(q))=\varnothing $ . Indeed, if there is $y\in C(x_1,R(q)\cap f^{-m}R(q))\cap C(x_2, R(q)\cap f^{-m}R(q)),$ by Definition 3.2, one sees $d(f^kx_j,f^ky)\leq \rho $ for any ${0\leq k\leq m}$ and $j=1,2$ . Thus, $d_m(x_1,x_2)\leq 2\rho $ . However, since $F_m\cap P$ is an $(n,2\rho )$ -separated set, we obtain $d_m(x_1,x_2)\geq d_n(x_1,x_2)>2\rho $ , which is a contradiction. Therefore, there are at least $\text {card}( F_m\cap P)$ disjoint s-rectangles in $R(q)$ , mapped by $f^m$ to $\text {card}(F_m\cap P)$ disjoint admissible u-rectangles in $R(q)$ .

Let

$$ \begin{align*}\Lambda^*=\bigcap\limits_{n\in\mathbb{Z}}f^{-mn}\bigg(\bigcup\limits_{x\in F_m\cap P}C(x,R(q)\cap f^{-m}R(q)) \bigg).\end{align*} $$

Then $f^m|_{\Lambda ^*}$ is conjugate to a full shift in $\text {card}(F_m\cap P)$ -symbols. Moreover, for any ${y\in \Lambda ^*,}$ any $n\in \mathbb {Z}$ , there exists $x_n\in F_m\cap P$ such that $f^{mn}(y)\in C(x_n,R(q)\cap f^{-m}R(q)).$ Note that the orbit of $y\in \Lambda ^*$ remains in the union of $R(q),\ldots ,R(f^m(q))$ . Let

$$ \begin{align*}\Lambda=\Lambda^*\cup f(\Lambda^*)\cup\cdots\cup f^{m-1}(\Lambda^*).\end{align*} $$

Then $\Lambda $ is a hyperbolic horseshoe.

It remains to show the conclusions (i)–(iv) of Theorem 2.2 hold for this $\Lambda $ .

(i) Since

$$ \begin{align*}h_{\mathrm{top}}(f|_{\Lambda})=\frac{1}{m}h_{\text {top}}(f^m|_{\Lambda^*})=\frac{1}{m}\log \text{card}(F_m\cap P),\end{align*} $$

by equation (4.4), $h_{\mathrm{top}}(f|_{\Lambda })\geq -{1}/{m}\log t + {n}/{m}(h_\mu (f)-2\varepsilon ).$ Since $({1}/{\varepsilon })\log t< n\leq m\leq n(1+\varepsilon )$ ,

$$ \begin{align*}h_{\mathrm{top}}(f|_{\Lambda})\geq -\varepsilon+\frac{1}{1+\varepsilon}(h_\mu(f)-2\varepsilon)\geq h_\mu(f)-( h_\mu(f)+3)\varepsilon.\end{align*} $$

By equation (4.4), we also have

$$ \begin{align*} h_{\mathrm{top}}(f|_{\Lambda})\leq \frac{n}{m}(h_\mu(f)+2\varepsilon)\leq h_\mu(f)+2\varepsilon.\end{align*} $$

(ii) By the construction of $\Lambda $ and the definition of $R(q)$ , for any $y\in \Lambda ,$ there exist $x\in F_m\cap P, 0\leq k\leq m-1$ such that $d(y,f^kx)\leq \rho <\varepsilon /2.$ Since $x\in \Lambda _\delta \subset \text {~supp}(\mu ),$ we conclude $\Lambda $ is contained in an $\varepsilon /2$ -neighborhood of supp $(\mu )$ . However, fix any $x_0\in F_m\cap P\subset \Lambda _{\delta ,m}$ . By the construction of $\Lambda _{\delta ,m}$ , for any $z\in \text {~supp}(\mu )$ , there exists $1\leq k\leq m$ , such that $d(z,f^kx_0)\leq \varepsilon /2$ . Take $y\in \Lambda ^*$ such that $y\in C(x_0,R(q)\cap f^{-m}R(q))$ , so then $d(f^kx_0,f^ky)\leq \rho <\varepsilon /2$ . Therefore, supp $(\mu )$ is contained in an $\varepsilon $ -neighborhood of  $\Lambda $ . Hence, $d_H(\Lambda ,\text {~supp}(\mu ))\leq \varepsilon .$

(iii) For any f-invariant measure $\nu $ supported on $\Lambda $ , we may assume $\nu $ is ergodic first. Since

$$ \begin{align*}D(\mu,\nu)\leq \sum\limits_{j=1}^{J}\frac{|\int \varphi_j\,d\mu-\int \varphi_j\,d\nu|}{2^j}+\frac{1}{2^{J-1}}\leq \sum\limits_{j=1}^{J}\frac{|\int \varphi_j\,d\mu-\int \varphi_j\,d\nu|}{2^j}+\frac{\varepsilon}{4},\end{align*} $$

it is enough to show: $|\int \varphi _j\,d\mu -\int \varphi _j\,d\nu |\leq \tfrac {3}{4}\varepsilon \text { for all } 1\leq j\leq J.$ Take $y\in \Lambda ^*$ and $s{\in \mathbb {N}}$ large enough such that

$$ \begin{align*}\bigg|\frac{1}{ms}\sum\limits_{k=0}^{ms-1}\varphi_j(f^ky)-\int \varphi_j \,d\nu\bigg|\leq \frac{\varepsilon}{4},\quad 1\leq j\leq J.\end{align*} $$

Then there exist $x_0,x_1,\ldots ,x_{s-1}\in F_m\cap P$ such that

$$ \begin{align*}d(f^{mk+t}y,f^tx_k)\leq \rho\quad\text{for all } 0\leq k \leq s-1, 0\leq t\leq m-1.\end{align*} $$

By equation (4.2) and the construction of $\Lambda _{\delta ,m}$ , we obtain $|\int \varphi _jd\mu -\int \varphi _jd\nu |\leq \tfrac {3}{4}\varepsilon $ , $1\leq j\leq J,$ which implies $D(\mu ,\nu )\leq \varepsilon $ .

If $\nu $ is not ergodic, by the ergodic decomposition theorem, $\nu $ -a.e. ergodic component is supported on $\Lambda $ . Hence,

$$ \begin{align*} D(\mu,\nu) = \sum\limits_{j=1}^{\infty}\frac{|\int \varphi_j\,d\mu-\int \varphi_j\,d\nu|}{2^j} & = \sum\limits_{j=1}^{\infty}\frac{|\int(\int \varphi_j\,d\mu-\int \varphi_j\,d\nu_x)\,d\nu(x)|}{2^j} \\ &\leq \int D(\mu,\nu_x) \,d\nu(x) \\ & \leq \varepsilon. \end{align*} $$

4.2 Dominated splitting for cocycles

The conclusion of Theorem 2.2 is contained in the following proposition.

Proposition 4.1. Under the condition of Theorem 2.2, for any $i\in \mathbb {N}_{k_0}$ and any $0<\varepsilon <\varepsilon _i$ , let $\Lambda $ be constructed as above. Then there exists a continuous $\mathcal {A}$ -invariant splitting on $\Lambda $

$$ \begin{align*}\mathfrak{X}=E_1(y)\oplus\cdots\oplus E_i(y)\oplus F_i(y),\end{align*} $$

with $\dim (E_j)=d_j$ for $1\leq j\leq i$ , such that for any $y\in \Lambda $ , one has

To prove this proposition, we require the following definitions. Let $\mathcal {G}(\mathfrak {X})$ denote the Grassmannian of closed subspaces of $\mathfrak {X}$ , endowed with the Hausdorff metric $d_H$ , defined by

$$ \begin{align*}d_H(E,F) = \max\bigg\{\sup\limits_{u\in S_E}\mathrm{dist}(u,S_F),\sup\limits_{v\in S_F}\mathrm{dist}(v,S_E) \bigg\}\quad\text{for all } E,F\in \mathcal{G}(\mathfrak{X}),\end{align*} $$

where $S_F=\{v\in F:\|v\|=1 \}$ , $\mathrm{dist}(u,S_F)=\inf \{\|u-v\|:v\in S_F\}$ . Denote by $\mathcal {G}_j(\mathfrak {X})$ , $\mathcal {G}^j({\mathfrak {X}})$ the Grassmannian of j-dimensional and j-codimensional closed subspaces, respectively. Then by [Reference Kato15, Ch. IV, §2.1], $(\mathcal {G}({\mathfrak {X}}),d_H)$ is a complete metric space, and $\mathcal {G}_j({\mathfrak {X}})$ , $\mathcal {G}^j(\mathfrak {X})$ are closed in $\mathcal {G}(\mathfrak {X})$ . To compute $d_H$ conveniently, we introduce the gap $\hat {\delta }$ , defined by

$$ \begin{align*} \hat{\delta}(E,F) = \max\bigg\{\sup\limits_{u\in S_E}\mathrm{dist}(u,F),\sup\limits_{v\in S_F}\mathrm{dist}(v,E) \bigg\}\quad\text{for all } E,F\in \mathcal{G}({\mathfrak{X}}).\end{align*} $$

Note that the gap is not a metric on $\mathcal {G}(\mathfrak {X})$ , but

$$ \begin{align*}\hat{\delta}(E,F)\leq d_H(E,F)\leq 2\hat{\delta}(E,F).\end{align*} $$

See [Reference Kato15, Ch. IV, §2.1] for a proof.

Before proving Proposition 4.1, we give the following lemma first.

Lemma 4.2. Let $A \in L(\mathfrak {X})$ . If there exists a splitting

(4.5) $$ \begin{align} \mathfrak{X}=A(H)\oplus F \end{align} $$

and $A|_{H}$ is a bijection, then

$$ \begin{align*}\mathfrak{X}=H \oplus A^{-1}(F). \end{align*} $$

We now prove Proposition 4.1.

Proof of Proposition 4.1

Fix any $i\in \mathbb {N}_{k_0}$ , $0<\varepsilon <\varepsilon _i$ , and $1\leq j\leq i$ . We first assume $y\in \Lambda ^*,$ and then by the definition of $\Lambda ^*$ , there exist $\{x_n\}_{n\in \mathbb {Z}}\subset F_m$ , such that

$$ \begin{align*}d(f^kx_{n},f^{k}(f^{nm}y))\leq \rho\cdot e^{-({\chi}/{2})\min\{k,m-k\}}\quad\text{for all } n\in \mathbb{Z},k=0,\ldots,m.\end{align*} $$

For any $n\in \mathbb {Z}$ , let $\mathfrak {X}=H_j(x_{n})\oplus F_j(x_{n})$ be the Oseledets decomposition at $x_n$ , where $H_j(x_{n})=E_1(x_{n})\oplus \cdots \oplus E_j(x_{n})$ . We first prove $\{\mathcal {A}_{f^{-mn}y}^{mn}H_j(x_{-n})\}_{n\geq 1}$ is a Cauchy sequence in $\mathcal {G}(\mathfrak {X})$ . Denote $\widetilde {H}_j(x_{-n})=\mathcal {A}_{f^{-m(n+1)}y}^{m}H_j(x_{-n-1}).$ Then,

$$ \begin{align*}\mathcal{A}_{f^{-mn}y}^{mn}\widetilde{H}_j(x_{-n})=\mathcal{A}_{f^{-m(n+1)}y}^{m(n+1)}H_j(x_{-n-1}).\end{align*} $$

By Lemma 3.5, $\widetilde {H}_j(x_{-n})\subset U_j(x_{-n},\theta _0)$ , $\mathcal {A}_{f^{-mn}y}^{mn}\widetilde {H}_j(x_{-n})\subset U_j(x_{0},\theta _0)$ , and $\mathcal {A}_{f^{-mn}y}^{mn}$ is injective on $ U_j(x_{-n},\theta _0)$ . Thus, we have

(4.6) $$ \begin{align} \mathfrak{X}=\mathcal{A}_{f^{-mn}y}^{mn}\widetilde{H}_j(x_{-n})\oplus F_j(x_0). \end{align} $$

Now, for any $u\in H_j(x_{-n})$ ,

(4.7) $$ \begin{align} & \mathrm{dist}\bigg(\frac{\mathcal{A}_{f^{-mn}y}^{mn}(u)}{\|\mathcal{A}_{f^{-mn}y}^{mn}(u)\|},\mathcal{A}_{f^{-m(n+1)}y}^{m(n+1)}H_j(x_{-n-1})\bigg)\nonumber\\ &\quad= \inf_{v\in \widetilde{H}_j(x_{-n}) } \bigg\|\frac{\mathcal{A}_{f^{-mn}y}^{mn}(u)}{\|\mathcal{A}_{f^{-mn}y}^{mn}(u)\|}- \mathcal{A}_{f^{-mn}y}^{mn}(v) \bigg\|\nonumber\\ &\quad= \frac{\|u\|}{\|\mathcal{A}_{f^{-mn}y}^{mn}(u)\|}\cdot\inf_{v\in \widetilde{H}_j(x_{-n}) }\bigg\|\mathcal{A}_{f^{-mn}y}^{mn}\bigg(\frac{u}{\|u\|}- v\bigg) \bigg\|. \end{align} $$

By equation (4.6), for $\mathcal {A}_{f^{-mn}y}^{mn}({u}/{\|u\|})\in \mathfrak {X}$ , there exists $v_n\in \widetilde {H}_j(x_{-n})$ such that

$$ \begin{align*} \mathcal{A}_{f^{-mn}y}^{mn}\bigg(\frac{u}{\|u\|}\bigg)-\mathcal{A}_{f^{-mn}y}^{mn}(v_n)\in F_j(x_0)\subset V_j(x_0,\theta_0). \end{align*} $$

Then it follows from equation (4.7) and Lemma 3.5 that

Claim. There exists $c=c(\delta )>0$ , such that $\|{u}/{\|u\|}- v_n\|\leq c.$

Proof of the claim

Using equation (4.6) and Lemma 4.2, we obtain

(4.8) $$ \begin{align} \mathfrak{X}=\widetilde{H}_j(x_{-n})\oplus \mathcal{A}_y^{-mn}F_j(x_0). \end{align} $$

Let $\pi _2$ be the projection operator associated with equation (4.8) onto $\mathcal {A}_y^{-mn}F_j(x_0)$ parallel to $\widetilde {H}_j(x_{-n})$ . Then,

$$ \begin{align*}\frac{u}{\|u\|}- v_n=\pi_2\bigg(\frac{u}{\|u\|}\bigg)\in \mathcal{A}_y^{-mn}F_j(x_0).\end{align*} $$

Since the splitting $\mathfrak {X}=H_j(x)\oplus F_j(x)$ is uniformly continuous on $\mathcal {R}^{\mathcal {A}}_\delta $ , there exists ${c=c(\delta )>0}$ such that for any $x\in \mathcal {R}^{\mathcal {A}}_\delta $ and any $H\subset U_j(x,\theta _0), F\subset V_j(x,\theta _0)$ with $\mathfrak {X}=H\oplus F$ , one has

(4.9) $$ \begin{align} \|\pi^H\|\leq c,\|\pi^F\|\leq c, \end{align} $$

where $\pi ^H, \pi ^F$ are the projections associated with the splitting $\mathfrak {X}=H\oplus F$ . Therefore,

$$ \begin{align*}\bigg\|\frac{u}{\|u\|}- v_n\bigg\|=\bigg\|\pi_2\bigg(\frac{u}{\|u\|}\bigg)\bigg\|\leq c. \end{align*} $$

This proves the claim.

The claim gives

Similarly, for any $v\in H_j(x_{-n-1})$ , we have

Thus,

which implies $\{\mathcal {A}_{f^{-mn}y}^{mn}H_j(x_{-n})\}_{n\geq 1}$ is a Cauchy sequence in $(\mathcal {G}(\mathfrak {X}),d_H)$ .

Now we prove $\{\mathcal {A}_{f^{mn}y}^{-mn}F_j(x_{n})\}_{n\geq 1}$ is a Cauchy sequence in $(\mathcal {G}(\mathfrak {X}),d_H)$ . By Lemma 3.5, $\mathcal {A}_{y}^{mn}{H}_j(x_{0})\subset U_j(x_n,\theta _0)$ and $\mathcal {A}_{y}^{mn}$ is injective on ${H}_j(x_{0})$ . Thus, $\mathfrak {X}=\mathcal {A}_{y}^{mn}{H}_j(x_{0})\oplus F_j(x_n).$ Using Lemma 4.2, we obtain

(4.10) $$ \begin{align} \mathfrak{X}={H}_j(x_{0})\oplus \mathcal{A}_{f^{mn}y}^{-mn}F_j(x_n). \end{align} $$

For any $u\in \mathcal {A}_{f^{m(n+1)}y}^{-m(n+1)}F_j(x_{n+1})$ with $\|u\|=1$ , choose $v_n\in \mathcal {A}_{f^{mn}y}^{-mn}F_j(x_n)$ such that

$$ \begin{align*}u-v_n\in {H}_j(x_{0}).\end{align*} $$

Then, by Lemma 3.5,

(4.11)

However, since $u\in \mathcal {A}_{f^{m(n+1)}y}^{-m(n+1)}F_j(x_{n+1})\subset \mathcal {A}_{f^{mn}y}^{-mn}V_j(x_n,\theta _0)$ and $v_n\in \mathcal {A}_{f^{mn}y}^{-mn}F_j(x_n)\subset \mathcal {A}_{f^{mn}y}^{-mn}V_j(x_n,\theta _0)$ , Lemma 3.5 gives

(4.12)

where $\pi ^F$ is the projection associated with equation (4.10) onto $\mathcal {A}_{f^{mn}y}^{-mn}F_j(x_n)$ parallel to $H_j(x_{0})$ and the constant c is given by equation (4.9). Therefore, equations (4.11) and (4.12) show that for any $u\in \mathcal {A}_{f^{m(n+1)}y}^{-m(n+1)}F_j(x_{n+1})$ with $\|u\|=1$ ,

Similarly, for any $v\in \mathcal {A}_{f^{mn}y}^{-mn}F_j(x_{n})$ with $\|v\|=1$ , we have

Thus,

which implies $\{\mathcal {A}_{f^{mn}y}^{-mn}F_j(x_{n})\}_{n\geq 1}$ is a Cauchy sequence in $(\mathcal {G}(\mathfrak {X}),d_H)$ .

Let

$$ \begin{align*} H_j(y):=\lim\limits_{n\to\infty} \mathcal{A}_{f^{-mn}y}^{mn}H_j(x_{-n}), \quad F_j(y):=\lim\limits_{n\to\infty}\mathcal{A}_{f^{mn}y}^{-mn}F_j(x_{n}). \end{align*} $$

Since $\mathfrak {X}= \mathcal {A}_y^{mn}H_j(x_{0})\oplus F_j(x_{n})$ , by Lemma 4.2, $\mathfrak {X}= H_j(x_{0})\oplus \mathcal {A}_{f^{mn}y}^{-mn} F_j(x_{n})$ . Thus,

$$ \begin{align*}\text{ codim}(\mathcal{A}_{f^{mn}y}^{-mn}F_j(x_{n}))=D_j=\dim(\mathcal{A}_{f^{-mn}y}^{mn}H_j(x_{-n})),\end{align*} $$

where $D_j=d_1+\cdots +d_j$ . It follows from $\mathcal {G}_{D_j}(\mathfrak {X})$ and $\mathcal {G}^{D_j}(\mathfrak {X})$ being closed subsets of $\mathcal {G}(\mathfrak {X})$ that $H_j(y)\in \mathcal {G}_{D_j}(\mathfrak {X}),F_j(y)\in \mathcal {G}^{D_j}(\mathfrak {X})$ . Notice that $H_j(y)\subset U_j(x_0,\theta _0)$ and $ F_j(y)\subset \mathcal {A}_{f^{m}y}^{-m}V_j(x_1,\theta _0)\subset V_j(x_0,\theta _0)$ . We conclude

$$ \begin{align*}\mathfrak{X}= H_j(y)\oplus F_j(y).\end{align*} $$

Additionally, by Lemma 3.5, we have

In general, for any $z\in \Lambda ,$ there exists $0\leq k\leq m-1,y\in \Lambda ^*$ such that $z=f^ky.$ Similar to the proof above, we can also get

$$ \begin{align*} H_j(z):=\lim\limits_{n\to\infty} \mathcal{A}_{f^{-mn}(f^ky)}^{mn}H_j(f^kx_{-n}), \end{align*} $$

$$ \begin{align*} F_j(z):=\lim\limits_{n\to\infty}\mathcal{A}_{f^{mn}(f^ky)}^{-mn}F_j(f^kx_{n}), \end{align*} $$

satisfying $\mathfrak {X}= H_j(z)\oplus F_j(z), H_j(z)\in \mathcal {G}_{D_j}(\mathfrak {X}),F_j(z)\in \mathcal {G}^{D_j}(\mathfrak {X})$ , and

(4.13)

(4.14)

Claim. The splitting $ \mathfrak {X}= H_j(z)\oplus F_j(z)$ is $\mathcal {A}$ -invariant on $\Lambda $ .

Proof of the claim

Before proving the invariance of the splitting, we show that for any ${y\in \Lambda ^*,0\leq k\leq m-1}$ and any $H_{n,k}\in \mathcal {G}_{D_j}(\mathfrak {X})$ satisfying $H_{n,k} \subset U_j(f^kx_{-n},\theta _0)$ , one has

$$ \begin{align*}H_j(f^ky)=\lim\limits_{n\to\infty} \mathcal{A}_{f^{-mn}(f^ky)}^{mn}{H_{n,k}}.\end{align*} $$

Indeed, recall that $\mathfrak {X}=\mathcal {A}_{f^{-mn}(f^ky)}^{mn}H_{n,k}\oplus F_j(x_0)$ . Thus for any $u\in H_j(f^kx_{-n})$ , similar to the estimation above, we have

which implies $d_H(\mathcal {A}_{f^{-mn}(f^ky)}^{mn}H_j(x_{-n}),\mathcal {A}_{f^{-mn}(f^ky)}^{mn}{H_{n,k}})\to 0.$ Hence,

(4.15) $$ \begin{align} H_j(f^ky)=\lim\limits_{n\to\infty} \mathcal{A}_{f^{-mn}(f^ky)}^{mn}{H_{n,k}} \quad\text{for all } 0\leq k\leq m-1. \end{align} $$

Now for any $z \in \Lambda ,$ we may assume $z=f^k(y)$ for some $y\in \Lambda ^*$ and $0\leq k\leq m-1$ . If $0\leq k\leq m-2$ , then by equation (3.9), $A(f^{-mn}z)H_j(f^kx_{-n})\subset U_j(f^{k+1}x_{-n},\theta _0)$ ; if $k=m-1$ , then by equation (3.17), $A(f^{-mn}z)H_j(f^{m-1}x_{-n})\subset U_j(x_{-n+1},\theta _0)$ . Thus, equation (4.15) gives

$$ \begin{align*}H_j(fz)=\lim\limits_{n\to\infty} \mathcal{A}_{f^{-mn}(fz)}^{mn}(A(f^{-mn}z)H_j(f^kx_{-n})). \end{align*} $$

Therefore, for any $u\in H_j(z),$

$$ \begin{align*} &\mathrm{dist}(A(z)u,\mathcal{A}_{f^{-mn}(fz)}^{mn}A(f^{-mn}z)H_j(f^kx_{-n}) )\\ &\quad\leq \|A(z)\|\cdot \mathrm{dist}(u,\mathcal{A}_{f^{-mn}(z)}^{mn}H_j(f^kx_{-n}) )\\ &\quad\leq \|A(z)\|\cdot d_H(H_j(z),\mathcal{A}_{f^{-mn}(z)}^{mn}H_j(f^kx_{-n}) )\to 0, \end{align*} $$

which implies $A(z)u\in H_j(fz)$ . Thus, $A(z)H_j(z)\subset H_j(fz)$ . It can be proved analogously that $A(z)F_j(z)\subset F_j(fz)$ . Since $\dim (A(z)H_j(z))= \dim (H_j(fz))$ , we have $A(z)H_j(z)=H_j(fz)$ . Then by the remark in §2.1, the splitting $ \mathfrak {X}= H_j(z)\oplus F_j(z)$ is $\mathcal {A}$ -invariant on $\Lambda $ .

Claim. The splitting $ \mathfrak {X}= H_j(z)\oplus F_j(z)$ is continuous on $\Lambda $ .

Proof of the claim

Let $\pi ^{H_j}_z$ , $\pi ^{F_j}_z$ be the projection operators associated with $ \mathfrak {X}= H_j(z)\oplus F_j(z)$ . Then for any $\tilde {z}\in \Lambda ,u\in H_j(\tilde {z})$ , let $w=\mathcal {A}_{\tilde {z}}^{-mn}(u)\in H_j(f^{-mn}\tilde {z})$ , by the invariance of the splitting and equations (4.9), (4.13), and (4.14), we may estimate

Thus,

which gives

Similarly, for $v\in F_j(\tilde {z})$ ,

Therefore,

Now for any $\tau>0,$ take n large enough such that $c\cdot e^{(\unicode{x3bb} _{j+1}-\unicode{x3bb} _j+8\varepsilon )mn}<\tau ,$ and then take $\delta>0$ small enough such that for any $\tilde {z}\in B(z,\delta ),$

Then we have $\|\pi ^{H_j}_z|_{F_j(\tilde {z})}\|\leq 2 \tau ,\|\pi ^{F_j}_z|_{H_j(\tilde {z})}\leq 2\tau .$ Thus by [Reference Blumenthal and Morris4, Remark 10],

$$ \begin{align*}d_H(F_j(z),F_j(\tilde{z}))&\leq 4D_j\|\pi^{H_j}_z|_{F_j(\tilde{z})}\|\leq 8D_j\tau,\\ d_H(H_j(z),H_j(\tilde{z}))&\leq 4D_j\|\pi^{F_j}_z|_{H_j(\tilde{z})}\|\leq 8D_j\tau,\end{align*} $$

which gives the continuity of $F_j(z)$ and $H_j(z)$ .

At last, for any $z\in \Lambda ,1\leq j\leq i$ , let $E_j(z):=H_j(z)\cap F_{j-1}(z)$ , where $F_0(z):={\mathfrak {X}}$ . Then, $\dim (E_j)=d_j$ . Since

$$ \begin{align*}A(z)E_j(z)\subset A(z)H_j(z)\cap A(z)F_{j-1}(z)\subset H_j(fz)\cap F_{j-1}(fz)=E_j(fz) \end{align*} $$

and $\dim (A(z)E_j(z))=d_j=\dim (E_j(fz))$ , we have $A(z)E_j(z)=E_j(fz).$ Therefore, we obtain a continuous $\mathcal {A}$ -invariant splitting on $\Lambda $ :

$$ \begin{align*}\mathfrak{X}=E_1(z)\oplus\cdots\oplus E_i(z)\oplus F_i(z),\end{align*} $$

with $\dim (E_j)=d_j \text { for all } 1\leq j\leq i$ . Additionally, by equations (4.13) and (4.14), we conclude

This completes the proof.