Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces^*

Let ${GL}({ \mathcal B })$ denote the group of invertible bounded linear operators on ${ \mathcal B }$, the metric ρ on ${GL}({ \mathcal B })$ is defined as follows

$$\begin{eqnarray}&&\rho (A,B)=\parallel A-B\parallel +\parallel {A}^{-1}-{B}^{-1}\parallel ,\end{eqnarray} \tag{ 1 }$$

then $({GL}({ \mathcal B }),\rho )$ is a complete metric space.

Let $A\,:X\to {GL}({ \mathcal B })$ be a continuous operator valued function, we have that $\parallel A\parallel := {\sup }_{x\in X}\parallel A(x)\parallel \lt \infty$ since X is compact. Consequently, we have that $\mathrm{log}\parallel A\parallel \in {L}_{1}(\mu )$. A map ${ \mathcal A }\,:X\times {\mathbb{Z}}\to {GL}({ \mathcal B })$ is called a linear cocycle over f generated by A, if

$$\begin{eqnarray*}{ \mathcal A }(x,n)=\left\{\begin{array}{ll}A({f}^{n-1}(x))\circ \cdots \circ A(f(x))\circ A(x), & n\gt 0\\ {Id}, & n=0\\ A{\left({f}^{-n}(x)\right)}^{-1}\circ \cdots \circ A{\left({f}^{-1}(x)\right)}^{-1}, & n\lt 0\end{array}\right.\end{eqnarray*}$$

for every x ∈ X. Clearly, ${ \mathcal A }(x,n+k)={ \mathcal A }({f}^{k}(x),n)\circ { \mathcal A }(x,k)$. Let

$$\begin{eqnarray*}&&\lambda (x):= \mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel { \mathcal A }(x,n)\parallel \end{eqnarray*}$$

whenever the limit exists.

Define the index of compactness (or Kuratowski measure of non-compactness) of a continuous linear operator $T\,:{ \mathcal B }\to { \mathcal B }$ as the number

$$\begin{eqnarray*}&&\parallel T{\parallel }_{\alpha }:= \inf \{k\gt 0\,:T({S}_{{ \mathcal B }})\ {\rm{can}}\,{\rm{be}}\,{\rm{covered}}\,{\rm{by}}\,{\rm{finitely}}\,{\rm{many}}\,{\rm{balls}}\,{\rm{of}}\,{\rm{radius}}k\},\end{eqnarray*}$$

where ${S}_{{ \mathcal B }}$ is the unit ball of ${ \mathcal B }$. It is easy to show that ∥T∥_α ≤ ∥T∥ and ∥T◦S∥_α ≤ ∥T∥_α ∥S∥_α for every $T,S\in {GL}({ \mathcal B })$. Given a linear cocycle ${ \mathcal A }\,:X\times {\mathbb{Z}}\to {GL}({ \mathcal B })$ over f generated by $A\,:X\to {GL}({ \mathcal B })$, the index of compactness at point x is defined as

$$\begin{eqnarray*}&&\alpha (x):= \mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel { \mathcal A }(x,n){\parallel }_{\alpha }\end{eqnarray*}$$

whenever the limit exists.

By the subadditive ergodic theorem, α(x) and λ(x) is well defined for μ-almost every x, and the function x ↦ α(x) and x ↦ λ(x) are measurable and f-invariant. Since μ is an f-invariant ergodic measure, α(x) and λ(x) are constants for μ-almost every x, denote the constants by α(A, μ) and λ(A, μ) respectively. Note that α(A, μ) ≤ λ(A, μ), and we call A is quasi-compact if α(A, μ) < λ(A, μ).

The following Oseledets multiplicative ergodic theorem of continuous cocycles on Banach spaces was proved by P. Thieullen [6].Multiplicative ergodic theorem.

Theorem 2.1 Let f be a homeomorphism on a compact metric space (X, d), and let $\mu$ be an f-invariant ergodic Borel probability measure. Given a linear cocycle ${ \mathcal A }$ over f generated by the quasi-compact and continuous operator valued function $A\,:X\to {GL}({ \mathcal B })$, where $({ \mathcal B },\parallel \cdot \parallel )$ is a Banach space. Then there exists a f-invariant subset ${X}_{0}\subset X$ of full $\mu$-measure such that one of the following cases hold: Case (1): There exist finite numbers

$$\begin{eqnarray*}&&{\lambda }_{1}\gt {\lambda }_{2}\gt \cdots \gt {\lambda }_{k}\end{eqnarray*}$$

with ${\lambda }_{1}=\lambda (A,\mu )$ and ${\lambda }_{k}\gt \alpha (A,\mu )$, and for every x ∈ X₀ there is a splitting

$$\begin{eqnarray*}&&{ \mathcal B }={E}_{1}(x)\oplus {E}_{2}(x)\oplus \cdots \oplus {E}_{k}(x)\oplus F(x)\end{eqnarray*}$$

such that

• (a)  
  For each $i=\{1,\cdots ,k\}$, $\dim {E}_{i}(x)={m}_{i}$ is finite and constant. Moreover, $A(x){E}_{i}(x)={E}_{i}(f(x))$, and for any $v\in {E}_{i}(x)\setminus \{0\}$ we have

  $$\begin{eqnarray*}&&{\lambda }_{i}=\mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel { \mathcal A }(x,n)v\parallel =-\mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel { \mathcal A }(x,-n)v\parallel ;\end{eqnarray*}$$
• (b)  
  The distribution $F(x)$ is closed and finite-codimensional, satisfies $A(x)F(x)=F(f(x))$ and

  $$\begin{eqnarray*}&&\alpha (A,\mu )\gt \mathop{\mathrm{lim}\,\sup }\limits_{n\to \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel { \mathcal A }(x,n){| }_{F(x)}\parallel ;\end{eqnarray*}$$
• (c)  
  The maps $x\mapsto {E}_{i}(x),x\mapsto F(x)$ are measurable;
• (d) Writing ${\pi }_{i}(x)$ for the projection of ${ \mathcal B }$ onto ${E}_{i}(x)(i=1,2,\cdots ,k)$ via the splitting at x, we have

  $$\begin{eqnarray*}&&\mathop{\mathrm{lim}}\limits_{n\to \pm \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel {\pi }_{i}({f}^{n}(x))\parallel =0.\end{eqnarray*}$$

Case (2): There exist infinite numbers

$$\begin{eqnarray*}&&{\lambda }_{1}\gt {\lambda }_{2}\gt \cdots \gt \alpha (A,\mu )\end{eqnarray*}$$

with ${\lambda }_{1}=\lambda (A,\mu )$ such that the following properties hold: for each x ∈ X₀ and each positive integer $k\in {\mathbb{N}}$ there is a splitting

$$\begin{eqnarray*}&&{ \mathcal B }={E}_{1}(x)\oplus {E}_{2}(x)\oplus \cdots \oplus {E}_{k}(x)\oplus {F}_{k+1}(x)\end{eqnarray*}$$

such that

• (a)  
  For each $i=\{1,\cdots ,k\}$, $\dim {E}_{i}(x)={m}_{i}$ is finite and constant for μ-almost every x. Moreover, $A(x){E}_{i}(x)={E}_{i}(f(x))$ and for every $v\in {E}_{i}(x)\setminus \{0\}$ we have

  $$\begin{eqnarray*}&&{\lambda }_{i}=\mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel { \mathcal A }(x,n)v\parallel =-\mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel { \mathcal A }(x,-n)v\parallel ;\end{eqnarray*}$$
• (b)  
  The distribution ${F}_{k+1}(x)$ is closed, finite-codimensional and $A(x){F}_{k+1}(x)={F}_{k+1}(f(x))$ and

  $$\begin{eqnarray*}&&{\lambda }_{k+1}\gt \mathop{\mathrm{lim}\,\sup }\limits_{n\to \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel { \mathcal A }(x,n){| }_{{F}_{k+1}(x)}\parallel ;\end{eqnarray*}$$
• (c)  
  The maps $x\mapsto {E}_{i}(x),x\mapsto {F}_{k+1}(x)$ are measurable;
• (d) Writing ${\pi }_{i}(x)$ for the projection of ${ \mathcal B }$ onto ${E}_{i}(x)$ via the splitting at x, we have

  $$\begin{eqnarray*}&&\mathop{\mathrm{lim}}\limits_{n\to \pm \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel {\pi }_{i}({f}^{n}(x))\parallel =0.\end{eqnarray*}$$

${ \mathcal A }$

We gather in this subsection some facts that are relevant to Banach spaces. The definitions of gap and distance are taken from Kato [17] (see also [7] and [18]).

Let E and F be two non-trivial closed linear subspaces of the Banach space ${ \mathcal B }$, and let S_E denote the unit ball of E. Put

$$\begin{eqnarray*}&&\delta (E,F)=\mathop{\sup }\limits_{v\in {S}_{E}}\mathrm{dist}(v,F)=\inf \{a\,:\mathrm{dist}(v,F)\leqslant a\parallel v\parallel \ {\rm{for}}\,{\rm{any}}\ v\in E\setminus \{0\}\}\end{eqnarray*}$$

where $\mathrm{dist}(v,F)={\inf }_{u\in F}\parallel v-u\parallel$, and define the gap between E and F as follows

$$\begin{eqnarray*}&&\hat{\delta }(E,F)=\max \{\delta (E,F),\delta (F,E)\}.\end{eqnarray*}$$

If ${ \mathcal B }$ is a Hilbert space, then $\hat{\delta }$ is a metric and coincides with the operator norm of the difference between orthogonal projections. However, in the case that ${ \mathcal B }$ is a Banach space, $\hat{\delta }$ is not a metric since it does not satisfies the triangle inequality in general [17].

The topology on the set of closed linear subspaces of the Banach space ${ \mathcal B }$ is the metric topology defined by the Hausdorff distance $\hat{d}$ between unit spheres

$$\begin{eqnarray*}&&\hat{d}(E,F)=\max \{\mathrm{dist}(E,F),\mathrm{dist}(F,E)\}\end{eqnarray*}$$

where $\mathrm{dist}(E,F)={\sup }_{v\in {S}_{E}}\mathrm{dist}(v,{S}_{F})$. The gap $\hat{\delta }$ and the distance $\hat{d}$ are related by the following inequality (see [17]):

$$\begin{eqnarray}&&\hat{\delta }(E,F)\leqslant \hat{d}(E,F)\leqslant 2\hat{\delta }(E,F).\end{eqnarray} \tag{ 2 }$$

Hence, in the following we will work with $\hat{\delta }(E,F)$ since it is more convenient.

The following lemma gives conditions under which complementation persists, see [18], lemma 3.3 for detailed proofs.

Lemma 2.1. Assume that E is a finite dimensional subspace of ${ \mathcal B }$, F is a closed subspace of ${ \mathcal B }$ and ${ \mathcal B }=E\oplus F$. Let ${\pi }_{E//F}$ be the projection operator $E\oplus F\to E$. If $E^{\prime}$ is a finite dimensional subspace of ${ \mathcal B }$ such that $\hat{d}(E,E^{\prime} )\leqslant \parallel {\pi }_{E//F}{\parallel }^{-1}$, then ${ \mathcal B }=E^{\prime} \oplus F$.

For a cocycle generated by an invertible bounded linear operator on the Banach space ${ \mathcal B }$ and an ergodic measure, this paper proves that the corresponding Oseledets subspaces varies continuously on a compact set of arbitrarily large measure. More precisely, if f : X → X is bi-Lipschitz on a compact metric space and $A\,:X\to {GL}({ \mathcal B })$ is Hölder continuous, then the Oseledets subspaces E_i (x), F_i (x) and F(x) are Hölder continuous on a compact set of measure close to 1. This extends the main results in [15] for linear cocycles on a Banach space. Dragičević et al [16] also proved the same result for possibly non-invertible cocycles on ${{\mathbb{R}}}^{d}$ as well as compact operator cocycles on Hilbert spaces.

Since there is no inner product in our case, the tools in the previous papers do not work here. For the invertible case, we use the methods in [18] to construct coordinate changes between two Oseledets space (see lemma 3.7), which play a key role in our proof. For the non-invertible case, we use the methods in [12] to overcome the lack of orthogonal complement, and get a new complement of Oseledets space and show that it has necessary properties for our purpose in proving the main results (see lemma 4.1).

Theorem 2.2. Let $f\,:X\to X$ be a bi-Lipschitz homeomorphism on a compact metric space X and $\mu$ an ergodic Borel probability measure on X, and let ${ \mathcal A }$ be a linear cocycle over f generated by a quasi-compact and ν-Hölder continuous function $A\,:X\to {GL}({ \mathcal B })$. Let ${\lambda }_{1}\gt {\lambda }_{2}\gt \cdots \gt {\lambda }_{k}\gt {\lambda }_{k+1}$ denote the distinct $k+1$ Lyapunov exponents, corresponding to the splitting ${ \mathcal B }={E}_{1}(x)\oplus \cdots \oplus {E}_{k}(x)\oplus {F}_{k+1}(x)$ defined for $\mu$-almost every $x\in X$. Then, for every $\gamma \gt 0$, there exist a compact subset ${{\rm{\Lambda }}}_{\gamma }$ of X with $\mu ({{\rm{\Lambda }}}_{\gamma })\gt 1-\gamma$, and constants $C=C({{\rm{\Lambda }}}_{\gamma })\gt 0$, ${\omega }_{i}={\omega }_{i}({\lambda }_{1},\cdots ,{\lambda }_{k+1})\lt 1,i\,=\,1,\cdots ,k+1$ and $\delta =\delta (\gamma ,{{\rm{\Lambda }}}_{\gamma },{\lambda }_{1},\cdots ,{\lambda }_{k+1})$ such that for all $x,y\in {{\rm{\Lambda }}}_{\gamma }$ with $d(x,y)\lt \delta$, we have that for $i=1,\cdots ,k$

$$\begin{eqnarray*}&&\hat{d}({E}_{i}(x),{E}_{i}(y))\leqslant {Cd}{\left(x,y\right)}^{\nu {\omega }_{i}}\ \mathrm{and}\ \ \hat{d}({F}_{k+1}(x),{F}_{k+1}(y))\leqslant {Cd}{\left(x,y\right)}^{\nu {\omega }_{k+1}}.\end{eqnarray*}$$

Remark 2.1. In the above theorem, k is the number of all finite dimensional subspaces and ${\lambda }_{k+1}:= \alpha (A,\mu ),{F}_{k+1}(x):= F(x)$ in Case (1), and k is any given positive integer in Case (2).

We first recall the definition of the Lyapunov norm. Given a sufficiently small number ε > 0, for each x ∈ X₀ and every $u={u}_{1}+\cdots +{u}_{k}+{u}_{k+1}\in { \mathcal B },{u}_{i}\in {E}_{i}(x)(i=1,\cdots ,k),{u}_{k+1}\in F(x)$, the Lyapunov norm is defined as

$$\begin{eqnarray*}&&\parallel u{\parallel }_{x,\varepsilon }:= \parallel u{\parallel }_{x}=\parallel {u}_{1}{\parallel }_{x}+\cdots +\parallel {u}_{k}{\parallel }_{x}+\parallel {u}_{k+1}{\parallel }_{x}\end{eqnarray*}$$

where $\parallel {u}_{i}{\parallel }_{x}=\sum _{n=-\infty }^{\infty }{e}^{-n{\lambda }_{i}-| n| \varepsilon }\parallel { \mathcal A }(x,n){u}_{i}\parallel$ for i = 1, ⋯ ,k and

$$\begin{eqnarray*}&&\parallel {u}_{k+1}{\parallel }_{x}=\displaystyle \sum _{n=0}^{\infty }{e}^{-n({\lambda }_{k+1}+\varepsilon )}\parallel { \mathcal A }(x,n){u}_{k+1}\parallel .\end{eqnarray*}$$

For every n > 0 and every u_i ∈ E_i (x)(i = 1, ⋯ ,k), one can easily show that

$$\begin{eqnarray}&&{e}^{n({\lambda }_{i}-\varepsilon )}\parallel {u}_{i}{\parallel }_{x}\leqslant \parallel { \mathcal A }(x,n){u}_{i}{\parallel }_{{f}^{n}(x)}\leqslant {e}^{n({\lambda }_{i}+\varepsilon )}\parallel {u}_{i}{\parallel }_{x},\end{eqnarray} \tag{ 3 }$$

and for every u_k+1 ∈ F(x)

$$\begin{eqnarray}&&\parallel { \mathcal A }(x,n){u}_{k+1}{\parallel }_{{f}^{n}(x)}\leqslant {e}^{n({\lambda }_{k+1}+\varepsilon )}\parallel {u}_{k+1}{\parallel }_{x}.\end{eqnarray} \tag{ 4 }$$

The following lemma provides some fundamental properties of the above Lyapunov norm, see [19], theorem 7.2.3 for detailed proofs.

Lemma 3.1. Given a small number $\varepsilon \gt 0$, there exists a measurable function ${D}_{\varepsilon }\,:{X}_{0}\to [1,\infty )$ such that for every $x\in {X}_{0}$

$$\begin{eqnarray}&&\parallel \cdot \parallel \leqslant \parallel \cdot {\parallel }_{x}\leqslant {D}_{\varepsilon }(x)\parallel \cdot \parallel \end{eqnarray} \tag{ 5 }$$

and for each $n\in {\mathbb{Z}}$

$$\begin{eqnarray*}&&{D}_{\varepsilon }({f}^{n}(x))\leqslant {e}^{| n| \varepsilon }{D}_{\varepsilon }(x).\end{eqnarray*}$$

In the setting of the space ${{\mathbb{R}}}^{d}$ or the Hilbert space, we have that the angels between two Oseledets subspaces decay sub-exponentially along orbits of x. However, in the case of Banach spaces, there is not such statement since the lack of reasonable definition of the 'angels' between two closed linear subspaces of Banach space. The fact that the norm of each projection operator π_i (x), i = 1, ⋯ ,k is temperate (see (d) of theorem 2.1) helps us to overcome this difficulty. For the multiplicative ergodic theorem of semi-invertible operator on Banach spaces (see [11]), it is also valid by the following lemma proved by Dragičević et al [16], lemma 1.

Lemma 3.2. Let ${\rm{\Lambda }}$ be an f-invariant set and let E(x) and $F(x),x\in {\rm{\Lambda }}$ be two families of closed subspaces of ${ \mathcal B }$. Assume that there exist numbers ${\chi }_{2}\lt {\chi }_{1},\varepsilon \gt 0$ with ${\chi }_{2}+3\varepsilon \leqslant {\chi }_{1}-2\varepsilon$ and measurable functions $C,\tilde{C}\,:{\rm{\Lambda }}\to [1,\infty )$ such that

• (1)  
  $A(x)E(x)\subset E(f(x)),A(x)F(x)\subset F(f(x))$ and $E(x)\cap F(x)=\{0\}$ for every $x\in {\rm{\Lambda }};$
• (2)  
  For every $x\in {\rm{\Lambda }},v\in E(x)\oplus F(x)$ and $n\geqslant 0$,

  $$\begin{eqnarray*}&&\parallel { \mathcal A }(x,n)v\parallel \leqslant \tilde{C}(x){e}^{({\chi }_{1}+\varepsilon )n}\parallel v\parallel ;\end{eqnarray*}$$
• (3)  
  For every $x\in {\rm{\Lambda }},v\in F(x)$ and $n\geqslant 0$,

  $$\begin{eqnarray*}&&\parallel { \mathcal A }(x,n)v\parallel \geqslant \displaystyle \frac{1}{\tilde{C}(x)}{e}^{({\chi }_{1}-\varepsilon )n}\parallel v\parallel ;\end{eqnarray*}$$
• (4)  
  For every $x\in {\rm{\Lambda }},v\in E(x)$ and $n\geqslant 0$,

  $$\begin{eqnarray*}&&\parallel { \mathcal A }(x,n)v\parallel \leqslant C(x){e}^{({\chi }_{2}+\varepsilon )n}\parallel v\parallel ;{and}\end{eqnarray*}$$
• (5)  
  For every $x\in {\rm{\Lambda }}$ and $m\in {\mathbb{Z}}$,

  $$\begin{eqnarray*}&&C({f}^{n}(x))\leqslant {e}^{| n| \varepsilon }C(x)\ {and}\ \tilde{C}({f}^{n}(x))\leqslant {e}^{| n| \varepsilon }\tilde{C}(x).\end{eqnarray*}$$

Then, there exists a measurable function $K\,:{\rm{\Lambda }}\to [1,\infty )$ satisfies

$$\begin{eqnarray*}&&K({f}^{n}(x))\leqslant {e}^{| n| \varepsilon }K(x)\ \mathrm{foreach}\ n\in {\mathbb{Z}}\ {and}\ x\in {\rm{\Lambda }}\end{eqnarray*}$$

and such that

$$\begin{eqnarray*}&&\parallel {v}_{1}\parallel \leqslant K(x)\parallel {v}_{1}+{v}_{2}\parallel \ {and}\ \parallel {v}_{2}\parallel \leqslant K(x)\parallel {v}_{1}+{v}_{2}\parallel \end{eqnarray*}$$

for ${v}_{1}\in E(x)$ and ${v}_{2}\in F(x)$.

Remark 3.1. Dragičević et al [16] proved the above lemma for the Euclidean space ${{\mathbb{R}}}^{d}$, it is also valid for the case of Banach space ${ \mathcal B }$, with a minor modification of the proof. Use the lemma above with some inductions, one can easily to show that (d) of theorem 2.1 is valid for semi-invertible cocycle on Banach space.

Theorem 3.1. Let ${ \mathcal A }$ be a cocycle over f with Lyapunov exponents as in Case (1), for each $i\in \{1,\cdots ,k\}$. Let

$$\begin{eqnarray*}&&{E}_{i}^{+}(x)=\underset{j=1}{\overset{i}{\displaystyle \bigoplus }}{E}_{j}(x)\ {and}\ {E}_{i}^{-}(x)=\left(\underset{j=i+1}{\overset{k}{\displaystyle \bigoplus }}{E}_{j}(x)\right)\bigoplus F(x).\end{eqnarray*}$$

Then there exists a full μ-measure, f-invariant subset Λ of X such that, for each $\varepsilon \gt 0$ small enough with $\varepsilon \lt {\min }_{i=1,\cdots ,k}\{({\lambda }_{i}-{\lambda }_{i+1})/100\}$, there are measurable functions $C,K\,:{\rm{\Lambda }}\to [1,\infty )$ with

$$\begin{eqnarray*}&&C({f}^{n}(x))\leqslant {e}^{| n| \varepsilon }C(x)\ {and}\ K({f}^{n}(x))\leqslant {e}^{| n| \varepsilon }K(x).\end{eqnarray*}$$

so that for every $x\in {\rm{\Lambda }}$ :

• (1)  
  For each $u\in {E}_{i}^{-}(x),v\in {E}_{i}^{+}(x)$ and $n\geqslant 0$,

  $$\begin{eqnarray*}&&\parallel { \mathcal A }(x,n)u\parallel \leqslant C(x){e}^{({\lambda }_{i+1}+\varepsilon )n}\parallel u\parallel \ {and}\ \parallel { \mathcal A }(x,n)v\parallel \geqslant \displaystyle \frac{1}{C(x)}{e}^{({\lambda }_{i}-\varepsilon )n}\parallel v\parallel ;\end{eqnarray*}$$
• (2)  
  For each $u\in {E}_{i}^{-}(x)$ and $v\in {E}_{i}^{+}(x)$,

  $$\begin{eqnarray*}&&\max \{\parallel u\parallel ,\parallel v\parallel \}\leqslant K(x)\parallel u+v\parallel .\end{eqnarray*}$$

Proof. Following the proof of proposition 3.2 in [20], we give the detailed proofs as follows. By (3), (4) and lemma 3.1, there exist a set Λ and a measurable function $C\,:{\rm{\Lambda }}\to [1,\infty )$ with $C({f}^{\pm }(x))\leqslant {e}^{\varepsilon }C(x)$ so that the first statement hold. To complete the proof of the theorem, it suffices to prove the second statement.

Let ${\pi }_{i}^{+}(x)$ and ${\pi }_{i}^{-}(x)$ denote the projections of ${ \mathcal B }$ onto ${E}_{i}^{+}(x)$ and ${E}_{i}^{-}(x)$ via the splitting ${ \mathcal B }={E}_{i}^{+}(x)\oplus {E}_{i}^{-}(x)$. By (d) of theorem 2.1, for every $x\in {\rm{\Lambda }}$ we have that

$$\begin{eqnarray*}&&\mathop{\mathrm{lim}}\limits_{n\to \pm \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel {\pi }_{i}^{+}({f}^{n}(x))\parallel =0,\mathop{\mathrm{lim}}\limits_{n\to \pm \infty }\displaystyle \frac{1}{n}\mathrm{log}\parallel {\pi }_{i}^{-}({f}^{n}(x))\parallel =0.\end{eqnarray*}$$

Therefore, define the function $K\,:{\rm{\Lambda }}\to [1,\infty )$ as follows :

$$\begin{eqnarray*}&&K(x)=\mathop{\sup }\limits_{n\in {\mathbb{Z}}}\displaystyle \frac{\max \{\parallel {\pi }_{i}^{-}({f}^{n}(x))\parallel ,\parallel {\pi }_{i}^{+}({f}^{n}(x))\parallel \}}{{e}^{| n| \varepsilon }}.\end{eqnarray*}$$

One can easily to show that K is a well defined function on Λ. Moreover, one has that $K(x)\geqslant \max \{\parallel {\pi }_{i}^{-}(x)\parallel ,\parallel {\pi }_{i}^{+}(x)\parallel \}$ and $K({f}^{\pm }(x))\leqslant {e}^{\varepsilon }K(x)$ for every $x\in {\rm{\Lambda }}$.

Finally, for each $u\in {E}_{i}^{-}(x)$ and each $v\in {E}_{i}^{+}(x)$ one has that

$$\begin{eqnarray}&&\parallel u\parallel =\parallel {\pi }_{i}^{-}(x)(u+v)\parallel \leqslant \parallel {\pi }_{i}^{-}(x)\parallel \cdot \parallel u+v\parallel \leqslant K(x)\parallel u+v\parallel \end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&\parallel v\parallel =\parallel {\pi }_{i}^{+}(x)(u+v)\parallel \leqslant \parallel {\pi }_{i}^{+}(x)\parallel \cdot \parallel u+v\parallel \leqslant K(x)\parallel u+v\parallel .\end{eqnarray} \tag{ 7 }$$

This completes the proof of the theorem. □

Fix a sufficiently small ε > 0. For every ${\ell }\in {\mathbb{N}}$, we define the regular set Λ_ℓ by

$$\begin{eqnarray*}&&{{\rm{\Lambda }}}_{{\ell }}=\{x\in {\rm{\Lambda }}\,:C(x)\leqslant {\ell }\ {and}\ K(x)\leqslant {\ell }\}.\end{eqnarray*}$$

One can easily show that each Λ_ℓ is compact, Λ_ℓ ⊂ Λ_ℓ+1 and ⋃_ℓ>0Λ_ℓ = Λ. Thus, for every γ > 0, we may choose a subset Λ_ℓ with μ(Λ_ℓ ) > 1 − γ.

Fix i ∈ {1, ⋯ ,k}, and let ${E}_{i}^{-}(x)$ and ${E}_{i}^{+}(x)$ be as in theorem 3.1, we now prove that the map $x\mapsto {E}_{i}^{-}(x)$ and $x\mapsto {E}_{i}^{+}(x)$ are (locally) Hölder continuous on Λ_ℓ .

The following two lemmas is useful in the proof of the main result. See [14] for the original versions of the finite dimensional case, we also refer the reader to lemmas 5.3.4 and 5.3.5 in [1] or lemmas 2.1 and 2.2 in [15].

Lemma 3.3. Assume that $A\,:X\to {GL}({ \mathcal B })$ is ν-Hölder continuous with Hölder constant a₁ and $f\,:X\to X$ is bi-Lipschitz with constant $L\geqslant 1$, then there exists a constant $a\gt {a}_{1}$ such that

$$\begin{eqnarray}&&\parallel { \mathcal A }(x,n)-{ \mathcal A }(y,n)\parallel \leqslant {a}^{| n| }d{\left(x,y\right)}^{\nu }\end{eqnarray} \tag{ 8 }$$

for every $x,y\in X$ and $n\in {\mathbb{Z}}$.

Proof. We follow the proof of lemma 2.2 in [15] and argue by induction on n.

For k = 1, by the Hölder continuity of the map $A\,:X\to {GL}({ \mathcal B })$ and (1) we have

$$\begin{eqnarray*}&&\parallel A(x)-A(y)\parallel \leqslant {a}_{1}d{\left(x,y\right)}^{\nu }.\end{eqnarray*}$$

Assume that there exists $a\gt {a}_{1}$ so that (8) hold for $k=1,2,\cdots ,n$. For $k=n+1$, we have that

$$\begin{eqnarray*}&&\begin{array}{l}\parallel { \mathcal A }(x,n+1)-{ \mathcal A }(y,n+1)\parallel \\ \,\,\leqslant \parallel A({f}^{n}(x))\parallel \cdot \parallel { \mathcal A }(x,n)-{ \mathcal A }(y,n)\parallel +\parallel { \mathcal A }(y,n)\parallel \cdot \parallel A({f}^{n}(x))-A({f}^{n}(y))\parallel \\ \,\,\leqslant (\mathop{\sup }\limits_{x\in X}\parallel A(x\parallel ){a}^{n}d{\left(x,y\right)}^{\nu }+{\left(\mathop{\sup }\limits_{x\in X}\parallel A(x)\parallel \right)}^{n}{a}_{1}d{\left({f}^{n}(x),{f}^{n}(y)\right)}^{\nu }\\ \,\,\leqslant [(\mathop{\sup }\limits_{x\in X}\parallel A(x\parallel ){a}^{n}+{\left(\mathop{\sup }\limits_{x\in X}\parallel A(x)\parallel \right)}^{n}{a}_{1}{L}^{n\nu }]d{\left(x,y\right)}^{\nu }\end{array}\end{eqnarray*}$$

where we use the fact that f is Lipschitz in the last inequality. To find the number a, all we need is that

$$\begin{eqnarray*}&&{a}^{n+1}\gt (\mathop{\sup }\limits_{x\in X}\parallel A\left(x\parallel \right){a}^{n}+{\left(\mathop{\sup }\limits_{x\in X}\parallel A(x)\parallel \right)}^{n}{a}_{1}{L}^{n\nu },\end{eqnarray*}$$

that is

$$\begin{eqnarray*}&&a\geqslant {a}_{1}{\left(\displaystyle \frac{\mathop{\sup }\limits_{x\in X}\parallel A(x)\parallel \cdot {L}^{\nu }}{a}\right)}^{n}+\mathop{\sup }\limits_{x\in M}\parallel A(x)\parallel .\end{eqnarray*}$$

This can be easily achieved by taking a sufficiently large a such that

$$\begin{eqnarray*}&&a\gt \max \{{a}_{1},\mathop{\sup }\limits_{x\in M}\parallel A(x)\parallel \cdot {L}^{\nu }\}.\end{eqnarray*}$$

The case for $n\lt 0$ can be proven in a similar fashion, this completes the proof. □

Lemma 3.4. Let ${\{{A}_{n}\}}_{n\geqslant 1},{\{{B}_{n}\}}_{n\geqslant 1}$ be two sequences of operators in ${GL}({ \mathcal B })$, such that, for some $0\lt {\alpha }_{2}\lt {\alpha }_{1}$ and ${\ell }\geqslant 1$, there exist closed subspaces $E,E^{\prime} ,F,F^{\prime}$ and ${{ \mathcal B }}_{0}$ of ${ \mathcal B }$ satisfying ${{ \mathcal B }}_{0}=E\oplus E^{\prime} =F\oplus F^{\prime}$ such that for some fixed n

• (i)  
  $\parallel {A}_{n}u\parallel \leqslant {\ell }{\alpha }_{2}^{n}\parallel u\parallel$ and $\parallel {A}_{n}v\parallel \geqslant {{\ell }}^{-1}{\alpha }_{1}^{n}\parallel v\parallel$ for every $u\in E$, $v\in E^{\prime} ;$
• (ii) $\parallel {B}_{n}u\parallel \leqslant {\ell }{\alpha }_{2}^{n}\parallel u\parallel$ and $\parallel {B}_{n}v\parallel \geqslant {{\ell }}^{-1}{\alpha }_{1}^{n}\parallel v\parallel$ for every $u\in F$, $v\in F^{\prime} ;$
• (iii) $\mathrm{Max}\{\parallel v\parallel ,\parallel w\parallel \}\leqslant {\ell }\parallel u\parallel$ for each $u=v+w,v\in E,w\in E^{\prime}$ or $v\in F,w\in F^{\prime}$.

Then for every $\delta \lt 1,a\geqslant {\alpha }_{1}$ satisfying

$$\begin{eqnarray*}&&{\left(\displaystyle \frac{{\alpha }_{2}}{a}\right)}^{n+1}\leqslant \delta \lt {\left(\displaystyle \frac{{\alpha }_{2}}{a}\right)}^{n}\ \mathrm{and}\ \parallel {A}_{n}-{B}_{n}\parallel \leqslant \delta {a}^{n},\end{eqnarray*}$$

we have that

$$\begin{eqnarray*}&&\hat{d}(E,F)\leqslant (4+2{\ell }){{\ell }}^{2}\displaystyle \frac{{\alpha }_{1}}{{\alpha }_{2}}{\delta }^{\mathrm{log}({\alpha }_{2}/{\alpha }_{1})/\mathrm{log}({\alpha }_{2}/a)}.\end{eqnarray*}$$

Proof. With a minor modification of the proof of lemma 2.1 in [15], we give the proof of the result in the following.

Let us define the cone $Q=\{u\in {{ \mathcal B }}_{0}\,:\parallel {A}_{n}u\parallel \leqslant 2{\ell }{\alpha }_{2}^{n}\parallel u\parallel \}$. For each $v\in F$, one has

$$\begin{eqnarray*}\begin{array}{rcl}\parallel {A}_{n}v\parallel & \leqslant & \parallel {A}_{n}-{B}_{n}\parallel \cdot \parallel v\parallel +\parallel {B}_{n}v\parallel \\ & & \leqslant (\delta {a}^{n}+{\ell }{\alpha }_{2}^{n})\parallel v\parallel \\ & & \leqslant 2{\ell }{\alpha }_{2}^{n}\parallel v\parallel .\end{array}\end{eqnarray*}$$

In consequence, $v\in Q$ and this implies that $F\subset Q$.

For each $v\in Q$, write $v={v}_{1}+{v}_{2}$ where ${v}_{1}\in E$ and ${v}_{2}\in E^{\prime}$. Recall that ${\ell }\parallel v\parallel \geqslant \max \{\parallel {v}_{1}\parallel ,\parallel {v}_{2}\parallel \}$, one has

$$\begin{eqnarray*}\begin{array}{rcl}2{\ell }{\alpha }_{2}^{n}\parallel v\parallel & \geqslant & \parallel {A}_{n}({v}_{1}+{v}_{2})\parallel \\ & & \geqslant \parallel {A}_{n}{v}_{2}\parallel -\parallel {A}_{n}{v}_{1}\parallel \\ & & \geqslant {{\ell }}^{-1}{\alpha }_{1}^{n}\parallel {v}_{2}\parallel -{\ell }{\alpha }_{2}^{n}\cdot {\ell }\parallel v\parallel .\end{array}\end{eqnarray*}$$

This together with the fact that ${\left({\alpha }_{2}/a\right)}^{n+1}\lt \delta \lt {\left({\alpha }_{2}/a\right)}^{n}$ imply that

$$\begin{eqnarray*}&&\mathrm{dist}(v,E)\leqslant \parallel {v}_{2}\parallel \leqslant (2+{\ell }){{\ell }}^{2}{\left(\displaystyle \frac{{\alpha }_{2}}{{\alpha }_{1}}\right)}^{n}\parallel v\parallel \leqslant (2+{\ell }){{\ell }}^{2}\displaystyle \frac{{\alpha }_{1}}{{\alpha }_{2}}{\delta }^{\mathrm{log}({\alpha }_{2}/{\alpha }_{1})/\mathrm{log}({\alpha }_{2}/a)}\parallel v\parallel .\end{eqnarray*}$$

Since $F\subset Q$, by the definition of $\delta (\cdot ,\cdot )$ one has

$$\begin{eqnarray*}&&\delta (F,E)\leqslant (2+{\ell }){{\ell }}^{2}\displaystyle \frac{{\alpha }_{1}}{{\alpha }_{2}}{\delta }^{\mathrm{log}({\alpha }_{2}/{\alpha }_{1})/\mathrm{log}({\alpha }_{2}/a)}.\end{eqnarray*}$$

Symmetrically, one can show that $\delta (E,F)\leqslant (2+{\ell }){{\ell }}^{2}\tfrac{{\alpha }_{1}}{{\alpha }_{2}}{\delta }^{\mathrm{log}({\alpha }_{2}/{\alpha }_{1})/\mathrm{log}({\alpha }_{2}/a)}$. Hence, using (2) we conclude that

$$\begin{eqnarray*}&&\hat{d}(E,F)\leqslant (4+2{\ell }){{\ell }}^{2}\displaystyle \frac{{\alpha }_{1}}{{\alpha }_{2}}{\delta }^{\mathrm{log}({\alpha }_{2}/{\alpha }_{1})/\mathrm{log}({\alpha }_{2}/a)}.\end{eqnarray*}$$

□

Now, we estimate $\hat{d}({E}_{i}^{-}(x),{E}_{i}^{-}(y))$ and $\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))$ for each x, y ∈ Λ_l with d(x, y) < 1, where ${E}_{i}^{-}(x),{E}_{i}^{+}(x)$ is the same as in theorem 3.1.

Lemma 3.5. For each $x,y\in {{\rm{\Lambda }}}_{{\ell }}$ with $d(x,y)\lt 1$, we have

$$\begin{eqnarray*}&&\hat{d}({E}_{i}^{-}(x),{E}_{i}^{-}(y))\leqslant {C}_{i}^{-}d{\left(x,y\right)}^{{\nu }_{i}^{-}}\end{eqnarray*}$$

where ${C}_{i}^{-}$ and ${\nu }_{i}^{-}$ are two constants.

Proof. Given $x,y\in {{\rm{\Lambda }}}_{{\ell }}$ with $d(x,y)\lt 1$, let ${A}_{n}={ \mathcal A }(x,n),{B}_{n}={ \mathcal A }(y,n),{\alpha }_{2}={e}^{{\lambda }_{i+1}+\varepsilon },{\alpha }_{1}={e}^{{\lambda }_{i}-\varepsilon },E={E}_{i}^{-}(x),F={E}_{i}^{-}(y),E^{\prime} ={E}_{i}^{+}(x)$ and $F^{\prime} ={E}_{i}^{+}(y)$. By theorem 3.1 and the definition of ${{\rm{\Lambda }}}_{{\ell }}$, the conditions (i), (ii), (iii) of lemma 3.4 hold for every $n\in {\mathbb{N}}$. Set $\delta =d{\left(x,y\right)}^{\nu }\lt 1$. By lemma 3.3, there exists a sufficiently large constant a such that $1\gt {e}^{{\lambda }_{i+1}+\varepsilon }/a$ and

$$\begin{eqnarray*}&&\parallel {A}_{n}-{B}_{n}\parallel \leqslant {a}^{n}d{\left(x,y\right)}^{\nu }={a}^{n}\delta \end{eqnarray*}$$

for each $n\in {\mathbb{N}}$. Moreover, there exists $n^{\prime} =n^{\prime} (\delta ,a,i)\in {\mathbb{N}}$ such that

$$\begin{eqnarray*}&&{\left(\displaystyle \frac{{e}^{{\lambda }_{i+1}+\varepsilon }}{a}\right)}^{n^{\prime} +1}\leqslant \delta \lt {\left(\displaystyle \frac{{e}^{{\lambda }_{i+1}+\varepsilon }}{a}\right)}^{n^{\prime} }.\end{eqnarray*}$$

It follows from lemma 3.4 that

$$\begin{eqnarray}\begin{array}{rcl}\hat{d}({E}_{i}^{-}(x),{E}_{i}^{-}(y)) & \leqslant & (4+2{\ell }){{\ell }}^{2}\displaystyle \frac{{\alpha }_{1}}{{\alpha }_{2}}{\delta }^{\mathrm{log}({\alpha }_{2}/{\alpha }_{1})/\mathrm{log}({\alpha }_{2}/a)}\\ & & =(4+2{\ell }){{\ell }}^{2}{e}^{{\lambda }_{i}-{\lambda }_{i+1}-2\varepsilon }d{\left(x,y\right)}^{\tfrac{\nu ({\lambda }_{i+1}-{\lambda }_{i}+2\varepsilon )}{{\lambda }_{i+1}+\varepsilon -\mathrm{log}a}}\\ & & ={C}_{i}^{-}d{\left(x,y\right)}^{{\nu }_{i}^{-}}\end{array}\end{eqnarray} \tag{ 9 }$$

where ${C}_{i}^{-}=(4+2{\ell }){{\ell }}^{2}{e}^{{\lambda }_{i}-{\lambda }_{i+1}-2\varepsilon }$ and ${\nu }_{i}^{-}=\nu ({\lambda }_{i}-{\lambda }_{i+1}-2\varepsilon )/(\mathrm{log}a-{\lambda }_{i+1}-\varepsilon )\lt \nu \lt 1.$ □

Remark 3.2. Notice the hypotheses that A(x) is invertible is not used in the estimate of $\hat{d}({E}_{i}^{-}(x),{E}_{i}^{-}(y))$, and theorem 3.1 is also valid for semi-invertible cocycles (see [20], proposition 3.2). Thus, the conclusion of the above lemma is true for semi-invertible operators on a Banach space.

Lemma 3.6. For each $x,y\in {{\rm{\Lambda }}}_{{\ell }}$ with $d(x,y)\lt 1$, we have

$$\begin{eqnarray*}&&\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\leqslant {C}_{i}^{+}d{\left(x,y\right)}^{{\nu }_{i}^{+}}\end{eqnarray*}$$

where ${C}_{i}^{+}$ and ${\nu }_{i}^{+}$ are two constants.

Proof. In order to estimate $\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))$ for every $x,y\in {{\rm{\Lambda }}}_{{\ell }}$ with $d(x,y)\lt 1$, set ${A}_{n}={ \mathcal A }(x,-n),{B}_{n}={ \mathcal A }(y,-n)$ for every $n\in {\mathbb{N}}$, ${\alpha }_{2}={e}^{-{\lambda }_{i}+2\varepsilon },{\alpha }_{1}={e}^{-{\lambda }_{i+1}-2\varepsilon },E\,={E}_{i}^{+}(x),F={E}_{i}^{+}(y),E^{\prime} ={E}_{i}^{-}(x)$ and $F^{\prime} ={E}_{i}^{-}(y)$. Take $u\in E^{\prime}$, since ${ \mathcal A }(x,-n)={\left({ \mathcal A }({f}^{-n}(x),n)\right)}^{-1}$, it follows from theorem 3.1 that

$$\begin{eqnarray*}\begin{array}{rcl}\parallel u\parallel & = & \parallel { \mathcal A }({f}^{-n}(x),n)({ \mathcal A }(x,-n)u)\parallel \\ & & \leqslant C({f}^{-n}(x)){e}^{{\lambda }_{i+1}+\varepsilon }\parallel { \mathcal A }(x,-n)u\parallel \\ & & \leqslant {e}^{n\varepsilon }{\ell }{e}^{{\lambda }_{i+1}+\varepsilon }\parallel { \mathcal A }(x,-n)u\parallel \end{array}\end{eqnarray*}$$

where we use the facts that $C({f}^{-n}(x))\leqslant {e}^{n\varepsilon }C(x)$ and $C(x)\leqslant {\ell }$. Consequently, one has

$$\begin{eqnarray*}&&\parallel {A}_{n}u\parallel \geqslant {{\ell }}^{-1}{e}^{n(-{\lambda }_{i+1}-2\varepsilon )}\parallel u\parallel ={{\ell }}^{-1}{\alpha }_{1}^{n}\parallel u\parallel .\end{eqnarray*}$$

Similarly, one can show that $\parallel {A}_{n}v\parallel \leqslant {\ell }{\alpha }_{2}^{n}\parallel v\parallel$ for every $v\in E$. Thus, the condition (i) of lemma 3.4 holds. Replace x with y, one can show that the condition (ii) of lemma 3.4 holds in similar fashion, and the condition (iii) follows from (2) of theorem 3.1. Let $\delta =d{\left(x,y\right)}^{\nu }\lt 1$ and let the constant a be as lemma 3.4. It follows from lemma 3.4 that

$$\begin{eqnarray}&&\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\leqslant {C}_{i}^{+}d{\left(x,y\right)}^{{\nu }_{i}^{+}}.\end{eqnarray} \tag{ 10 }$$

where ${C}_{i}^{+}=(4+2{\ell }){{\ell }}^{2}{e}^{{\lambda }_{i}-{\lambda }_{i+1}-4\varepsilon }$ and ${\nu }_{i}^{+}=\nu ({\lambda }_{i}-{\lambda }_{i+1}-4\varepsilon )/(\mathrm{log}a+{\lambda }_{i}-2\varepsilon )\lt \nu \lt 1.$ □

In this section, we will give the proof of the main result of this paper.

For every γ > 0, fix Λ_ℓ so that μ(Λ_ℓ ) > 1 − γ. Take a point x ∈ Λ_ℓ , then ${ \mathcal B }={E}_{i}^{-}(x)\oplus {E}_{i}^{+}(x)$, here ${E}_{i}^{-}(x),{E}_{i}^{+}(x)$ are the same as in theorem 3.1. By (10), choose a small number δ₁ ∈ (0, 1) such that

$$\begin{eqnarray*}&&\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\leqslant 1/{\ell }\end{eqnarray*}$$

for each x, y ∈ Λ_ℓ with d(x, y) < δ₁. By the definition of Λ_ℓ and (7), the norm of the projection operator ${\pi }_{i}^{+}(x)\,:{E}_{i}^{+}(x)\oplus {E}_{i}^{-}(x)\to {E}_{i}^{+}(x)$ is no larger than ℓ, thus

$$\begin{eqnarray*}&&\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\leqslant 1/{\ell }\lt \parallel {\pi }_{i}^{+}(x){\parallel }^{-1}.\end{eqnarray*}$$

By lemma 2.1, this yields that ${ \mathcal B }={E}_{i}^{+}(y)\oplus {E}_{i}^{-}(x)$. So, there exists a linear operator ${L}_{x,y}\,:{E}_{i}^{+}(x)\to {E}_{i}^{-}(x)$ such that the graph of L_x,y is equivalent to the subspace ${E}_{i}^{+}(y)$, that is,

$$\begin{eqnarray*}&&{E}_{i}^{+}(y)=\{u+{L}_{x,y}(u)\,:u\in {E}_{i}^{+}(x)\}.\end{eqnarray*}$$

Lemma 3.7. For each $x,y\in {{\rm{\Lambda }}}_{{\ell }}$ with $d(x,y)\lt {\delta }_{1}$, we have

$$\begin{eqnarray}&&\displaystyle \frac{\parallel {L}_{x,y}\parallel }{{\ell }(1+\parallel {L}_{x,y}\parallel )}\leqslant \hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\leqslant 2{\ell }\parallel {L}_{x,y}\parallel .\end{eqnarray} \tag{ 11 }$$

Proof. For simplicity of presentation, denote ${L}_{x,y}$ by L. First, for each $u\in {E}_{i}^{+}(x)$, the fact that $u+{Lu}\in {E}_{i}^{+}(y)$ implies

$$\begin{eqnarray*}&&\mathrm{dist}(u,{E}_{i}^{+}(y))\leqslant \parallel u-(u+{Lu})\parallel \leqslant \parallel L\parallel \cdot \parallel u\parallel .\end{eqnarray*}$$

Then, we have that $\delta ({E}_{i}^{+}(x),{E}_{i}^{+}(y))\leqslant \parallel L\parallel$. Next, given $v\in {E}_{i}^{+}(y)$, there exist $u\in {E}_{i}^{+}(x)$ such that $v=u+{Lu}$. Since $u\in {E}_{i}^{+}(x),{Lu}\in {E}_{i}^{-}(x)$, it follows from (2) of theorem 3.1 that $\parallel u\parallel \leqslant {\ell }\parallel v\parallel$. Thus, one has

$$\begin{eqnarray*}&&\mathrm{dist}(v,{E}_{i}^{+}(x))\leqslant \parallel v-u\parallel \leqslant \parallel L\parallel \cdot \parallel u\parallel \leqslant {\ell }\parallel L\parallel \cdot \parallel v\parallel .\end{eqnarray*}$$

Therefore, $\delta ({E}_{i}^{+}(y),{E}_{i}^{+}(x))\leqslant {\ell }\cdot \parallel L\parallel$. This together with (2) yield that

$$\begin{eqnarray*}&&\hat{d}({E}_{i}^{+}(y),{E}_{i}^{+}(x))\leqslant 2\hat{\delta }({E}_{i}^{+}(y),{E}_{i}^{+}(x))\leqslant 2{\ell }\parallel L\parallel .\end{eqnarray*}$$

To prove the other inequality in (11), note that for each $\beta \gt \delta ({E}_{i}^{+}(y),{E}_{i}^{+}(x))$ and for every $u\in {E}_{i}^{+}(x)\setminus \{0\}$, one has

$$\begin{eqnarray*}&&\mathrm{dist}(u+{Lu},{E}_{i}^{+}(x))\lt \beta \cdot \parallel u+{Lu}\parallel .\end{eqnarray*}$$

Fix such a $u\in {E}_{i}^{+}(x)\setminus \{0\}$, there exists $u^{\prime} \in {E}_{i}^{+}(x)$ such that $\parallel u+{Lu}-u^{\prime} \parallel \lt \beta \cdot \parallel u+{Lu}\parallel$. Since $u-u^{\prime} \in {E}_{i}^{+}(x)$, ${Lu}\in {E}_{i}^{-}(x)$ and $K(x)\leqslant {\ell }$, by (2) of theorem 3.1 we have that

$$\begin{eqnarray*}&&\parallel {Lu}\parallel \lt {\ell }\parallel u-u^{\prime} +{Lu}\parallel \lt {\ell }\beta \cdot \parallel u+{Lu}\parallel \lt {\ell }\beta (1+\parallel L\parallel )\cdot \parallel u\parallel .\end{eqnarray*}$$

Thus, we conclude that

$$\begin{eqnarray*}&&\beta \gt \displaystyle \frac{\parallel {Lu}\parallel }{{\ell }(1+\parallel L\parallel )\cdot \parallel u\parallel }\end{eqnarray*}$$

for every $u\in {E}_{i}^{+}(x)\setminus \{0\}$. This implies that

$$\begin{eqnarray*}&&\beta \geqslant \displaystyle \frac{\parallel L\parallel }{{\ell }(1+\parallel L\parallel )}.\end{eqnarray*}$$

By the arbitrariness of β and (2), one has

$$\begin{eqnarray*}&&\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\geqslant \delta ({E}_{i}^{+}(x),{E}_{i}^{+}(y))\geqslant \displaystyle \frac{\parallel L\parallel }{{\ell }(1+\parallel L\parallel )}.\end{eqnarray*}$$

This completes the proof of the lemma. □

Next, we shall prove theorem 2.2.

Proof of theorem 2.2. By lemma 3.7, we have that

$$\begin{eqnarray*}&&\parallel {L}_{x,y}\parallel \to 0\ \mathrm{whenever}\ d(x,y)\to 0.\end{eqnarray*}$$

Hence, there exists ${\delta }_{2}\in (0,{\delta }_{1})$ with ${\delta }_{2}^{\nu }\lt 1/4$ such that $\parallel {L}_{x,y}\parallel \lt 1/2$ for any $x,y\in {{\rm{\Lambda }}}_{{\ell }}$ with $d(x,y)\lt {\delta }_{2}$.

Fix $x,y\in {{\rm{\Lambda }}}_{{\ell }}$ with $d(x,y)\lt {\delta }_{2}$. Let ${{\rm{\Phi }}}_{x,y}={Id}+{L}_{x,y}$ be the isomorphism from ${E}_{i}^{+}(x)$ to ${E}_{i}^{+}(y)$. It is easy to see that $\parallel {{\rm{\Phi }}}_{x,y}\parallel \leqslant 1+\parallel {L}_{x,y}\parallel$. Since $\parallel {L}_{x,y}\parallel \lt 1/2$, we obtain that ${{\rm{\Phi }}}_{x,y}^{-1}={Id}+{\sum }_{k=1}^{\infty }{\left(-{L}_{x,y}\right)}^{k}$. We write it as ${{\rm{\Phi }}}_{x,y}^{-1}={Id}+{\hat{L}}_{x,y}$, where ${\hat{L}}_{x,y}\,:{E}_{i}^{+}(y)\to {E}_{i}^{-}(x)$ and one can show that $\parallel {\hat{L}}_{x,y}\parallel \leqslant \parallel {L}_{x,y}\parallel {\left(1-\parallel {L}_{x,y}\parallel \right)}^{-1}$.

For simplicity of notations, denote ${L}_{x,y},{\hat{L}}_{x,y}$ and ${{\rm{\Phi }}}_{x,y}^{\pm }$ by $L,\hat{L}$ and ${{\rm{\Phi }}}^{\pm }$ respectively. Note that

$$\begin{eqnarray*}&&{E}_{i}^{+}(x)={E}_{i}(x)\oplus {E}_{i-1}^{+}(x)={{\rm{\Phi }}}^{-1}{E}_{i}(y)\oplus {{\rm{\Phi }}}^{-1}{E}_{i-1}^{+}(y).\end{eqnarray*}$$

By the triangle inequality we have

$$\begin{eqnarray}&&\hat{d}({E}_{i}(x),{E}_{i}(y))\leqslant \hat{d}({E}_{i}(x),{{\rm{\Phi }}}^{-1}{E}_{i}(y))+\hat{d}({{\rm{\Phi }}}^{-1}{E}_{i}(y),{E}_{i}(y)).\end{eqnarray} \tag{ 12 }$$

We first estimate $\hat{d}({{\rm{\Phi }}}^{-1}{E}_{i}(y),{E}_{i}(y))$ in (12). For each $u\in {E}_{i}(y)$ with $\parallel u\parallel =1$, we have that

$$\begin{eqnarray*}&&\mathrm{dist}(u,{{\rm{\Phi }}}^{-1}{E}_{i}(y))\leqslant \parallel u-{{\rm{\Phi }}}^{-1}u\parallel \leqslant \parallel \hat{L}u\parallel \leqslant \displaystyle \frac{\parallel L\parallel }{1-\parallel L\parallel }\leqslant 2\parallel L\parallel .\end{eqnarray*}$$

Similarly, for each $v\in {{\rm{\Phi }}}^{-1}{E}_{i}(y)$ with $\parallel v\parallel =1$ one has that

$$\begin{eqnarray*}&&\mathrm{dist}(v,{E}_{i}(y))\leqslant \parallel v-{\rm{\Phi }}v\parallel \leqslant \parallel {Lv}\parallel \leqslant \parallel L\parallel .\end{eqnarray*}$$

Consequently, we obtain that $\hat{\delta }({{\rm{\Phi }}}^{-1}{E}_{i}(y),{E}_{i}(y))\leqslant 2\parallel L\parallel$. By (2) and lemma 3.7, we have

$$\begin{eqnarray*}&&\hat{d}({{\rm{\Phi }}}^{-1}{E}_{i}(y),{E}_{i}(y))\leqslant 4\parallel L\parallel \leqslant 4{\ell }(1+\parallel L\parallel )\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\leqslant 6{\ell }\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\end{eqnarray*}$$

where the last inequality use the fact that $\parallel L\parallel \leqslant 1$. Combining the above inequality and (10), we have that

$$\begin{eqnarray}&&\hat{d}({{\rm{\Phi }}}^{-1}{E}_{i}(y),{E}_{i}(y))\leqslant 6{\ell }{C}_{i}^{+}d{\left(x,y\right)}^{{\nu }_{i}^{+}}.\end{eqnarray} \tag{ 13 }$$

Next we estimate $\hat{d}({E}_{i}(x),{{\rm{\Phi }}}^{-1}{E}_{i}(y))$. Let ${A}_{n}={ \mathcal A }(x,n),{B}_{n}={ \mathcal A }(y,n)\circ {\rm{\Phi }},E={E}_{i}(x),F={{\rm{\Phi }}}^{-1}{E}_{i}(y)$ and $E^{\prime} ={E}_{i-1}^{+}(x),F^{\prime} ={{\rm{\Phi }}}^{-1}{E}_{i-1}^{+}(y)$.

For each $u\in F={{\rm{\Phi }}}^{-1}{E}_{i}(y)$ and each $n\in {\mathbb{N}}$, by (1) of theorem 3.1

$$\begin{eqnarray}\begin{array}{rcl}\parallel {B}_{n}u\parallel =\parallel { \mathcal A }(y,n){\rm{\Phi }}u\parallel & \leqslant & {\ell }{e}^{({\lambda }_{i}+\varepsilon )n}\cdot \parallel {\rm{\Phi }}u\parallel \\ & & \leqslant \parallel {\rm{\Phi }}\parallel {\ell }{e}^{({\lambda }_{i}+\varepsilon )n}\cdot \parallel u\parallel \\ & & \leqslant (1+\parallel L\parallel ){\ell }{e}^{({\lambda }_{i}+\varepsilon )n}\cdot \parallel u\parallel \\ & & \leqslant 2{\ell }{e}^{({\lambda }_{i}+\varepsilon )n}\cdot \parallel u\parallel \end{array}\end{eqnarray} \tag{ 14 }$$

where we use the fact that $\parallel L\parallel \lt 1/2$ in the last inequality.

For each $u\in F^{\prime} ={{\rm{\Phi }}}^{-1}{E}_{i-1}^{+}(y)$ and each $n\in {\mathbb{N}}$, by (1) of theorem 3.1 we have

$$\begin{eqnarray}\begin{array}{rcl}\parallel {B}_{n}u\parallel & = & \parallel { \mathcal A }(y,n){\rm{\Phi }}u\parallel \\ & & \geqslant {{\ell }}^{-1}{e}^{({\lambda }_{i-1}-\varepsilon )n}\cdot \parallel {\rm{\Phi }}u\parallel \\ & & \geqslant \parallel {{\rm{\Phi }}}^{-1}{\parallel }^{-1}{{\ell }}^{-1}{e}^{({\lambda }_{i-1}-\varepsilon )n}\cdot \parallel u\parallel \\ & & \geqslant (1-\parallel L\parallel ){{\ell }}^{-1}{e}^{({\lambda }_{i-1}-\varepsilon )n}\cdot \parallel u\parallel \\ & & \geqslant {\left(2{\ell }\right)}^{-1}{e}^{({\lambda }_{i-1}-\varepsilon )n}\cdot \parallel u\parallel \end{array}\end{eqnarray} \tag{ 15 }$$

where the third inequality uses the fact that $\parallel {{\rm{\Phi }}}^{-1}\parallel \leqslant 1+\parallel \hat{L}\parallel \leqslant {\left(1-\parallel L\parallel \right)}^{-1}$.

For every $u\in F,v\in F^{\prime}$, then ${\rm{\Phi }}u\in {E}_{i}(y),{\rm{\Phi }}v\in {E}_{i-1}^{+}(y)$ and, by (2) of theorem 3.1 one has that

$$\begin{eqnarray*}&&{\ell }\parallel {\rm{\Phi }}u+{\rm{\Phi }}v\parallel \geqslant \max \{\parallel {\rm{\Phi }}u\parallel ,\parallel {\rm{\Phi }}v\parallel \}.\end{eqnarray*}$$

This yields that

$$\begin{eqnarray*}&&{\ell }\parallel {\rm{\Phi }}\parallel \cdot \parallel u+v\parallel \geqslant \parallel {{\rm{\Phi }}}^{-1}{\parallel }^{-1}\cdot \max \{\parallel u\parallel ,\parallel v\parallel \}.\end{eqnarray*}$$

Since $\parallel L\parallel \lt 1/2$, we have that $\parallel {\rm{\Phi }}\parallel \cdot \parallel {{\rm{\Phi }}}^{-1}\parallel \leqslant (1+\parallel L\parallel ){\left(1-\parallel L\parallel \right)}^{-1}\leqslant 3$. Therefore, one has that

$$\begin{eqnarray}&&\max \{\parallel u\parallel ,\parallel v\parallel \}\leqslant 3{\ell }\parallel u+v\parallel .\end{eqnarray} \tag{ 16 }$$

Consider ${\alpha }_{1}={e}^{{\lambda }_{i-1}-\varepsilon },{\alpha }_{2}={e}^{{\lambda }_{i}+\varepsilon }$ and replace ℓ by $3{\ell }$. Clearly, the sequence $\{{A}_{n}\}$ satisfies condition (i) of lemma 3.4, and it follows from (14), (15) and (16) that the condition (ii) and (iii) of lemma 3.4 hold for every $n\in {\mathbb{N}}$.

At last, consider a constant $a\gt {\sup }_{x\in X}\parallel A(x)\parallel$ as in lemma 3.3, one can show that

$$\begin{eqnarray*}\begin{array}{rcl}\parallel {A}_{n}-{B}_{n}\parallel & \leqslant & \parallel { \mathcal A }(x,n)-{ \mathcal A }(x,n){\rm{\Phi }}\parallel +\parallel ({ \mathcal A }(x,n)-{ \mathcal A }(y,n)){\rm{\Phi }}\parallel \\ & & \leqslant \parallel { \mathcal A }(x,n)\parallel \cdot \parallel L\parallel +\parallel { \mathcal A }(x,n)-{ \mathcal A }(y,n)\parallel \cdot (1+\parallel L\parallel )\\ & & \leqslant {a}^{n}\parallel L\parallel +2{a}^{n}d{\left(x,y\right)}^{\nu }\\ & & \leqslant {a}^{n}(\parallel L\parallel +2d{\left(x,y\right)}^{\nu }).\end{array}\end{eqnarray*}$$

Let $\delta =\parallel L\parallel +2d{\left(x,y\right)}^{\nu }\lt 1$, since $a\gt {\alpha }_{1}\gt {\alpha }_{2}$ there exist $\hat{n}\in {\mathbb{N}}$ such that

$$\begin{eqnarray*}&&{\left(\displaystyle \frac{{\alpha }_{2}}{a}\right)}^{\hat{n}+1}\leqslant \delta \lt {\left(\displaystyle \frac{{\alpha }_{2}}{a}\right)}^{\hat{n}}.\end{eqnarray*}$$

It follows from lemma 3.4 that

$$\begin{eqnarray}&&\hat{d}({E}_{i}(x),{{\rm{\Phi }}}^{-1}{E}_{i}(y))\leqslant {\hat{C}}_{i}{\left(\parallel L\parallel +2d{\left(x,y\right)}^{\nu }\right)}^{{\hat{\nu }}_{i}}\end{eqnarray} \tag{ 17 }$$

where ${\hat{C}}_{i}=(4+6{\ell }){\left(3{\ell }\right)}^{2}{e}^{{\lambda }_{i-1}-{\lambda }_{i}-2\varepsilon }$ and ${\hat{\nu }}_{i}=({\lambda }_{i-1}-{\lambda }_{i}-2\varepsilon )/(\mathrm{log}a-{\lambda }_{i}-\varepsilon )\lt 1$. By lemma 3.7 and (10), there exist constants ${C}_{i}^{+}\gt 0$ and ${\nu }_{i}^{+}\in (0,\nu )$ so that

$$\begin{eqnarray*}&&\parallel L\parallel \leqslant {\ell }(1+\parallel L\parallel )\hat{d}({E}_{i}^{+}(x),{E}_{i}^{+}(y))\leqslant 2{\ell }{C}_{i}^{+}d{\left(x,y\right)}^{{\nu }_{i}^{+}}.\end{eqnarray*}$$

Since ${\nu }_{i}^{+}\lt \nu$ implies $d{\left(x,y\right)}^{{\nu }_{i}^{+}}\geqslant d{\left(x,y\right)}^{\nu }$, this together with (17) one has

$$\begin{eqnarray}&&\hat{d}({E}_{i}(x),{{\rm{\Phi }}}^{-1}{E}_{i}(y))\leqslant 2{\hat{C}}_{i}({\ell }{C}_{i}^{+}+1)d{\left(x,y\right)}^{{\nu }_{i}^{+}{\hat{\nu }}_{i}}.\end{eqnarray} \tag{ 18 }$$

Combining (12), (13) and (18), we have that

$$\begin{eqnarray}\begin{array}{rcl}\hat{d}({E}_{i}(x),{E}_{i}(y)) & \leqslant & 6{\ell }{C}_{i}^{+}d{\left(x,y\right)}^{{\nu }_{i}^{+}}+2{\hat{C}}_{i}({\ell }{C}_{i}^{+}+1)d{\left(x,y\right)}^{{\nu }_{i}^{+}{\hat{\nu }}_{i}}\\ & & \leqslant {C}_{i}d{\left(x,y\right)}^{{\nu }_{i}}\end{array}\end{eqnarray} \tag{ 19 }$$

where ${\nu }_{i}={\nu }_{i}^{+}{\hat{\nu }}_{i}$ and ${C}_{i}=6{\ell }{C}_{i}^{+}+2{\hat{C}}_{i}({\ell }{C}_{i}^{+}+1)$. This completes the proof. □

Let $({ \mathcal B },\parallel \cdot \parallel )$ be a separable Banach space, and let $f\,:{ \mathcal B }\to { \mathcal B }$ be a C² Fréchet differentiable map. We denote by Df(x) the derivative of f at point x and assume that the map x ↦ Df(x) is quasi-compact. This class of mapping includes time-t maps of semi-flows defined by periodically forced nonlinear dissipative parabolic PDEs (see [18] for details). We consider a compact attractor A, that is f(A) ⊂ A. Note that dynamical systems defined by dissipative PDEs are known to have attractors [22]. Let μ be an f-ergodic invariant Borel probability measure supported on A.

We consider the derivative cocycles on Banach space ${ \mathcal B }$ defined by

$$\begin{eqnarray*}&&F\,:A\times { \mathcal B }\to A\times { \mathcal B },(x,v)\mapsto (f(x),{Df}(x)v).\end{eqnarray*}$$

By theorem 4.1, for each ε > 0, there exists an f-invariant subset A₀ of full μ-measure such that there exist $k\in {\mathbb{N}}$ numbers λ₁ > ⋯ > λ_k > α(A, μ) + ε. For each x∈A₀, there is a filtration

$$\begin{eqnarray*}&&{ \mathcal B }={V}_{1}(x)\supset {V}_{2}(x)\supset \cdots \supset {V}_{k+1}(x)\end{eqnarray*}$$

satisfying theorem 4.1.

Therefore, by theorem 4.2 we have that the maps x ↦ V_i (x) (i = 1, ⋯ ,k + 1) are (locally) Hölder continuous on a compact set of arbitrarily large μ-measure. Moreover, if f(A) = A and f is a C² Fréchet diffeomorphisms on A. Then, for each x∈A₀, there is a splitting

$$\begin{eqnarray*}&&{ \mathcal B }={E}_{1}(x)\oplus {E}_{2}(x)\oplus \cdots {E}_{k}(x)\oplus F(x)\end{eqnarray*}$$

satisfying theorem 2.1. By theorem 2.2 we have that x ↦ E_i (x) (i = 1, ⋯ ,k) and x ↦ F(x) are (locally) Hölder continuous on a compact subset of arbitrarily large μ-measure.

Let ${ \mathcal B }$ be a Banach space, and let ${\{{A}_{i}\}}_{i=0}^{n-1}$ be a sequence of operators in ${GL}({ \mathcal B })$. We consider the compact metric space ${\rm{\Sigma }}=\{0,\cdots ,n-1\}{}^{{\mathbb{Z}}}$ with the following metric d :

$$\begin{eqnarray*}&&d(\{{a}_{i}\},\{{b}_{i}\})={\beta }^{\max \{n\,:{a}_{i}={b}_{i}\ \mathrm{forevery}\ | i| \leqslant n\}}\end{eqnarray*}$$

where 0 < β < 1 is a constant and {a_i }, {b_i } ∈ Σ. Let θ be the left shift map on Σ, and let μ be a θ-ergodic probability measure on Σ. Define the operator valued function $A\,:{\rm{\Sigma }}\to {GL}({ \mathcal B })$ by $A(\{{a}_{n}\})={A}_{{a}_{0}}$. Note that A is Hölder continuous since it is locally constant. Assume further that A is quasi-compact.

For each ε > 0, by theorem 2.2 there exists a θ-invariant subset Σ₀ of full μ-measure and $k\in {\mathbb{N}}$ numbers λ₁ > ⋯ > λ_k > α(A, μ) + ε such that for each x ∈ Σ₀, there is a splitting

$$\begin{eqnarray*}&&{ \mathcal B }={E}_{1}(x)\oplus {E}_{2}(x)\oplus \cdots {E}_{k}(x)\oplus F(x)\end{eqnarray*}$$

satisfying theorem 2.1. Applying theorem 2.2 one can obtain that the maps x ↦ E_i (x) (i = 1, ⋯ ,k) and x ↦ F(x) are (locally) Hölder continuous on a compact subset of arbitrarily large μ-measure.