2.1 A class of anisotropic Banach spaces introduced by Gouëzel and Liverani

The purpose of this subsection is to briefly summarize the main results from [Reference Gonzalez-Tokman and Quas26]. More precisely, we recall the properties of the so-called scale of anisotropic Banach spaces, on which the transfer operator associated to a transitive Anosov diffeomorphism has a spectral gap. The discussion we present here is relevant when building examples under which the abstract results of the present paper are applicable.

Let M denote a $C^\infty $ compact and connected Riemannian manifold. Furthermore, let T be a transitive Anosov diffeomorphism on M of class $C^{r+1}$ for $r>1$ . We denote the transfer operator associated with T by $\mathcal L_T$ . We recall that the action of $\mathcal L_T$ on smooth functions $h\in C^r(M, \mathbb R)$ is given by

$$ \begin{align*} \mathcal L_T h=(h\lvert \det (DT)\rvert^{-1})\circ T^{-1}. \end{align*} $$

Let us now briefly summarize the main results from [Reference Gonzalez-Tokman and Quas26]. Take $p\in \mathbb {N}$ , $p\le r$ and $q>0$ such that $p+q<r$ . It is proved in [Reference Gonzalez-Tokman and Quas26] that there exist Banach spaces $\mathcal B^{p,q}=(\mathcal B^{p,q}, \lVert \cdot \rVert _{p,q})$ and $\mathcal B^{p-1, q+1}=(\mathcal B^{p-1, q+1}, \lVert \cdot \rVert _{p-1, q+1})$ with the following properties.

• • By construction, $C^r(M,\mathbb {R})$ is dense in $\mathcal B^{i,j}$ for $(i,j)=\{(p,q), (p-1, q+1)\}$ .

• • By [Reference Gonzalez-Tokman and Quas26, Lemma 2.1], $\mathcal B^{p,q}$ can be embedded in $\mathcal B^{p-1, q+1}$ and the unit ball of $\mathcal B^{p,q}$ is relatively compact in $\mathcal B^{p-1, q+1}$ .

• • By [Reference Gonzalez-Tokman and Quas26, Proposition 4.1], elements of $\mathcal B^{p,q}$ are distributions of order at most q.

• • By [Reference Gonzalez-Tokman and Quas26, Lemma 3.2], multiplication by a $C^{k+q}$ function, $1\le k\le p$ , induces a bounded operator on $\mathcal B^{p,q}$ . Moreover, the action of a $C^r$ vector field induces a bounded operator from $\mathcal B^{p,q}$ to $\mathcal B^{p-1,q+1}$ .

• • Here $\mathcal L_T$ acts as a bounded operator on $\mathcal B^{i,j}$ for $(i,j)=\{(p,q), (p-1, q+1)\}$ . Moreover, for each $h\in \mathcal B^{i,j}$ and $\varphi \in C^j(M, \mathbb {R})$ , we have that

  $$ \begin{align*} (\mathcal L_T h)(\varphi)=h(\varphi \circ T), \end{align*} $$
  where we denote the action of a distribution h on a test function $\varphi $ by $h(\varphi )$ .

• • By [Reference Gonzalez-Tokman and Quas26, Lemma 2.2], there exist $A>0$ and $a\in (0,1)$ such that

  $$ \begin{align*} \lVert \mathcal L_T^n h\rVert_{p-1, q+1} \le A\lVert h\rVert_{p-1, q+1} \quad \text{for }n\in \mathbb{N} \text{ and }h\in \mathcal B^{p-1, q+1} \end{align*} $$
  and
  $$ \begin{align*} \lVert \mathcal L_T^n h\rVert_{p,q} \le Aa^n \lVert h\rVert_{p,q}+A\lVert h\rVert_{p-1, q+1} \quad \text{for }n\in \mathbb{N}\text{ and }h\in \mathcal B^{p, q}. \end{align*} $$

• • By [Reference Gonzalez-Tokman and Quas26, Theorem 2.3], $\mathcal L_T$ is a quasi-compact operator on $\mathcal B^{p,q}$ with spectral radius 1. Moreover, $1$ is the only eigenvalue of $\mathcal L_T$ on the unit circle. Finally, $1$ is a simple eigenvalue of $\mathcal L_T$ and the corresponding eigenspace is spanned by the unique S.R.B. measure for T.

2.2 Regularity assumptions

In this section, we state precisely our regularity assumptions and our main theorem. We start by fixing, once and for all, the system of $C^\infty $ coordinates chart to be $(\psi _i)_{i=1,\ldots ,N}$ , where $\psi _i:(-r_i,r_i)^d\to M$ , and such that the $X_i=\psi _i((-r_i/2,r_i/2)^d)$ cover M are given by the anisotropic norm construction (see [Reference Gonzalez-Tokman and Quas26]). We also let $\delta $ be the Lebesgue number of the previous cover. Recall the following fact: if T and S are $C^{r+1}$ maps from M to itself, such that $\sup _{x\in M}d_M(Tx,Sx)\le {\delta }/{2}$ , then one has: for any $i\in \{1,\ldots ,N\}$ ,

$$ \begin{align*}\mathcal J_S(i):=\{j\in\{1,\ldots,N\},~S(X_i)\cap X_j\not=\emptyset \}=\mathcal J_T(i),\end{align*} $$

and one may write

$$ \begin{align*}d_{C^{r+1}}(T,S)=\sum_{i=1}^N\sum_{j\in\mathcal J(i)}\|T_{ij}-S_{ij}\|_{C^{r+1}},\end{align*} $$

where $\mathcal J(i)=\mathcal J_S(i)=\mathcal J_T(i)$ and $T_{ij}=\psi _j^{-1}\circ T\circ \psi _i:(-r_i,r_i)^d\to (-r_j,r_j)^d$ is a map between open sets in $\mathbb R^d$ .

For an interval $0\in I\subset \mathbb R$ , we consider a $C^s$ mapping $\mathcal T:I\to C^{r+1}(M,M)$ , such that $T_0:=\mathcal T(0)(\cdot )$ is a $C^{r+1}$ , transitive Anosov diffeomorphism. Up to shrinking I, we may and do assume that for all $\varepsilon \in I$ , $T_\varepsilon :=\mathcal T(\varepsilon )(\cdot )$ is a $C^{r+1}$ Anosov diffeomorphism, and that $\sup _{\varepsilon \in I}d_{C^{r+1}}(T_\varepsilon ,T_0)\le {\delta }/{4}$ . In particular, for any $i\in \{1,\ldots ,N\}$ , the set

$$ \begin{align*}\mathcal J_\varepsilon(i):=\{j\in\{1,\ldots,N\},~T_\varepsilon(X_i)\cap X_j\not=\emptyset \}\end{align*} $$

is independent of $\varepsilon $ . We informally refer to this property by saying that ‘the maps $T_\varepsilon $ may be read in the same charts’.

Consider now a $\Delta>0$ , and set $V:=B_{C^s(I,C^{r+1}(M,M))}(\mathcal T,\Delta )$ , that is, we consider a small ball, in $C^s(I,C^{r+1}(M,M))$ topology, centered at $\mathcal T$ . Up to shrinking $\Delta $ , we may assume that for any $\mathcal S\in V$ , any $\varepsilon \in I$ , $ S_\varepsilon :=\mathcal S(\varepsilon )(\cdot )$ is an Anosov diffeomorphism, and that $\sup _{\varepsilon \in I}d_{C^{r+1}}(T_\varepsilon , S_\varepsilon )\le {\delta }/{4}$ . In particular, for any $i\in \{1,\ldots ,N\}$ , the sets

$$ \begin{align*}\mathcal J_{\mathcal S}(i):=\{j\in\{1,\ldots,N\},~ S_\varepsilon(X_i)\cap X_j\not=\emptyset \}\end{align*} $$

are independent of $\varepsilon $ and $\mathcal S$ both (i.e. they only depend on V).

We may now describe the type of perturbed cocycle we will consider in the following:

Hypothesis 1. Let $r>4$ , $s>1$ , and $0\in I\subset \mathbb R$ an interval; let $\mathcal T$ and $V\subset C^s(I,C^{r+1}(M,M))$ be as described previously. Furthermore, let $(\Omega ,\mathcal F, \mathbb P)$ be a probability space, $\sigma \colon \Omega \to \Omega $ an invertible, ergodic $\mathbb P$ -preserving transformation and consider a measurable mapping

$$ \begin{align*}\mathbf{T}\colon \Omega \to V\end{align*} $$

Set $T_{\omega ,\varepsilon }:=\mathbf {T}(\omega )(\varepsilon )( \cdot )$ , $\omega \in \Omega $ and $\varepsilon \in I$ .

Let us make a few comments on this assumption, based on the previous discussion.

• • We choose the neighborhood V sufficiently small so that for any $\omega \in \Omega $ , any $\varepsilon \in I$ , the collection of $T_{\omega ,\varepsilon }$ can all be read in the same coordinate charts and share the same set of admissible leaves. In particular, one may study their transfer operators on the same anisotropic Banach spaces.

• • Our assumption is tailored so that for each fixed $\omega \in \Omega $ , $\varepsilon \mapsto T_{\omega ,\varepsilon }$ is a smooth curve of Anosov diffeomorphisms, all close-by to a fixed one (namely, $T_{\omega , 0}$ ).

We are now in position to formulate our main result.

Theorem 1. Let $(T_{\omega ,\varepsilon })_{\omega \in \Omega ,\varepsilon \in I}$ be a parametrized cocycle of Anosov diffeomorphisms, satisfying Hypothesis 1. Then, by shrinking I if necessary, there exists a triplet of Banach spaces

$$ \begin{align*}\mathcal B_{ss} \subset \mathcal B_s\subset \mathcal B_w, \end{align*} $$

and for each $\varepsilon \in I$ a unique family $(h_{\omega }^\varepsilon )_{\omega \in \Omega }\subset \mathcal B_{ss}$ with the following properties:

• • $\omega \mapsto h_\omega ^\varepsilon $ is measurable for each $\varepsilon \in I$ ;

• • $h_\omega ^\varepsilon $ is a probability measure for $\varepsilon \in I$ and $\omega \in \Omega $ ;

• • $\mathcal L_{\omega , \varepsilon } h_\omega ^\varepsilon =h_{\sigma \omega }^\varepsilon $ for $\varepsilon \in I$ and $\omega \in \Omega $ , where $\mathcal L_{\omega , \varepsilon }$ denotes the transfer operator of $T_{\omega , \varepsilon }$ ;

• • the map $I\ni \varepsilon \mapsto h_{\omega }^\varepsilon \in L^\infty (\Omega ,\mathcal B_w)$ is differentiable at $0$ , and for $\phi \in C^{r}(M)$ , we have that

  (3) $$ \begin{align} \partial_\varepsilon\bigg[\int_M\phi \,dh_\omega^\varepsilon \bigg] \bigg{\rvert}_{\varepsilon=0}=\sum_{n=0}^\infty\int_M \partial_{\varepsilon}[\phi\circ T_{\sigma^{-n}\omega}^{n}\circ T_{\sigma^{-n-1}\omega,\varepsilon}] \bigg{\rvert}_{\varepsilon=0}\,dh_{\sigma^{-n-1}\omega}, \end{align} $$
  where $h_\omega :=h_\omega ^0$ , $\omega \in \Omega $ .

2.3 Examples

Here we give explicit examples of systems satisfying Hypothesis 1. In all instances, $r>4$ and $s>1$ .

Example 2. Let $q\in \mathbb N$ , $\Omega =\{1,\ldots ,q\}^{\mathbb Z}$ , endowed with a Bernoulli measure. Consider a family $(T_1,\ldots ,T_q)$ of (close-enough) $C^{r+1}$ Anosov diffeomorphisms of the d-dimensional torus $\mathbb {T}^d$ , where $p:\mathbb T^d\to \mathbb T^d$ is a $C^{r+1}$ mapping and $0\in I\subset \mathbb {R}$ is an interval. We set

(4) $$ \begin{align} \mathbf T(\omega)(\varepsilon,x):=T_i(x)+\varepsilon p(x),\quad\text{if}~\omega_0=i, \end{align} $$

where $x\in \mathbb T^d$ , $\varepsilon \in I$ , and $\omega =(\omega _n)_{n\in \mathbb Z}\in \Omega $ .

Example 3. Let $q\in \mathbb N$ , $\Omega =\{1,\ldots ,q\}^{\mathbb Z}$ , endowed with a Bernoulli measure. Consider a $C^{r+1}$ Anosov diffeomorphism T of $\mathbb {T}^d$ . Moreover, consider $p_1,\ldots ,p_q\ C^{r+1}$ mappings of $\mathbb T^d$ and $0\in I\subset \mathbb {R}$ an interval, Then we define, for $\varepsilon \in I$ , $x\in \mathbb T^d$ , and $\omega =(\omega _n)_{n\in \mathbb Z}\in \Omega $ , the random map

(5) $$ \begin{align} \mathbf T(\omega)(\varepsilon,x)= T(x)+\varepsilon p_i(x) \quad \text{if}~\omega_0=i. \end{align} $$

In both Examples 2 and 3, for each $\omega \in \Omega $ , $\mathbf T(\omega )\in C^s(I,C^{r+1}(M,M))$ . Furthermore, because for each $i\in \{1,\ldots ,q\}$ , the set $\{\mathbf {T}(\omega )=T_i+\varepsilon p\}$ (respectively, $\{\mathbf {T}(\omega )=T+\varepsilon p_i\}$ ) is the 1-cylinder $\{\omega _0=i\}$ , one easily checks that the map is measurable.

Example 4. We now consider the following setting: for $\delta>0$ , $\omega \in B_{\mathbb {R}^d}(0,\delta )$ (that is, randomly chosen with respect to Lebesgue measure) and $\varepsilon _0>0$ , we consider a $C^s$ -smooth curve of Anosov diffeomorphisms

$$ \begin{align*} I:=(-\varepsilon_0,\varepsilon_0) \ni \varepsilon \to T_\varepsilon\in C^{r+1}(\mathbb T^d,\mathbb T^d).\end{align*} $$

Finally, set

$$ \begin{align*}\mathbf T(\omega)(\varepsilon,x):= T_\varepsilon(x)+\omega, \quad x\in \mathbb T^d.\end{align*} $$

In this last instance, one easily checks that the map $\Omega \ni \omega \mapsto \mathbf T(\omega )\in C^s(I,C^{r+1}(M,M))$ is continuous and, thus, measurable.

3.1 Quenched statistical stability for random systems

In this section, we formulate an abstract result regarding the statistical stability of certain random dynamical systems that applies, in particular, to random hyperbolic dynamics.

Let $(\Omega , \mathcal F, \mathbb P)$ be a probability space and consider an invertible transformation $\sigma \colon \Omega \to \Omega $ which preserves $\mathbb P$ . Furthermore, let $\mathbb P$ be ergodic.

Moreover, let $\mathcal B_w=(\mathcal B_w, \lVert \cdot \rVert _w)$ and $\mathcal B_s=(\mathcal B_s, \lVert \cdot \rVert _s)$ be two Banach spaces such that $\mathcal B_s$ is embedded in $\mathcal B_w$ and that $\lVert \cdot \rVert _w \le \lVert \cdot \rVert _s$ on $\mathcal B_s$ . Suppose that for each $\omega \in \Omega $ , $\mathcal L_\omega $ is a bounded operator both on $\mathcal B_w$ and $\mathcal B_s$ . In addition, assume that $\omega \to \mathcal L_\omega $ is strongly measurable on $\mathcal B_s$ , that is, that the map $\omega \mapsto \mathcal L_\omega h$ is measurable for each $h\in \mathcal B_s$ . For $\omega \in \Omega $ and $n\in \mathbb {N}$ , set

$$ \begin{align*} \mathcal L_\omega^n:=\mathcal L_{\sigma^{n-1} \omega}\circ \cdots \circ \mathcal L_{\sigma \omega}\circ \mathcal L_\omega. \end{align*} $$

We consider a fixed, non-zero $\psi \in \mathcal B_s^{\prime }$ that admits a bounded extension to $\mathcal B_w$ that we still denote by $\psi $ , and assume that there exist $D, \lambda>0$ such that

(6) $$ \begin{align} \lVert \mathcal L_\omega^n h\rVert_s \le De^{-\lambda n} \lVert h\rVert_s, \end{align} $$

for $\mathbb P$ -almost every $\omega \in \Omega $ , $n\in \mathbb {N}$ , and $h\in \mathcal B_s^0$ , where

(7) $$ \begin{align} \mathcal B_s^0=\{h\in \mathcal B_s: \psi(h)=0\}. \end{align} $$

Obviously, $\mathcal B_s^0$ depends on the choice of $\psi $ . However, this dependence has no bearing on our results (see Remark 5), so we do not make it explicit in the notation itself.

Consider now an interval $I\subset \mathbb {R}$ around $0\in \mathbb {R}$ and suppose that for $\varepsilon \in I$ , we have a family $(\mathcal L_{\omega , \varepsilon })_{\omega \in \Omega }$ of bounded linear operators on spaces $\mathcal B_s$ and $\mathcal B_w$ . Moreover, assume that $\omega \mapsto \mathcal L_{\omega , \varepsilon }$ is strongly measurable on $\mathcal B_s$ for each $\varepsilon \in I$ . Analogously to $\mathcal L_\omega ^n$ , for $\omega \in \Omega $ , $\varepsilon \in I$ , and $n\in \mathbb {N}$ , we define

$$ \begin{align*} \mathcal L_{\omega, \varepsilon}^n:=\mathcal L_{\sigma^{n-1} \omega, \varepsilon}\circ \cdots \circ \mathcal L_{\sigma \omega, \varepsilon}\circ \mathcal L_{\omega, \varepsilon}. \end{align*} $$

We set $\mathcal L_{\omega , 0}=\mathcal L_\omega $ and we suppose that there exist $C>0$ , $\lambda _1 \in (0,1)$ , and a measurable $\Omega ^{\prime } \subset \Omega $ satisfying $\mathbb P(\Omega ^{\prime })=1$ such that for each $\varepsilon \in I$ :

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , $n\in \mathbb {N}$ , and $h\in \mathcal B_s$ ,

  (8) $$ \begin{align} \lVert \mathcal L_{\omega, \varepsilon}^nh\rVert_s \le C\lambda_1^n \lVert h\rVert_s+C\lVert h\rVert_w; \end{align} $$

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , and $h\in \mathcal B_s$ ,

  (9) $$ \begin{align} \lVert (\mathcal L_{\omega, \varepsilon} -\mathcal L_\omega)h\rVert_w\le C\lvert \varepsilon \rvert \cdot \lVert h\rVert_s; \end{align} $$

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , and $n\in \mathbb {N}$ ,

  (10) $$ \begin{align} \lVert \mathcal L_{\omega, \varepsilon}^n\rVert_w \le C; \end{align} $$

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , we have that

  (11) $$ \begin{align} \psi(\mathcal L_{\omega, \varepsilon} h)=\psi(h) \quad \text{for each }h\in \mathcal B_s. \end{align} $$

We can assume without any loss of generality that $\Omega ^{\prime }$ is contained in a full measure set on which (6) holds.

We first show that the above assumptions imply that all the perturbed cocycles $(\mathcal L_{\omega , \varepsilon })_{\omega \in \Omega }$ also satisfy the condition of the type (6) whenever $\lvert \varepsilon \rvert $ is sufficiently small. More precisely, we have the following auxiliary result.

Proposition 6. There exist $\varepsilon _0, D^{\prime }>0$ and $\lambda ^{\prime }>0$ such that

(12) $$ \begin{align} \lVert \mathcal L_{\omega, \varepsilon}^n h\rVert_s \le D^{\prime}e^{-\lambda^{\prime} n} \lVert h\rVert_s, \end{align} $$

for $\varepsilon \in I$ satisfying $\lvert \varepsilon \rvert \le \varepsilon _0$ , $\omega \in \Omega ^{\prime }$ , $n\in \mathbb {N}$ , and $h\in \mathcal B_s^0$ .

Proof. Let $\varepsilon _0>0$ be such that

(13) $$ \begin{align} \frac{C^4}{1-\lambda_1}\varepsilon_0<1/2, \end{align} $$

and take an arbitrary $\varepsilon \in I$ satisfying $\lvert \varepsilon \rvert \le \varepsilon _0$ .

As

$$ \begin{align*} \mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n=\sum_{k=1}^n \mathcal L_{\sigma^k \omega, \varepsilon}^{ n-k}(\mathcal L_{\sigma^{k-1} \omega, \varepsilon}-\mathcal L_{\sigma^{k-1} \omega})\mathcal L_\omega^{k-1}, \end{align*} $$

it follows from (8), (9), and (10) that

$$ \begin{align*} \begin{split} \lVert (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n) h\rVert_w &\le \sum_{k=1}^n \lVert \mathcal L_{\sigma^k \omega, \varepsilon}^{ n-k}(\mathcal L_{\sigma^{k-1} \omega, \varepsilon}-\mathcal L_{\sigma^{k-1} \omega})\mathcal L_\omega^{k-1}h\rVert_w\\ &\le C\sum_{k=1}^n \lVert (\mathcal L_{\sigma^{k-1} \omega, \varepsilon}-\mathcal L_{\sigma^{k-1} \omega})\mathcal L_\omega^{k-1}h\rVert_w\\ &\le C^2 \lvert \varepsilon \rvert \sum_{k=1}^n \lVert \mathcal L_\omega^{k-1}h\rVert_s\\ &\le C^2 \lvert \varepsilon \rvert \sum_{k=1}^n(C\lambda_1^{k-1}\lVert h\rVert_s+C\lVert h\rVert_w) \\ &\le C^3 \lvert \varepsilon \rvert \bigg{(}\frac{1}{1-\lambda_1} \lVert h\rVert_s+n \lVert h\rVert_w \bigg{)}, \end{split} \end{align*} $$

and, thus,

(14) $$ \begin{align} \lVert (\mathcal L_{\omega, \varepsilon}^n-\mathcal L_\omega^n) h\rVert_w \le C^3 \lvert \varepsilon \rvert \bigg{(}\frac{1}{1-\lambda_1} \lVert h\rVert_s+n \lVert h\rVert_w \bigg{)}, \end{align} $$

for $n\in \mathbb {N}$ , $\omega \in \Omega ^{\prime }$ , and $h\in \mathcal B_s$ . Thus, (6), (8), and (14) imply that

$$ \begin{align*} \begin{split} \lVert \mathcal L_{\omega, \varepsilon}^{ n+m} h\rVert_s &=\lVert \mathcal L_{\sigma^m \omega, \varepsilon}^n \mathcal L_{\omega, \varepsilon}^m h\rVert_s \\ &\le C\lambda_1^n \lVert \mathcal L_{\omega, \varepsilon}^m h\rVert_s +C\lVert \mathcal L_{\omega, \varepsilon}^m h\rVert_w\\ &\le C\lambda_1^n (C\lambda_1^m \lVert h\rVert_s+C\lVert h\rVert_w)+C(\lVert \mathcal L_\omega^{ m} h\rVert_w+\lVert (\mathcal L_{\omega, \varepsilon}^m-\mathcal L_\omega^m) h\rVert_w)\\ &\le C^2\lambda_1^{n+m}\lVert h\rVert_s+C^2 \lambda_1^n \lVert h\rVert_s+CDe^{-\lambda m} \lVert h\rVert_s+C^4 \lvert \varepsilon \rvert \bigg{(}\frac{1}{1-\lambda_1} +m \bigg{)} \lVert h\rVert_s, \end{split} \end{align*} $$

for $n, m\in \mathbb {N}$ , $\omega \in \Omega ^{\prime }$ , and $h\in \mathcal B_s^0$ . Hence (recall also (13)), we can find (by decreasing $\varepsilon _0$ if necessary) $a\in (0, 1)$ and $N_0\in \mathbb {N}$ (independent of $\varepsilon $ and $\omega $ ) such that

(15) $$ \begin{align} \lVert \mathcal L_{\omega, \varepsilon}^{ N_0}h \rVert_s\le a \lVert h\rVert_s, \end{align} $$

for $\omega \in \Omega ^{\prime }$ and $h\in \mathcal B_s^0$ .

On the other hand, it follows readily from (8) that

(16) $$ \begin{align} \lVert \mathcal L_{\omega, \varepsilon}^n \rVert_s \le 2C \quad \text{for }n\in \mathbb{N} \text{ and } \omega \in \Omega^{\prime}. \end{align} $$

Take now an arbitrary $n\in \mathbb {N}$ and write it as $n=mN_0+k$ for $m, k \in \mathbb {N} \cup \{0\}$ , $0\le k<N_0$ . It follows from (15) and (16) that

$$ \begin{align*} \begin{split} \lVert \mathcal L_{\omega, \varepsilon}^n h\rVert_s =\lVert \mathcal L_{\omega, \varepsilon}^{mN_0+k}h\rVert_s &\le 2C a^m \lVert h\rVert_s \\ &=2Ce^{-m \log a^{-1}}\lVert h\rVert_s \\ &=2Ce^{({k}/{N_0})\log a^{-1}}e^{-({n}/{N_0})\log a^{-1}}\lVert h\rVert_s \\ &\le 2Ce^{\log a^{-1}}e^{-({n}/{N_0})\log a^{-1}}\lVert h\rVert_s, \end{split} \end{align*} $$

for $\omega \in \Omega ^{\prime }$ , $n\in \mathbb {N}$ , $h\in \mathcal B_s^0$ . We conclude that (12) holds with

$$ \begin{align*} \lambda^{\prime}=\log a^{-1}/N_0>0 \quad \text{and} \quad D^{\prime}=2Ce^{\log a^{-1}}>0,\end{align*} $$

which are independent on $\varepsilon $ . The proof of the proposition is completed.

We are now in position to establish the existence of a random fixed point for the cocycle $(\mathcal L_{\omega , \varepsilon })_{\omega \in \Omega }$ whenever $\lvert \varepsilon \rvert \le \varepsilon _0$ .

Proof. Let $\mathcal Y$ denote the set of all measurable functions $v\colon \Omega ^{\prime } \to \mathcal B_s$ such that

$$ \begin{align*} \lVert v\rVert_\infty=\sup_{\omega \in \Omega^{\prime}} \lVert v(\omega)\rVert_s<\infty. \end{align*} $$

Then, $(\mathcal Y, \lVert \cdot \rVert _\infty )$ is a Banach space. Set

$$ \begin{align*} \mathcal Z:=\{v\in \mathcal Y: \psi(v(\omega))=1 \ \text{for }\omega \in \Omega^{\prime} \}. \end{align*} $$

Observe that $\mathcal Z$ is nonempty. Indeed, because $\psi $ is non-zero, there exists $g\in \mathcal B_s$ such that $\psi (g)=1$ . Set $v_0\colon \Omega ^{\prime } \to \mathcal B_s$ by $v_0(\omega )=g$ for $\omega \in \Omega ^{\prime }$ . Then, $v_0\in \mathcal Z$ . We claim that $\mathcal Z$ is a closed subset of $\mathcal Y$ . Indeed, let $(v_n)_n$ be a sequence in $\mathcal Z$ that converges to some $v\in \mathcal Y$ . Then, we have that

$$ \begin{align*} \lvert \psi(v_n(\omega))-\psi(v(\omega))\rvert \le \lVert \psi \rVert_s\cdot \lVert v_n(\omega)-v(\omega)\rVert_s\le \lVert \psi \rVert_s \cdot \lVert v_n-v\rVert_\infty, \end{align*} $$

for $n\in \mathbb {N}$ and $\omega \in \Omega ^{\prime }$ , where $\|\psi \|_s$ denotes the norm of $\psi \in \mathcal B_s^{\prime }$ . Hence, $\psi (v(\omega ))=1$ for $\omega \in \Omega ^{\prime }$ and, thus, $v\in \mathcal Z$ .

For $\lvert \varepsilon \rvert \le \varepsilon _0$ , we define a linear operator $\mathbb L^\varepsilon \colon \mathcal Y \to \mathcal Y$ by

$$ \begin{align*} (\mathbb L^\varepsilon v)(\omega)=\mathcal L_{\sigma^{-1} \omega, \varepsilon} v(\sigma^{-1}\omega), \quad \omega \in \Omega^{\prime}. \end{align*} $$

It follows from (16) (together with our assumption that $\omega \mapsto \mathcal L_{\omega , \varepsilon }$ is strongly measurable on $\mathcal B_s$ for each $\varepsilon $ ) that $\mathbb L^\varepsilon $ is a well-defined and bounded operator. Moreover, $\mathbb L^\varepsilon \mathcal Z\subset \mathcal Z$ . Indeed, for each $v\in \mathcal Z$ we have (using (11)) that

$$ \begin{align*} \psi((\mathbb L^\varepsilon v)(\omega))=\psi(\mathcal L_{\sigma^{-1} \omega, \varepsilon} v(\sigma^{-1}\omega))=\psi(v(\sigma^{-1}\omega))=1, \end{align*} $$

for $\omega \in \Omega ^{\prime }$ . Thus, $\mathbb L^\varepsilon v \in \mathcal Z$ .

Let us now choose $N\in \mathbb {N}$ such that $D^{\prime }e^{-\lambda ^{\prime }N}<1$ . It follows from (12) that

$$ \begin{align*} \begin{split} \lVert (\mathbb L^\varepsilon)^Nv_1-(\mathbb L^\varepsilon)^Nv_2\rVert_\infty &=\sup_{\omega \in \Omega^{\prime}} \lVert \mathcal L_{\sigma^{-N} \omega, \varepsilon}^N (v_1(\sigma^{-N} \omega)-v_2(\sigma^{-N} \omega))\rVert_s \\ &\le D^{\prime}e^{-\lambda^{\prime}N}\sup_{\omega \in \Omega^{\prime}} \lVert v_1(\sigma^{-N} \omega)-v_2(\sigma^{-N} \omega)\rVert_s \\ &\le D^{\prime}e^{-\lambda^{\prime} N} \lVert v_1-v_2\rVert_\infty, \end{split} \end{align*} $$

for $\lvert \varepsilon \rvert \le \varepsilon _0$ and $v_1, v_2\in \mathcal Z$ . Hence, $(\mathbb L^\varepsilon )^N$ is a contraction on $\mathcal Z$ and therefore, $\mathbb L^\varepsilon $ has a unique fixed point $v^\varepsilon \in \mathcal Z$ . Thus, the family $(h_\omega ^\varepsilon )_{\omega \in \Omega ^{\prime }}$ defined by $h_\omega ^\varepsilon :=v^\varepsilon (\omega )$ satisfies (17), (18), and (19).

In order to establish the uniqueness, it is sufficient to note that each family $(h_\omega ^\varepsilon )_{\omega \in \Omega ^{\prime }}$ satisfying (17), (18), and (19) gives rise to a fixed point of $\mathbb L^\varepsilon $ in $\mathcal Z$ , which is unique. The proof of the proposition is complete.

Set

$$ \begin{align*} h_\omega :=h_\omega^0 \quad \omega \in \Omega^{\prime}. \end{align*} $$

The following is our statistical stability result.

Theorem 8. Let $\varepsilon \in I$ , $|\varepsilon |\le \varepsilon _0$ . Then

(20) $$ \begin{align} \sup_{\omega \in \Omega^{\prime}} \lVert h_\omega^{\varepsilon} -h_\omega \rVert_{w} \le C|\varepsilon|\lvert\log(|\varepsilon|)\rvert, \end{align} $$

where $C>0$ is independent on $\varepsilon $ .

Before we establish Theorem 8, we need the following auxiliary result. Let $h^\varepsilon $ denote the family $(h_\omega ^\varepsilon )_{\omega \in \Omega }$ given by Proposition 7.

Lemma 9. We have that

(21) $$ \begin{align} \sup_{\lvert \varepsilon \rvert \le \varepsilon_0}\sup_{\omega \in \Omega^{\prime}} \lVert h_\omega^\varepsilon \rVert_s < \infty. \end{align} $$

Proof. We use the same notation as in the proof of Proposition 7. Take an arbitrary $u\in \mathcal Z$ . It follows from Banach’s contraction principle that

$$ \begin{align*} h^\varepsilon =\lim_{k\to \infty} ( \mathbb L^\varepsilon)^{kN} u, \end{align*} $$

for $\lvert \varepsilon \rvert \le \varepsilon _0$ . Fix now any $\varepsilon $ such that $\lvert \varepsilon \rvert \le \varepsilon _0$ . There exists $k_0\in \mathbb {N}$ such that

$$ \begin{align*} \lVert h^\varepsilon -( \mathbb L^\varepsilon)^{k_0N} u \rVert_\infty <1. \end{align*} $$

Hence, using (8) we have that

$$ \begin{align*} \lVert h^\varepsilon \rVert_\infty \le 1+ \lVert ( \mathbb L^\varepsilon)^{k_0N} u \rVert_\infty \le 2C\lVert u\rVert_\infty+1, \end{align*} $$

which readily implies the conclusion of the lemma.

We are now in a position to prove Theorem 8.

Proof of Theorem 8

Take an arbitrary $\varepsilon \in I$ such that $|\varepsilon |\le \varepsilon _0$ . Observe that

(22) $$ \begin{align} \lVert h_\omega^{\varepsilon} -h_\omega \rVert_w &=\lVert \mathcal L_{\sigma^{-n} \omega, \varepsilon}^n h_{\sigma^{-n} \omega}^{\varepsilon} -\mathcal L_{\sigma^{-n} \omega}^n h_{\sigma^{-n} \omega}\rVert_w \nonumber\\ &\le \lVert \mathcal L_{\sigma^{-n} \omega, \varepsilon}^n h_{\sigma^{-n} \omega}^{\varepsilon} -\mathcal L_{\sigma^{-n} \omega}^n h_{\sigma^{-n} \omega}^{\varepsilon}\rVert_w+\lVert \mathcal L_{\sigma^{-n} \omega}^n (h_{\sigma^{-n} \omega}^{\varepsilon}-h_{\sigma^{-n} \omega})\rVert_w, \end{align} $$

for each $n\in \mathbb {N}$ and $\omega \in \Omega ^{\prime }$ . It follows from (6) and (21) that there exists $\tilde D>0$ such that

(23) $$ \begin{align} \lVert \mathcal L_{\sigma^{-n} \omega}^n (h_{\sigma^{-n} \omega}^{\varepsilon}-h_{\sigma^{-n} \omega})\rVert_w \le \lVert \mathcal L_{\sigma^{-n} \omega}^n (h_{\sigma^{-n} \omega}^{\varepsilon}-h_{\sigma^{-n} \omega})\rVert_s \le \tilde De^{-\lambda n}, \end{align} $$

for $n\in \mathbb {N}$ and $\omega \in \Omega ^{\prime }$ .

On the other hand, it follows from (8), (9), and (10) that

$$ \begin{align*} \begin{split} &\lVert \mathcal L_{\sigma^{-n} \omega, \varepsilon}^n h_{\sigma^{-n} \omega}^{\varepsilon} -\mathcal L_{\sigma^{-n} \omega}^n h_{\sigma^{-n} \omega}^{\varepsilon}\rVert_w\\ &\quad\le \sum_{j=1}^n \lVert \mathcal L_{\sigma^{-n+j} \omega}^{n-j} (\mathcal L_{\sigma^{-n+j-1}\omega}-\mathcal L_{\sigma^{-n+j-1}\omega, \varepsilon})\mathcal L_{\sigma^{-n} \omega, \varepsilon}^{ j-1}h_{\sigma^{-n} \omega}^{\varepsilon} \rVert_w\\ &\quad\le C\sum_{j=1}^n \lVert (\mathcal L_{\sigma^{-n+j-1}\omega}-\mathcal L_{\sigma^{-n+j-1}\omega, \varepsilon})\mathcal L_{\sigma^{-n} \omega, \varepsilon}^{ j-1}h_{\sigma^{-n} \omega}^{\varepsilon}\rVert_w\\ &\quad\le C^2\lvert \varepsilon \rvert \sum_{j=1}^n \lVert \mathcal L_{\sigma^{-n} \omega, \varepsilon}^{ j-1}h_{\sigma^{-n} \omega}^{\varepsilon} \rVert_s\\ &\quad\le 2nC^3\lvert \varepsilon \rvert \cdot \lVert h_{\sigma^{-n} \omega}^{\varepsilon}\rVert_s. \end{split} \end{align*} $$

Hence, by (21) we have that

(24) $$ \begin{align} \lVert \mathcal L_{\sigma^{-n} \omega, \varepsilon}^n h_{\sigma^{-n} \omega}^{\varepsilon} -\mathcal L_{\sigma^{-n} \omega}^n h_{\sigma^{-n} \omega}^{\varepsilon}\rVert_w \le 2nC^3 \lvert \varepsilon\rvert \sup_{|\varepsilon|\le\varepsilon_0} \sup_{\omega \in \Omega^{\prime}} \lVert h_\omega^{\varepsilon} \lVert_s, \end{align} $$

for $\omega \in \Omega ^{\prime }$ and $n\in \mathbb {N}$ . We conclude from (22), (23), and (24) that

$$ \begin{align*} \begin{split} \sup_{\omega \in \Omega^{\prime}} \lVert h_\omega^{\varepsilon} -h_\omega \rVert_w &\le 2nC^3 \lvert \varepsilon \rvert \sup_{|\varepsilon|\le\varepsilon_0} \sup_{\omega \in \Omega^{\prime}} \lVert h_\omega^{\varepsilon} \lVert_s+\tilde De^{-\lambda n}, \end{split} \end{align*} $$

for $n\in \mathbb {N}$ . Taking $n={\lfloor } {\lvert \log (\lvert \varepsilon \rvert ) \rvert }/{\lambda }{\rfloor }$ , we conclude that (20) holds.

3.2 Quenched linear response for random dynamics

Observe that Theorem 8 gives the continuity (in the appropriate sense) of the map $\varepsilon \mapsto (h_\omega ^\varepsilon )_{\omega \in \Omega }$ in $\varepsilon =0$ . We are now concerned with formulating sufficient conditions under which the same map is differentiable in $\varepsilon =0$ .

In addition to requiring the existence of spaces $\mathcal B_w$ and $\mathcal B_s$ as in §3.1, we also require the existence of a third space $\mathcal B_{ss}=(\mathcal B_{ss}, \lVert \cdot \rVert _{ss})$ that can be embedded in $\mathcal B_s$ and such that $\lVert \cdot \rVert _s \le \lVert \cdot \rVert _{ss}$ on $\mathcal B_{ss}$ . As in §3.1, we assume that $\psi $ is a non-zero functional on $\mathcal B_s$ , and we shall also assume that it admits a bounded extension to $\mathcal B_w$ . We still denote its restriction (respectively, extension) to $\mathcal B_{ss}$ (respectively, $\mathcal B_w$ ) by $\psi $ . Furthermore, we let $(\mathcal L_{\omega ,\varepsilon })_{\omega \in \Omega ,\varepsilon \in I}$ be a family such that each $\mathcal L_{\omega ,\varepsilon }$ is a bounded operator on each of those three spaces. In addition, suppose that $\omega \mapsto \mathcal L_{\omega , \varepsilon }$ is strongly measurable on both $\mathcal B_s$ and $\mathcal B_{ss}$ for each $\varepsilon \in I$ .

In addition to (6), we also require that

(25) $$ \begin{align} \lVert \mathcal L_\omega^n h\rVert_{ss} \le De^{-\lambda n} \lVert h\rVert_{ss}, \end{align} $$

for $\mathbb P$ -almost every $\omega \in \Omega $ , $n\in \mathbb {N}$ , and $h\in \mathcal B_{ss}^0$ , where

$$ \begin{align*} \mathcal B_{ss}^0=\{h\in \mathcal B_{ss}: \psi(h)=0\}. \end{align*} $$

We define $\mathcal B_s^0$ and $\mathcal B_w^0$ in a similar manner. In particular, $\mathcal B_s^0$ is the same as in (7).

In addition, we also assume that there exist $C>0$ , $\lambda _1 \in (0, 1)$ , and a measurable $\Omega ^{\prime } \subset \Omega $ with the property that $\mathbb P(\Omega ^{\prime })=1$ and:

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , $n\in \mathbb {N}$ , and $h\in \mathcal B_s$ , (8) holds;

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , and $h\in \mathcal B_s$ , (9) holds;

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , and $n\in \mathbb {N}$ , (10) holds;

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , $n\in \mathbb {N}$ , and $h\in \mathcal B_{ss}$ ,

  (26) $$ \begin{align} \lVert \mathcal L_{\omega, \varepsilon}^n h\rVert_{ss} \le C\lambda_1^n \lVert h\rVert_{ss}+C\lVert h\rVert_s; \end{align} $$

• • for each $\varepsilon \in I$ , $\omega \in \Omega ^{\prime }$ , and $h\in \mathcal B_{ss}$ ,

  (27) $$ \begin{align} \lVert (\mathcal L_{\omega, \varepsilon} -\mathcal L_\omega)h\rVert_{s}\le C\lvert \varepsilon \rvert \lVert h\rVert_{ss}; \end{align} $$

• • for each $\varepsilon \in I$ and $\omega \in \Omega ^{\prime }$ , we have that for $h\in \mathcal B_{s}$ (and, thus, also for $h\in \mathcal B_{ss}$ )

  (28) $$ \begin{align} \psi(\mathcal L_{\omega, \varepsilon} h)=\psi(h). \end{align} $$

As before, we can assume that $\Omega ^{\prime }$ is contained in a full-measure set on which (6) and (25) hold and that $\Omega ^{\prime }$ is $\sigma $ -invariant.

The following is a direct consequence of Proposition 6 (applied for the pairs $(\mathcal B_s, \mathcal B_{ss})$ and $(\mathcal B_w, \mathcal B_s)$ ).

Lemma 10. There exist $\varepsilon _0, D^{\prime }>0$ and $\lambda ^{\prime }>0$ such that for $\varepsilon \in I$ satisfying $\lvert \varepsilon \rvert \le \varepsilon _0$ , $\omega \in \Omega ^{\prime }$ , and $n\in \mathbb {N}$ , we have that

(29) $$ \begin{align} \lVert \mathcal L_{\omega,\varepsilon}^n h\rVert_{ss} \le D^{\prime}e^{-\lambda^{\prime} n} \lVert h\rVert_{ss} \quad \text{for }h\in \mathcal B_{ss}^0, \end{align} $$

and

(30) $$ \begin{align} \lVert \mathcal L_{\omega,\varepsilon}^n h\rVert_{s} \le D^{\prime}e^{-\lambda^{\prime} n} \lVert h\rVert_{s} \quad \text{for }h\in \mathcal B_{s}^0. \end{align} $$

By applying Proposition 7 for $\mathcal B_{ss}$ instead of $\mathcal B_s$ , we deduce the following result.

Let us now introduce some additional assumptions. We suppose that for $\omega \in \Omega ^{\prime }$ , there exists a bounded linear operator $\hat {\mathcal L}_\omega \colon \mathcal B_{ss} \to \mathcal B_s$ , admitting a bounded extension (which will also be denoted by $\hat {\mathcal {L}}_{\omega }$ ) from $\mathcal B_s$ to $\mathcal B_w$ , and such that

(34) $$ \begin{align} \left \{ \begin{aligned} \sup_{\omega \in \Omega^{\prime}} \lVert \hat{\mathcal L}_\omega\rVert_{\mathcal B_{ss} \to \mathcal B_s}&<\infty,\\ \sup_{\omega \in \Omega^{\prime}} \lVert \hat{\mathcal L}_\omega\rVert_{\mathcal B_{s} \to \mathcal B_w}&<\infty, \end{aligned} \right. \end{align} $$

and we suppose that there is a function $\alpha :I\to \mathbb R_+$ , $\lim _{\varepsilon \to 0} \alpha (\varepsilon )=0$ such that for $\omega \in \Omega ^{\prime }$ ,

(35) $$ \begin{align} \bigg{\lVert } \frac{1}{\varepsilon}(\mathcal L_{\omega, \varepsilon}-\mathcal L_\omega)h-\hat{\mathcal L}_\omega h \bigg{\rVert}_{w}\le \alpha(\varepsilon)\|h\|_{ss} \quad \text{for }h\in \mathcal B_{ss}\text{ and }\varepsilon \in I\setminus \{0\}. \end{align} $$

We emphasize that the inequality (35) only holds in $\mathcal {B}_w$ -topology. Obviously, $\hat {\mathcal L}_\omega \mathcal B_{ss}^0\subset \mathcal B_s^0$ , for $\omega \in \Omega ^{\prime }$ , but it also follows from (35) and boundedness of $\psi $ on $\mathcal B_w$ that $\hat {\mathcal {L}}_\omega :\mathcal B_{ss}\to \mathcal B_s^0$ .

Finally, we assume that for $\omega \in \Omega ^{\prime }$ and every $n\in \mathbb {N}$ ,

(36) $$ \begin{align} \lVert \mathcal L_{\omega}^ nh\rVert_{w}\le D^{\prime}e^{-\lambda^{\prime}n}\|h\|_w \quad \text{for }h\in \mathcal B_{w}^0. \end{align} $$

We continue to denote $h_\omega ^0$ simply by $h_\omega $ . For $\omega \in \Omega ^{\prime }$ , set

(37) $$ \begin{align} \hat{h}_\omega:=\sum_{j=0}^\infty \mathcal L_{\sigma^{-j}\omega}^j \hat{\mathcal L}_{\sigma^{-(j+1)} \omega}h_{\sigma^{-(j+1)}\omega}. \end{align} $$

It follows from (6), (31), (34), and the previous discussion that $\hat {h}_\omega \in \mathcal B_s^0$ for $\omega \in \Omega ^{\prime }$ . In addition,

(38) $$ \begin{align} \sup_{\omega \in \Omega^{\prime}}\lVert \hat h_\omega\rVert_s <\infty. \end{align} $$

The following is our linear response result.

Theorem 12. We have that

(39) $$ \begin{align} \lim_{\varepsilon \to 0}\sup_{\omega \in \Omega^{\prime}}\bigg{\lVert} \frac{1}{\varepsilon}(h_\omega^{\varepsilon}-h_\omega) -\hat{h}_\omega \bigg{\rVert}_{w}=0. \end{align} $$

Proof. Let us begin by introducing some auxiliary notation. Set

$$ \begin{align*} \tilde{h}^{\varepsilon}_\omega:=h_\omega^{\varepsilon}-h_\omega \quad \text{and} \quad \tilde{\mathcal L}_{\omega, \varepsilon}:=\mathcal L_{\omega, \varepsilon}-\mathcal L_\omega. \end{align*} $$

It follows easily from (33) that

$$ \begin{align*} \tilde{h}_\omega^{\varepsilon}-\mathcal L_{\sigma^{-1} \omega} \tilde{h}_{\sigma^{-1} \omega}^{\varepsilon} =\tilde{\mathcal L}_{\sigma^{-1} \omega, \varepsilon} h_{\sigma^{-1} \omega}^\varepsilon, \end{align*} $$

and, thus,

(40) $$ \begin{align} \tilde{h}_\omega^{\varepsilon}=\sum_{j=0}^\infty \mathcal L_{\sigma^{-j} \omega}^ j\tilde{\mathcal L}_{\sigma^{-(j+1)}\omega, \varepsilon} h_{\sigma^{-(j+1)} \omega}^\varepsilon, \end{align} $$

for $\omega \in \Omega ^{\prime }$ . By (37) and (40), we have that

(41) $$ \begin{align} \bigg{\lVert} \frac{1}{\varepsilon}\tilde{h}_\omega^{\varepsilon} -\hat{h}_\omega \bigg{\rVert}_{w} &=\bigg{\lVert}\frac{1}{\varepsilon}\sum_{j=0}^\infty \mathcal L_{\sigma^{-j} \omega}^ j\tilde{\mathcal L}_{\sigma^{-(j+1)}\omega, \varepsilon} h_{\sigma^{-(j+1)} \omega}^\varepsilon-\hat{h}_\omega \bigg{\rVert}_{w} \nonumber\\ &\le \bigg{\lVert}\sum_{j=0}^\infty\mathcal L_{\sigma^{-j} \omega}^j \bigg{(} \frac{1}{\varepsilon} \tilde{\mathcal L}_{\sigma^{-(j+1)}\omega, \varepsilon} -\hat{\mathcal L}_{\sigma^{-(j+1)} \omega} \bigg{)}h_{\sigma^{-(j+1)}\omega}^\varepsilon \bigg{\rVert}_{w} \nonumber\\ &\phantom{\le}+\bigg{\lVert} \sum_{j=0}^\infty\mathcal L_{\sigma^{-j} \omega}^j\hat{\mathcal L}_{\sigma^{-(j+1)} \omega} \bigg{(}h_{\sigma^{-(j+1)}\omega}^\varepsilon-h_{\sigma^{-(j+1)}\omega}\bigg{)} \bigg{\rVert}_{w}. \end{align} $$

By applying Lemma 9, we have

$$ \begin{align*}\sup_{|\varepsilon|\le\varepsilon_0}\sup_{\omega \in \Omega^{\prime}}\|h_\omega^\varepsilon\|_{ss}<\infty. \end{align*} $$

This, together with (35) and (36) implies that

(42) $$ \begin{align} &\bigg{\lVert}\sum_{j=0}^\infty\mathcal L_{\sigma^{-j} \omega}^j \bigg{(} \frac{1}{\varepsilon} \tilde{\mathcal L}_{\sigma^{-(j+1)}\omega, \varepsilon} -\hat{\mathcal L}_{\sigma^{-(j+1)} \omega} \bigg{)}h_{\sigma^{-(j+1)}\omega}^\varepsilon \bigg{\rVert}_{w} \nonumber\\ &\quad\le \sum_{j=0}^\infty D^{\prime}e^{-\lambda^{\prime} j}\bigg{\lVert} \bigg{(} \frac{1}{\varepsilon} \tilde{\mathcal L}_{\sigma^{-(j+1)}\omega, \varepsilon} -\hat{\mathcal L}_{\sigma^{-(j+1)} \omega} \bigg{)}h_{\sigma^{-(j+1)}\omega}^\varepsilon \bigg{\rVert}_w \nonumber\\ &\quad\le \tilde D\alpha(\varepsilon)\sup_{|\varepsilon|\le\varepsilon_0}\sup_{\omega\in\Omega^{\prime}}\|h_\omega^\varepsilon\|_{ss}, \end{align} $$

for $\omega \in \Omega ^{\prime }$ , where $\tilde D>0$ does not depend on $\omega $ and $\varepsilon $ . On the other hand, we have by (34) and (36) that

$$ \begin{align*} \begin{aligned} &\bigg{\lVert} \sum_{j=0}^\infty\mathcal L_{\sigma^{-j} \omega}^j\hat{\mathcal L}_{\sigma^{-(j+1)} \omega} {(}h_{\sigma^{-(j+1)}\omega}^\varepsilon-h_{\sigma^{-(j+1)}\omega}{)} \bigg{\rVert}_{w} \\ &\quad\le \sum_{j=0}^{\infty}D^{\prime}e^{-\lambda^{\prime}j}{\lVert} \hat{\mathcal L}_{\sigma^{-(j+1)} \omega} (h_{\sigma^{-(j+1)}\omega}^\varepsilon-h_{\sigma^{-(j+1)}\omega} ) {\rVert}_w \\ &\quad\le\sup_{\omega \in \Omega^{\prime}}\|\hat{\mathcal{L}}_\omega\|_{\mathcal B_s\to \mathcal B_w}\sum_{j=0}^{\infty}D^{\prime}e^{-\lambda^{\prime}j}{\lVert} h_{\sigma^{-(j+1)}\omega}^\varepsilon-h_{\sigma^{-(j+1)}\omega}{\rVert}_s. \end{aligned} \end{align*} $$

Now, our assumptions ensure that we may apply Theorem 8 for the pair $(\mathcal B_{s},\mathcal B_{ss})$ . Hence, we obtain

(43) $$ \begin{align} \bigg{\lVert}\sum_{j=0}^\infty\mathcal L_{\sigma^{-j} \omega}^j\hat{\mathcal L}_{\sigma^{-(j+1)} \omega}(h_{\sigma^{-(j+1)}\omega}^\varepsilon- h_{\sigma^{-(j+1)}\omega})\bigg{\rVert}_w\le C^{\prime}|\varepsilon|\lvert\log|\varepsilon\rvert \end{align} $$

for $\omega \in \Omega ^{\prime }$ , where $C^{\prime }>0$ is independent on $\omega $ and $\varepsilon $ . It follows readily from (41), (42), and (43) that (39) holds, which completes the proof of the theorem.

Remark 13. The purpose of this remark is to interpret Theorem 8 (as well as Theorem 12) in the context of the multiplicative ergodic theory. In order to do so, we first need to introduce two additional assumptions. Namely, we require that:

• • $\mathcal B_s$ is separable;

• • the inclusion $\mathcal B_s \hookrightarrow \mathcal B_w$ is compact.

We denote the largest Lyapunov exponent of the cocycle $(\mathcal L_{\omega , \varepsilon })_{\omega \in \Omega }$ , for $\varepsilon \in I$ , by $\Lambda (\varepsilon )\in \mathbb R\cup \{-\infty \}$ . We stress that the existence of $\Lambda (\varepsilon )$ is a direct consequence of (8) (applied to $n=1$ ) and the subadditive ergodic theorem. Moreover, we recall that

$$ \begin{align*} \Lambda(\varepsilon)=\lim_{n\to \infty} \frac 1 n \log \lVert \mathcal L_{\omega, \varepsilon}^n \rVert_s \quad \text{for }\mathbb P\text{-almost every }\omega \in \Omega. \end{align*} $$

By using (8) together with Proposition 7, it is easy to show (see the proof of [Reference Demers and Zhang19, Lemma 3.5]) that $\Lambda (\varepsilon )=0$ , for $\varepsilon \in I$ with $|\varepsilon | \le \varepsilon _0$ . Moreover, for each such $\varepsilon $ , the cocycle $(\mathcal L_{\omega , \varepsilon })_{\omega \in \Omega }$ is quasi-compact (in the sense of [Reference Galatolo and Sedro25, Definition 2.7]). Hence, it follows from the multiplicative ergodic theorem (see [Reference Galatolo and Sedro25, Theorem A]) that for each $\varepsilon \in I$ with $|\varepsilon | \le \varepsilon _0$ , there exists:

• • $1\le l=l(\varepsilon )\le \infty $ and a sequence of exceptional Lyapunov exponents

  $$ \begin{align*}0=\Lambda (\varepsilon)=\lambda_1(\varepsilon)>\lambda_2(\varepsilon)>\cdots>\lambda_l(\varepsilon)>\kappa (\varepsilon)\end{align*} $$
  or in the case $l=\infty $ ,
  $$ \begin{align*}0=\Lambda (\varepsilon)=\lambda_1(\varepsilon)>\lambda_2(\varepsilon)>\cdots \quad \text{with }\lim_{n\to\infty} \lambda_n(\varepsilon)=\kappa (\varepsilon); \end{align*} $$

• • a unique measurable Oseledets splitting

  $$ \begin{align*}\mathcal{B}_s=\bigg(\bigoplus_{j=1}^l Y_j^\varepsilon(\omega)\bigg)\oplus V^\varepsilon(\omega),\end{align*} $$
  where each component of the splitting is equivariant under $\mathcal {L}_{\omega , \varepsilon }$ , that is, $\mathcal L_{\omega , \varepsilon }(Y_j^\varepsilon (\omega ))= Y_j^\varepsilon (\sigma \omega )$ and $\mathcal L_{\omega , \varepsilon }(V^\varepsilon (\omega ))\subset V^\varepsilon (\sigma \omega )$ ; the subspaces $Y_j^\varepsilon (\omega )$ are finite-dimensional and for each $y\in Y_j^\varepsilon (\omega )\setminus \{0\}$ ,
  $$ \begin{align*}\lim_{n\to\infty}\frac 1n\log\|\mathcal L_{\omega, \varepsilon}^n y\|=\lambda_j(\varepsilon);\end{align*} $$
  moreover, for $y\in V(\omega )$ , $\lim _{n\to \infty }(1/n)\log \|\mathcal L_{\omega , \varepsilon }^n y\|\le \kappa (\varepsilon )$ .

It follows easily from Proposition 6 (see the proof of [Reference Demers and Zhang19, Proposition 3.6]) that $Y_1^\varepsilon (\omega )$ is one-dimensional and is spanned by $h_\omega ^\varepsilon $ , for each $\varepsilon \in I$ such that $|\varepsilon | \le \varepsilon _0$ .

Hence, Theorem 8 can be interpreted as a regularity result for the top Oseledets space of $(\mathcal L_{\omega , \varepsilon })_{\omega \in \Omega }$ . Namely, it shows that it is continuous in appropriate sense in $\varepsilon =0$ . Taking into account that Lyapunov exponents and corresponding Oseledets subspaces represent non-autonomous versions of the classical notions of an eigenvalue and the corresponding eigenspace, we conclude that Theorem 8 is a natural extension of statistical stability results concerned with deterministic systems. In a similar manner, Theorem 12 can be viewed as a non-autonomous generalization linear reponse result.

5.1 Annealed linear response for hyperbolic dynamics

As a first application, we establish a form of an annealed linear response.

For $F \in L^\infty (\Omega ,C^r(M))$ and $\varepsilon \in I$ , we set

(52) $$ \begin{align} R(\varepsilon, F)=\int_\Omega \int_M F(\omega,x)\, dh_{\omega}^\varepsilon(x) \, d \mathbb P(\omega). \end{align} $$

The following is our annealed linear response result.

Theorem 15. The map $R:I\times L^\infty (\Omega ,C^r(M))\to \mathbb {R}$ is differentiable at every $(0, F)$ , $F \in L^\infty (\Omega ,C^r(M))$ . Furthermore, one has

(53) $$ \begin{align} \partial_\varepsilon[R(\varepsilon,F)] \rvert_{\varepsilon=0}&=\sum_{n=0}^\infty\int_\Omega\int_M \partial_\varepsilon [F_\omega\circ T^{n}_{\sigma^{-n}\omega}\circ T_{\sigma^{-n-1}\omega,\varepsilon}] {\rvert}_{\varepsilon=0}\, dh_{\sigma^{-n-1}\omega} \, d \mathbb P(\omega). \end{align} $$

Remark 16. The previous result can be interpreted as linear response for the stationary measure of the skew product

$$ \begin{align*}S_\varepsilon(\omega,x):=(\sigma\omega,T_{\omega,\varepsilon}x),\end{align*} $$

acting on $\Omega \times M$ . Indeed, the stationary measure $\mu _\varepsilon $ of this skew-product classically admits the disintegration along fibers

$$ \begin{align*}\mu_\varepsilon(A\times B)=\int_{A} h^\varepsilon_{\omega}(B)\,d\mathbb P(\omega),\end{align*} $$

for measurable $A\subset \Omega $ , $B\subset M$ . In particular, this justifies the ‘annealed’ terminology, because in the independent and identically distributed case, the measure defined on M by $\tilde \mu _\varepsilon (\cdot )=\mu _\varepsilon (\Omega \times \cdot )$ corresponds to the invariant measure of the Markov chain associated with our cocycle.

We also point out that one may use this interpretation to establish a linear response for a class of deterministic partially hyperbolic skew products: let us set $\Omega =\mathbb S^1$ , $\mathbb P=\text {Lebesgue}$ , and $\sigma (\omega )=\omega +\alpha \mod 1$ for some $\alpha \in \mathbb {R}\backslash \mathbb Q$ . Then, consider a family $(T_{\omega ,\varepsilon })_{\omega \in \mathbb S^1,\varepsilon \in I}$ of Anosov diffeomorphisms of $\mathbb T^2$ , for example,

$$ \begin{align*}T_{\omega,\varepsilon}(x_1,x_2):= \begin{pmatrix} 2 & 1\\ 1 & 1 \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} + \begin{pmatrix} \omega\\ \omega \end{pmatrix} +\varepsilon \begin{pmatrix} \sin 2\pi x_1\\ \sin2\pi x_2 \end{pmatrix}. \end{align*} $$

This system clearly satisfies our Hypothesis 1 (note that it belongs to the class of Examples 4), and the skew product $S_\varepsilon $ acting on $\mathbb S^1\times \mathbb T^2\simeq \mathbb T^3$ is clearly a partially hyperbolic system (with central direction tangent to the first coordinate), exhibiting linear response by Theorem 15 and the previous discussion.

Proof of Theorem 15

Fix an arbitrary $F_0 \in L^\infty (\Omega ,C^r(M))$ . We claim that the derivative of R in $(0, F_0)$ is given by

(54) $$ \begin{align} DR(0, F_0)(\varepsilon, H)=\varepsilon\int_\Omega \hat h_\omega (F_0(\omega))\, d\mathbb P(\omega)+\int_\Omega h_\omega( H(\omega))\, d\mathbb P(\omega), \end{align} $$

for $(\varepsilon , H)\in \mathbb {R} \times L^\infty (\Omega ,C^r(M))$ , where $\hat h_\omega $ is given by (51). Indeed, observe that

$$ \begin{align*} \begin{split} & R(\varepsilon, F_0+H)-R(0, F_0)-\varepsilon\int_\Omega \hat h_\omega (F_0(\omega))\, d\mathbb P(\omega)-\int_\Omega h_\omega( H(\omega))\, d\mathbb P(\omega) \\ &\quad=\int_\Omega (h_\omega^\varepsilon-h_\omega-\varepsilon \hat h_\omega )(F_0(\omega))\, d\mathbb P(\omega)+\int_\Omega (h_\omega^\varepsilon-h_\omega)(H(\omega))\, d\mathbb P(\omega). \end{split} \end{align*} $$

Furthermore, the continuous embedding $\mathcal B^{p,q}\hookrightarrow \mathcal D^{\prime }_q$ entails that there is $C>0$ (independent on both $\omega $ and $\varepsilon $ ) such that

$$ \begin{align*} \begin{split} &\bigg |\frac{1}{\varepsilon} \int_\Omega (h_\omega^\varepsilon-h_\omega-\varepsilon \hat h_\omega )(F_0(\omega))\, d\mathbb P(\omega) \bigg |\\ &\quad \le C \lVert F_0\rVert_{L^\infty(\Omega,C^r(M))} \cdot \sup_{\omega \in \Omega}\bigg{\lVert} \frac{1}{\varepsilon}(h_\omega^{\varepsilon}-h_\omega) -\hat{h}_\omega \bigg{\rVert}_{w},\end{split} \end{align*} $$

and, thus, Theorem 12 implies that

$$ \begin{align*} \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} \int_\Omega (h_\omega^\varepsilon-h_\omega-\varepsilon \hat h_\omega )(F_0(\omega))\, d\mathbb P(\omega)=0. \end{align*} $$

In addition,

$$ \begin{align*} \bigg\lvert\int_\Omega (h_\omega^\varepsilon-h_\omega)(H(\omega))\, d\mathbb P(\omega)\bigg\rvert \le C\lVert H\rVert_{L^\infty(\Omega,C^r(M))} \cdot \sup_{\omega \in \Omega} \|h_\omega^\varepsilon-h_\omega \|_w, \end{align*} $$

and, consequently, by applying Theorem 8 (for the pair $(\mathcal B_{s},\mathcal B_{w})$ ), we obtain that

$$ \begin{align*} \lim_{(\varepsilon,H)\to (0,0)} \frac{1}{\lVert H\rVert_{L^\infty(\Omega,C^0(M))}}\bigg\lvert\int_\Omega (h_\omega^\varepsilon-h_\omega)(H(\omega))\, d\mathbb P(\omega)\bigg\rvert=0. \end{align*} $$

Thus, (54) holds and the proof of the theorem is completed. In order to establish (53), one can argue as in the proof of formula (3).

5.2 Regularity of the variance in the central limit theorem for random hyperbolic dynamics

In this section, we provide an application of Theorem 12 to the problem of the regularity of the variance (under suitable perturbations) in the quenched version of the central limit theorem for random hyperbolic dynamics.

Let F be as in the previous subsection. For $\omega \in \Omega $ and $\varepsilon \in I$ , set

$$ \begin{align*}f_{\omega,\varepsilon}:= F_\omega- h_{\omega}^\varepsilon (F_\omega)=F_\omega -\int_M F_\omega \, dh_\omega^\varepsilon.\end{align*} $$

Set

(55) $$ \begin{align} \Sigma^2_\varepsilon &:=\!\int_\Omega\int_M f_{\omega,\varepsilon}^2(x)\,dh_{\omega}^\varepsilon(x)\,d\mathbb P(\omega)\nonumber\\&\quad+2\!\sum_{n=1}^\infty\int_\Omega\int_M f_{\omega,\varepsilon}(x)f_{\sigma^n\omega,\varepsilon}(T^n_{\omega,\varepsilon}x)\,dh_{\omega}^\varepsilon(x)\,d\mathbb P(\omega). \end{align} $$

Observe that $\Sigma ^2_\varepsilon \ge 0$ and that $\Sigma ^2_\varepsilon $ does not depend on $\omega $ . It is proved in [Reference Demers and Zhang19, Theorem B] that if $\Sigma _\varepsilon ^2>0$ , the process $(f_{\omega ,\varepsilon }\circ T^n_{\omega ,\varepsilon })$ satisfies $\mathbb P$ -almost surely a quenched central limit theorem. More precisely, for every bounded and continuous $\phi :\mathbb R\to \mathbb R$ and $\mathbb P$ -almost every $\omega \in \Omega $ , we have that

$$ \begin{align*}\lim_{n\to\infty}\int \phi\bigg(\frac{S_n(f_{\omega,\varepsilon})}{\sqrt n}\bigg ) \,dh_{\omega}^\varepsilon=\int\phi\, d\mathcal N(0,\Sigma^2_\varepsilon), \end{align*} $$

where

$$ \begin{align*}S_n(f_{\omega, \varepsilon}):=\sum_{k=0}^{n-1}f_{\sigma^k\omega, \varepsilon}\circ T_{\omega,\varepsilon}^k,\end{align*} $$

and $\mathcal N(0, \Sigma ^2_\varepsilon )$ denotes the normal distribution with parameters $0$ and $\Sigma _\varepsilon $ . Our goal is to establish the following result.

We start the proof by making a few remarks related to the map $\varepsilon \mapsto (f_{\omega ,\varepsilon })_{\omega \in \Omega }\in C^r(M)$ .

• • For each $\varepsilon $ , $\omega \mapsto f_{\omega , \varepsilon }$ is an element of $L^\infty (\Omega ,C^r(M))$ . Moreover, by Lemma 9 we have that

  (56) $$ \begin{align} \sup_{|\varepsilon|\le \varepsilon_0}\operatorname{\mathrm{esssup}}_{\omega\in \Omega}\|f_{\omega,\varepsilon}\|_{C^r}\leq \Big(1+\sup_{|\varepsilon|\le\varepsilon_0}\operatorname{\mathrm{esssup}}_{\omega\in\Omega}\|h_\omega^\varepsilon\|_{ss}\Big)\operatorname{\mathrm{esssup}}_{\omega\in\Omega}\|F_\omega\|_{C^r}. \end{align} $$

• • It is differentiable at $\varepsilon =0$ . Indeed, we have

  $$ \begin{align*}\frac{1}{\varepsilon}( f_{\omega,\varepsilon}-f_\omega)= \frac{1}{\varepsilon} ( h_\omega-h_\omega^\varepsilon) (F_\omega),\end{align*} $$
  which yields
  (57) $$ \begin{align} \operatorname{\mathrm{esssup}}_{\omega\in\Omega}\bigg|\frac{1}{\varepsilon}( f_{\omega,\varepsilon}-f_\omega)+\hat h_\omega (F_\omega)\bigg|\to 0, \end{align} $$
  as $\varepsilon \to 0$ , via Theorem 12. Here, we write $f_\omega $ instead of $f_{\omega , 0}$ .

The above observations together with Theorem 12 easily imply the following lemma.

Lemma 18. The map

$$ \begin{align*} \varepsilon \mapsto \int_\Omega\int_M f_{\omega,\varepsilon}^2(x)\,dh_{\omega}^\varepsilon(x)\,d\mathbb P(\omega) \end{align*} $$

is differentiable at $\varepsilon =0$ .

Proof. For $\varepsilon $ sufficiently close to zero, let $H(\varepsilon )\in L^\infty (\Omega ,C^r(M))$ be defined by

$$ \begin{align*} H(\varepsilon)(\omega)=f_{\omega, \varepsilon}^2, \quad \omega \in \Omega. \end{align*} $$

Then, the discussion preceding the statement of the lemma implies that the map H is differentiable at $\varepsilon =0$ . Now the conclusion of the lemma follows from Theorem 15 and the simple observation that

$$ \begin{align*} \int_\Omega\int_M f_{\omega,\varepsilon}^2(x)\,dh_{\omega}^\varepsilon(x)\,d\mathbb P(\omega)=R(\varepsilon, H(\varepsilon)), \end{align*} $$

with R given by (52).

We recall that (see §2.1) that for $h\in \mathcal B^{p,q}$ and $f\in C^q (M)$ , we can define $f\cdot h \in \mathcal B^{p,q}$ whose action as a distribution is given by

$$ \begin{align*} (f\cdot h)(\phi)=h(f\phi), \quad\text{for }\phi \in C^q(M). \end{align*} $$

Moreover, there exists $C>0$ (depending only on M) such that

$$ \begin{align*} \lVert f\cdot h\rVert_{p,q}\le C \lVert h\rVert_{p,q} \cdot \lVert f\rVert_{C^q}. \end{align*} $$

The above inequality is frequently used in what follows and, thus, we do not explicitly refer to it. Moreover, in what follows, $C>0$ denotes a constant which is independent on all parameters ( $\omega $ , n, etc.) involved.

Observe that

$$ \begin{align*}( f_{\omega,\varepsilon} \cdot h_{\omega}^\varepsilon )(f_{\sigma^n\omega,\varepsilon}\circ T_{\omega,\varepsilon}^{n})= \mathcal{L}_{\omega,\varepsilon}^n (f_{\omega,\varepsilon} \cdot h_{\omega}^\varepsilon) (f_{\sigma^n\omega,\varepsilon}). \end{align*} $$

In addition, $( f_{\omega ,\varepsilon } \cdot h_{\omega }^\varepsilon )(1)=h_{\omega }^\varepsilon (f_{\omega ,\varepsilon })=0$ . We now write

(58) $$ \begin{align} \frac{1}{\varepsilon}(\mathcal{L}^{n}_{\omega,\varepsilon}(f_{\omega,\varepsilon}\cdot h_{\omega}^\varepsilon) (f_{\sigma^n\omega,\varepsilon} )- \mathcal{L}_\omega^n(f_\omega \cdot h_\omega )(f_{\sigma^n\omega}))=(I)_{n,\omega, \varepsilon}+(II)_{n,\omega, \varepsilon}+(III)_{n,\omega, \varepsilon}, \end{align} $$

where

$$ \begin{align*} (I)_{n,\omega, \varepsilon}&:=\mathcal{L}_\omega^n (f_\omega \cdot h_\omega) \bigg ( \frac{1}{\varepsilon}(f_{\sigma^n\omega,\varepsilon}-f_{\sigma^n\omega}) \bigg),\\ (II)_{n,\omega, \varepsilon}&:= \frac{1}{\varepsilon}(\mathcal{L}_{\omega,\varepsilon}^n-\mathcal{L}_{\omega}^n)(f_{\omega}\cdot h_\omega) (f_{\sigma^n\omega,\varepsilon}),\\ (III)_{n,\omega, \varepsilon}&:= \mathcal{L}_{\omega,\varepsilon}^n\bigg ( \frac{f_{\omega,\varepsilon}\cdot h_{\omega}^\varepsilon-f_\omega\cdot h_\omega}{\varepsilon}\bigg )(f_{\sigma^n\omega,\varepsilon}). \end{align*} $$

Lemma 19. For each $n\in \mathbb {N}$ ,

$$ \begin{align*} \lim_{\varepsilon \to 0}\operatorname{\mathrm{esssup}}_{\omega \in \Omega} \lvert (I)_{n,\omega, \varepsilon}-\hat h_{\sigma^n \omega}(F_{\sigma^n \omega})\mathcal{L}_\omega^n (f_\omega \cdot h_\omega) (1) \rvert=0. \end{align*} $$

In addition, for $\varepsilon $ sufficiently close to zero, we have that

$$ \begin{align*} \operatorname{\mathrm{esssup}}_{\omega \in \Omega} \lvert (I)_{n, \omega, \varepsilon} \rvert \le Ce^{-\lambda n}. \end{align*} $$

Proof. The first assertion follows directly from (25), (56), and (57). In addition, observe that for $\varepsilon $ sufficiently close to zero,

$$ \begin{align*} \operatorname{\mathrm{esssup}}_{\omega \in \Omega} \lvert (I)_{n, \omega, \varepsilon}- \hat h_{\sigma^n \omega}(F_{\sigma^n \omega})\mathcal{L}_\omega^n (f_\omega \cdot h_\omega) (1) \rvert \le Ce^{-\lambda n}. \end{align*} $$

On the other hand, (25), (38), and (57) imply that

$$ \begin{align*} \operatorname{\mathrm{esssup}}_{\omega \in \Omega} \lvert \hat h_{\sigma^n \omega}(F_{\sigma^n \omega})\mathcal{L}_\omega^n (f_\omega \cdot h_\omega) (1) \rvert \le Ce^{-\lambda n}. \end{align*} $$

The above two estimates readily give the second assertion of the lemma.

Lemma 20. For each $n\in \mathbb {N}$ ,

(59) $$ \begin{align} \lim_{\varepsilon \to 0} \operatorname{\mathrm{esssup}}_{\omega\in\Omega}|(II)_{n,\omega, \varepsilon}- \hat{\mathcal{L}}_{n,\omega}(f_\omega \cdot h_\omega) (f_{\sigma^n\omega})|= 0, \end{align} $$

where

$$ \begin{align*}\hat{\mathcal{L}}_{n,\omega}=\sum_{k=1}^n\mathcal{L}_{\sigma^k\omega}^{n-k}\hat{\mathcal{L}}_{\sigma^{k-1}\omega}\mathcal{L}^{k-1}_{\omega}.\end{align*} $$

Furthermore, for $\varepsilon $ sufficiently close to zero, we have that

$$ \begin{align*} \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\lvert (II)_{n, \omega, \varepsilon} \rvert \le Cne^{-\lambda^{\prime} n}. \end{align*} $$

Proof. In order to prove (59), we first claim that

(60) $$ \begin{align} \bigg\|\frac{1}{\varepsilon}(\mathcal{L}_{\omega,\varepsilon}^n-\mathcal{L}_\omega^n)(f_\omega \cdot h_\omega)-\hat{\mathcal{L}}_{n,\omega}(f_\omega \cdot h_\omega)\bigg\|_w\leq \tilde\alpha(\varepsilon), \end{align} $$

with $\tilde \alpha (\varepsilon )\to 0$ when $\varepsilon \to 0$ . Observe that

$$ \begin{align*} \frac{1}{\varepsilon}(\mathcal{L}_{\omega,\varepsilon}^n-\mathcal{L}_\omega^n)=\sum_{k=1}^n \mathcal{L}_{\sigma^k\omega,\varepsilon}^{n-k}\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}\mathcal{L}^{k-1}_{\omega}, \end{align*} $$

and, therefore,

$$ \begin{align*} \frac{1}{\varepsilon}(\mathcal{L}_{\omega,\varepsilon}^n-\mathcal{L}_\omega^n)-\hat{\mathcal{L}}_{n,\omega}&=\sum_{k=1}^n\bigg[\mathcal{L}_{\sigma^k\omega,\varepsilon}^{n-k}\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}-\mathcal{L}_{\sigma^k\omega}^{n-k}\hat{\mathcal{L}}_{\sigma^{k-1}\omega}\bigg]\mathcal{L}^{k-1}_{\omega}\\ &=\sum_{k=1}^n\bigg[(\mathcal{L}_{\sigma^k\omega,\varepsilon}^{n-k}-\mathcal{L}_{\sigma^k\omega}^{n-k})\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}\\ &\quad+ \mathcal{L}_{\sigma^k\omega}^{n-k}\bigg(\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}-\hat{\mathcal{L}}_{\sigma^{k-1}\omega}\bigg)\bigg]\mathcal{L}^{k-1}_{\omega}. \end{align*} $$

By the arguments in the proof of Proposition 6, (25), (27), and (56), we have that

(61) $$ \begin{align} &\bigg\|(\mathcal{L}_{\sigma^k\omega,\varepsilon}^{n-k}-\mathcal{L}_{\sigma^k\omega}^{n-k})\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\bigg\|_w\nonumber\\ &\quad\leq C|\varepsilon|(n-k)\bigg\|\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\bigg\|_s\nonumber\\ &\quad\leq C|\varepsilon|(n-k)e^{-\lambda(k-1)}\operatorname{\mathrm{esssup}}_{\omega\in\Omega}\|f_\omega \cdot h_\omega\|_{ss} \nonumber \\ &\quad\le C|\varepsilon |(n-k)e^{-\lambda k}. \end{align} $$

Similarly, using (25), (31), (35), (36), and (56), we obtain that

(62) $$ \begin{align} &\bigg\|\mathcal{L}_{\sigma^k\omega}^{n-k}\bigg(\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}-\hat{\mathcal{L}}_{\sigma^{k-1}\omega}\bigg)\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\bigg\|_{w}\nonumber\\ &\quad\leq Ce^{-\lambda^{\prime}(n-k)}\bigg\|\bigg(\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}-\hat{\mathcal{L}}_{\sigma^{k-1}\omega}\bigg)\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\bigg\|_{w}\nonumber\\ &\quad\leq Ce^{-\lambda^{\prime}(n-k)}\alpha(\varepsilon)\|\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\|_{ss} \nonumber \\ &\quad\leq Ce^{-\lambda^{\prime}n}\alpha(\varepsilon)\operatorname{\mathrm{esssup}}_{\omega\in\Omega}\|f_\omega \cdot h_\omega\|_{ss} \nonumber\\ &\quad\le C\alpha(\varepsilon)e^{-\lambda^{\prime} n}. \end{align} $$

Then, (61) and (62) imply (60).

Furthermore, (6), (25), (31), (34), and (56) imply that

(63) $$ \begin{align} \lVert \hat{\mathcal{L}}_{\omega,n}(f_\omega \cdot h_\omega) \rVert_w &\leq \sum_{k=1}^n\|\mathcal{L}_{\sigma^k\omega}^{n-k}\hat{\mathcal{L}}_{\sigma^{k-1}\omega}\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\|_{s}\nonumber\\ &\leq C\sum_{k=1}^n e^{-\lambda(n-k)}\operatorname{\mathrm{esssup}}_{\omega\in\Omega} (\|\hat{\mathcal{L}}_{\sigma^{k-1}\omega}\|_{B_{ss}\to B_s} \cdot \|\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\|_{ss})\nonumber\\ &\leq C\sum_{k=1}^n e^{-\lambda(n-k)}e^{-\lambda(k-1)}\operatorname{\mathrm{esssup}}_{\omega\in\Omega}\|f_\omega \cdot h_\omega\|_{ss} \nonumber \\ &\leq Cne^{-\lambda n}. \end{align} $$

Using Theorem 8, (56), (60), and (63), we have that

$$ \begin{align*} \begin{split} &\operatorname{\mathrm{esssup}}_{\omega\in\Omega}|(II)_{n,\omega, \varepsilon}- \hat{\mathcal{L}}_{n,\omega}(f_\omega \cdot h_\omega) (f_{\sigma^n\omega})| \\ &\quad\le \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\bigg |\frac{1}{\varepsilon}(\mathcal{L}_{\omega,\varepsilon}^n-\mathcal{L}_\omega^n)(f_\omega \cdot h_\omega)(f_{\sigma^n \omega, \varepsilon})-\hat{\mathcal{L}}_{n,\omega}(f_\omega \cdot h_\omega)(f_{\sigma^n \omega, \varepsilon})\bigg | \\ &\qquad+\operatorname{\mathrm{esssup}}_{\omega \in \Omega} | \hat{\mathcal{L}}_{n,\omega}(f_\omega \cdot h_\omega) (f_{\sigma^n \omega, \varepsilon}-f_{\sigma^n \omega}) |\\ &\quad\le \tilde\alpha(\varepsilon)\operatorname{\mathrm{esssup}}_{\omega \in \Omega}\lVert f_{\sigma^n \omega, \varepsilon}\rVert_{C^r}+Cne^{-\lambda n} \operatorname{\mathrm{esssup}}_{\omega \in \Omega} | (h_\omega^\varepsilon -h_\omega)(F_\omega)| \\ &\quad\le C\tilde \alpha (\varepsilon)+Cn e^{-\lambda n} |\varepsilon | |\log (|\varepsilon |)|, \end{split} \end{align*} $$

which implies the first assertion of the lemma.

On the other hand, using (36) (which also persists under small perturbations), (31), (34), and (56), we have that for $\varepsilon $ sufficiently small,

(64) $$ \begin{align} \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\bigg\|(\mathcal{L}_{\sigma^k\omega,\varepsilon}^{n-k}-\mathcal{L}_{\sigma^k\omega}^{n-k})\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\bigg\|_w \le Ce^{-\lambda^{\prime}n}. \end{align} $$

Moreover, from (62) it follows that for $\varepsilon $ sufficiently small,

(65) $$ \begin{align} \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\bigg\|\mathcal{L}_{\sigma^k\omega}^{n-k}\bigg(\frac{\mathcal{L}_{\sigma^{k-1}\omega,\varepsilon}-\mathcal{L}_{\sigma^{k-1}\omega}}{\varepsilon}-\hat{\mathcal{L}}_{\sigma^{k-1}\omega}\bigg)\mathcal{L}^{k-1}_{\omega}(f_\omega \cdot h_\omega)\bigg\|_{w} \le Ce^{-\lambda^{\prime} n}. \end{align} $$

By (64) and (65), we have that for sufficiently small $\varepsilon $ ,

$$ \begin{align*} \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\bigg\|\frac{1}{\varepsilon}(\mathcal{L}_{\omega,\varepsilon}^n-\mathcal{L}_\omega^n)(f_\omega \cdot h_\omega)-\hat{\mathcal{L}}_{n,\omega}(f_\omega \cdot h_\omega)\bigg\|_w \le Cne^{-\lambda^{\prime} n}. \end{align*} $$

The above estimate together with (63) easily implies that the second assertion of the lemma also holds.

By using similar arguments, one can establish the following lemma.

Lemma 21. For each $n\in \mathbb {N}$ ,

$$ \begin{align*} \lim_{\varepsilon \to 0} \operatorname{\mathrm{esssup}}_{\omega \in \Omega} \lvert (III)_{n,\omega, \varepsilon} - \mathcal{L}_{\omega}^n(\hat h_\omega (F_\omega) h_\omega+f_\omega \cdot \hat h_\omega) (f_{\sigma^n\omega}) \rvert =0. \end{align*} $$

Moreover, for $\varepsilon $ sufficiently small, we have that

$$ \begin{align*} \operatorname{\mathrm{esssup}}_{\omega \in \Omega}\lvert (III)_{n, \omega, \varepsilon} \rvert \le Ce^{-\lambda^{\prime} n}. \end{align*} $$

The conclusion of Theorem 17 follows from previous lemmas and the dominated convergence theorem.

6.1 Random uniformly expanding dynamics

In this subsection, let us describe the application of Theorems 8 and 12 to a simple class of fiberwise perturbations of random compositions of uniformly expanding maps of the unit circle $\mathbb S^1$ . The setting is close to [Reference Gouëzel and Liverani24, §6]: consider a family $(D_\varepsilon )_{\varepsilon \in I}$ of diffeomorphisms of $\mathbb S^1$ (where $0\in I\subset \mathbb {R}$ is an interval), satisfying

$$ \begin{align*}D_\varepsilon= \text{Id}+\varepsilon S,\end{align*} $$

where $S:\mathbb S^1\to \mathbb {R}$ is a $C^{4}$ map. Letting $(\Omega ,\mathcal F,\mathbb P)$ be a probability space, endowed with an invertible, measure-preserving and ergodic map $\sigma :\Omega \circlearrowleft $ . We consider a measurable map $\omega \in \Omega \mapsto T_{\omega }\in C^4(\mathbb S^1,\mathbb S^1)$ such that:

1. (1) there exists $\lambda>1$ such that for $\mathbb P$ -almost every $\omega \in \Omega $ , $\inf _{x\in \mathbb S^1}|T^{\prime }_\omega (x)|\ge \lambda $ ;

2. (2) $\operatorname {\mathrm {esssup}}_{\omega \in \Omega }\|T_\omega \|_{C^4}\le \Delta $ for some small $\Delta>0$ .

We then set

$$ \begin{align*}T_{\omega,\varepsilon}:= D_\varepsilon\circ T_\omega \quad \text{for }\varepsilon \in I\text{ and } \omega \in \Omega,\end{align*} $$

and we review the assumptions of Theorems 8 and 12 for the spaces $\mathcal {B}_{ss}=W^{3,1}(\mathbb S^1)$ , $\mathcal {B}_s=W^{2,1}(\mathbb S^1)$ , and $\mathcal {B}_w=W^{1,1}(\mathbb S^1)$ .

• • Equations (8) and (10) are established in [Reference Crimmins and Nakano14, §5].

• • Equation (9) follows from [Reference Gouëzel and Liverani24, Proposition 35].

• • By applying [Reference Crimmins13, Proposition 2.10] (provided that $\Delta $ is sufficiently small), we conclude that (6) holds on $(\mathcal {B}_{ss},\mathcal {B}_s,\mathcal {B}_w)$ .

• • To define the derivative operator $\hat {\mathcal {L}}_\omega $ , we start by remarking that because $\mathcal {L}_{\omega ,\varepsilon }=\mathcal {L}_{D_\varepsilon }\mathcal {L}_\omega $ , one has (see [Reference Gouëzel and Liverani24, Eq. (51)]) that

  $$ \begin{align*}\hat{\mathcal{L}}_\omega=\bigg[\frac{d\mathcal{L}_{D_\varepsilon}}{d\varepsilon}\bigg] \bigg \rvert_{\varepsilon=0}\mathcal{L}_\omega=-(\mathcal{L}_\omega(\cdot)S)^{\prime}.\end{align*} $$
  It is easy to see that $\hat {\mathcal {L}}_\omega $ defines a bounded operator from $\mathcal {B}_{ss}$ to $\mathcal {B}_s$ (respectively, from $\mathcal {B}_s$ to $\mathcal {B}_w$ ) and satisfies (34).

  As for condition (35), we have for $\phi \in \mathcal {B}_{s}$

  $$ \begin{align*} \hspace{-3pt}\|\varepsilon^{-1}(\mathcal{L}_{\omega,\varepsilon}-\mathcal{L}_\omega)\phi-\hat{\mathcal{L}}_\omega\phi\|_w &\le \|\varepsilon^{-1}(\mathcal{L}_{D_\varepsilon}-\text{Id})+(\cdot S)^{\prime}\|_{\mathcal{B}_{s}\to\mathcal{B}_w}\operatorname{\mathrm{esssup}}_{\omega\in\Omega}\|\mathcal{L}_\omega\phi\|_{s}\\ &\le C\alpha(\varepsilon)\|\phi\|_{s} \end{align*} $$
  by using (8), with $\alpha (\varepsilon )=\|\varepsilon ^{-1}(\mathcal {L}_{D_\varepsilon }-\text {Id})+(\cdot S)^{\prime }\|_{\mathcal {B}_s\to \mathcal {B}_w}$ , which goes to zero as $\varepsilon \to 0$ by [Reference Gouëzel and Liverani24, Proposition 36].

6.2 Random piecewise hyperbolic dynamics

Let us discuss the application of Theorem 8 to random compositions of close-by piecewise hyperbolic maps, defined on a two-dimensional compact Riemann manifold X, as described in [Reference Demers and Zhang19, §10] and [Reference Demers and Liverani16, §2]. It is noteworthy that one cannot directly apply Theorem 8, because, as noted in [Reference Demers and Zhang19, §10.2.1], the transfer operator map $\omega \mapsto \mathcal {L}_\omega $ is not strongly measurable. Still, the conclusion of Theorem 8 holds; let us explain why.

In [Reference Demers and Liverani16, §2.4], the set $\Gamma _A$ of maps T satisfying the assumptions of [Reference Demers and Liverani16, §2], with second derivative $|D^2T|<A$ is introduced, as well as the distance $\gamma $ between two such maps.

Let us fix a (small enough) $\varepsilon _0>0$ , a $T\in \Gamma _A$ and let $X_{\varepsilon _0}:=\{S\in \Gamma _A: \gamma (T,S)<\varepsilon _0\}$ . We let $\mathcal {B}_s$ and $\mathcal {B}_w$ be the Banach spaces defined in [Reference Demers and Liverani16, §2.2] (where $\mathcal {B}_s$ is denoted $\mathcal {B}$ ). In particular, we recall that elements of $\mathcal {B}_s$ are distributions of order at most one. Letting $I:=[-\varepsilon _0/2,\varepsilon _0/2]$ , we set, for a fixed $L>0$ :

$$ \begin{align*}B_{\varepsilon_0,L}:=\{\mathcal T:I\to X_{\varepsilon_0},~\gamma(\mathcal T(\varepsilon),\mathcal T(\varepsilon^{\prime}))\le L|\varepsilon-\varepsilon^{\prime}|,~\text{ for all }\varepsilon,\varepsilon^{\prime}\in I\}.\end{align*} $$

This can be viewed as a ball of Lipschitz (with respect to the distance $\gamma $ ) curves from I to $X_{\varepsilon _0}$ . We now consider a measurable, countably valued mapping $\mathbf {T}:\Omega \to B_{\varepsilon _0,L}$ . As before, $(\Omega ,\mathcal F,\mathbb P)$ is a probability space endowed with an invertible, measure-preserving, and ergodic map $\sigma $ and we use the notation $T_{\omega ,\varepsilon }:=\mathbf {T}(\omega )(\varepsilon ,\cdot )$ .

We claim that for any $\varepsilon \in I$ , there exists a measurable family $(h_\omega ^\varepsilon )_{\omega \in \Omega } \subset \mathcal {B}_s$ such that $\mathcal {L}_{\omega ,\varepsilon }h_{\omega }^\varepsilon =h_{\sigma \omega }^\varepsilon $ for $\mathbb P$ -almost every $\omega \in \Omega $ and

$$ \begin{align*}\operatorname{\mathrm{esssup}}_{\omega\in\Omega}\|h_\omega^\varepsilon-h_\omega\|_{w}\le C\varepsilon^\beta|\log(|\varepsilon|)|,\end{align*} $$

for some $C>0$ , $0<\beta <1$ , independent on $\omega $ and $\varepsilon $ , with $h_\omega :=h_\omega ^0$ .

Let us review the assumptions for Theorem 8 in this context.

• • Equation (6) holds by [Reference Demers and Zhang19, Eq. (70)], where $\psi \in \mathcal {B}_s^{\prime }$ is given by $\psi (h)=h(1)$ , $h\in \mathcal {B}_s$ .

• • Up to shrinking $\varepsilon _0$ , we have (8) and (10) by [Reference Demers and Zhang19, Eq. (71)].

• • Up to replacing $\varepsilon $ by $\varepsilon ^\beta $ , (9) follows from the definition of $B_{\varepsilon _0,L}$ and [Reference Demers and Liverani16, Lemma 6.1].

• • As usual, (11) holds as $\mathcal {L}_{\omega ,\varepsilon }$ is a transfer operator associated with $T_{\omega ,\varepsilon }$ .

In particular, Proposition 6 (uniform in $\varepsilon $ and $\omega $ exponential decay of correlations) holds in the present setting. We cannot use here the fixed-point construction of Proposition 7, because we do not know whether the cocycle of transfer operators $(\mathcal L_{\omega , \varepsilon })_{\omega \in \Omega }$ is strongly measurable. However, we can use (8), (10), and that, for each $\varepsilon \in I$ , $T_{\omega ,\varepsilon }$ is countably valued to apply the version of the MET for the so-called $\mathbb P$ -continuous cocycles (see [Reference Froyland, Lloyd and Quas22, Theorem 17]): this gives us, as in Remark 13, that for each $\varepsilon \in I$ there exists:

• • $1\le l=l(\varepsilon )\le \infty $ and a sequence of exceptional Lyapunov exponents

  $$ \begin{align*}0=\Lambda (\varepsilon)=\lambda_1(\varepsilon)>\lambda_2(\varepsilon)>\cdots>\lambda_l(\varepsilon)>\kappa (\varepsilon)\end{align*} $$
  or in the case $l=\infty $ ,
  $$ \begin{align*}0=\Lambda (\varepsilon)=\lambda_1(\varepsilon)>\lambda_2(\varepsilon)>\cdots \quad \text{with }\lim_{n\to\infty} \lambda_n(\varepsilon)=\kappa (\varepsilon); \end{align*} $$

• • a full-measure set $\Omega _\varepsilon $ such that for each $\omega \in \Omega _\varepsilon $ , there is a unique measurable Oseledets splitting

  $$ \begin{align*}\mathcal{B}_s=\bigg(\bigoplus_{j=1}^l Y_j^\varepsilon(\omega)\bigg)\oplus V^\varepsilon(\omega),\end{align*} $$
  where each component of the splitting is equivariant under $\mathcal {L}_{\omega , \varepsilon }$ , that is, $\mathcal L_{\omega , \varepsilon }(Y_j^\varepsilon (\omega ))= Y_j^\varepsilon (\sigma \omega )$ and $\mathcal L_{\omega , \varepsilon }(V^\varepsilon (\omega ))\subset V^\varepsilon (\sigma \omega )$ . The subspaces $Y_j^\varepsilon (\omega )$ are finite-dimensional and for each $y\in Y_j^\varepsilon (\omega )\setminus \{0\}$ ,
  $$ \begin{align*}\lim_{n\to\infty}\frac 1n\log\|\mathcal L_{\omega, \varepsilon}^n y\|_s=\lambda_j(\varepsilon).\end{align*} $$
  Moreover, for $y\in V(\omega )$ , $\lim _{n\to \infty }(1/n)\log \|\mathcal L_{\omega , \varepsilon }^n y\|_s\le \kappa (\varepsilon )$ .

It follows easily from Proposition 6 (see the proof of [Reference Demers and Zhang19, Proposition 3.6]) that $Y_1^\varepsilon (\omega )$ is one-dimensional: for each $\varepsilon \in I$ , we may, thus, consider a generator $h_{\omega }^\varepsilon $ , normalized by $\psi (h_\omega ^\varepsilon )=1$ , which satisfies $\mathcal {L}_{\omega ,\varepsilon }h_\omega ^\varepsilon =h_{\sigma \omega }^\varepsilon $ . We now claim that

(66) $$ \begin{align} \sup_{\varepsilon\in I}\operatorname{\mathrm{esssup}}_{\omega\in\Omega}\|h_\omega^\varepsilon\|_s<+\infty. \end{align} $$

In order to establish (66), we start by observing that using (12) we have that

(67) $$ \begin{align} \| h_\omega^\varepsilon-\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^n 1\|_s=\|\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^n(h_{\sigma^{-n} \omega}^\varepsilon-1)\|_s \le D^{\prime}e^{-\lambda^{\prime} n} \|h_{\sigma^{-n} \omega}^\varepsilon-1\|_s, \end{align} $$

for $n\in \mathbb {N}$ , $\omega \in \Omega $ and $\varepsilon \in I$ . Furthermore, because $\lambda _1(\varepsilon )=0$ , we have that the random variable $\omega \mapsto \|h_\omega ^\varepsilon \|_s$ is tempered (we recall that a random variable $K\colon \Omega \to (0, +\infty )$ is tempered if $\lim _{n\to \pm \infty }(1/n) \log K(\sigma ^n \omega )=0$ for $\mathbb P$ -almost every $\omega \in \Omega $ ) for each $\varepsilon \in I$ . Hence, by [Reference Arnold1, Proposition 4.3.3] for each $\varepsilon \in I$ , there exists a random variable $K_\varepsilon \colon \Omega \to (0, +\infty )$ such that

(68) $$ \begin{align} \|h^\varepsilon_\omega -1\|_s \le K_\varepsilon (\omega) \quad \text{and} \quad K_\varepsilon (\sigma^n \omega) \le e^{{\lambda^{\prime} |n|}/{2} }K_\varepsilon (\omega), \end{align} $$

for $\mathbb P$ -almost every $\omega \in \Omega $ and $n\in \mathbb Z$ . By (67) and (68), we obtain that

$$ \begin{align*} \| h_\omega^\varepsilon-\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^n 1\|_s \le D^{\prime}K_\varepsilon (\omega)e^{-({\lambda^{\prime} n}/{2})} \quad \text{for }\mathbb P\text{-almost every }\omega \in \Omega\text{ and }n\in \mathbb{N}, \end{align*} $$

which implies that for $\varepsilon \in I$ ,

(69) $$ \begin{align} h_\omega^\varepsilon=\lim_{n\to \infty} \mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^n 1 \quad \text{in }\mathcal{B}_s, \text{for }\mathbb P\text{-almost every }\omega \in \Omega. \end{align} $$

Clearly, (66) follows readily from (8) and (69). From there, we can reproduce the proof of Theorem 8, to obtain the announced result.