2.1 RDSs

Let $(\Omega , \mathcal F , \mathbb {P})$ be a probability space and let $\sigma :\Omega \to \Omega $ be a measurably invertible, measure-preserving map. For a measurable space $(\Sigma , \mathcal G)$ , we say that a measurable map $\Phi : {\mathbb N}_0 \times \Omega \times \Sigma \to \Sigma $ is an RDS on $\Sigma $ over the driving system $\sigma $ if

$$ \begin{align*} \varphi ^{(0)} _{\omega} = \mathrm{id} _{\Sigma} \quad\text{and}\quad \varphi ^{(n+m)} _ \omega = \varphi ^{(n)}_{ \sigma ^m\omega }\circ \varphi ^{(m)}_{\omega} \end{align*} $$

for each $n, m \in {\mathbb N}_0$ and $\omega \in \Omega $ , with the notation $\varphi ^{(n)}_{\omega } =\Phi (n,\omega ,\cdot )$ and $\sigma \omega =\sigma (\omega )$ , where ${\mathbb N}_0 =\{0\} \cup {\mathbb N}$ . A standard reference for RDSs is the monograph by Arnold [Reference Arnold1]. It is easy to check that

(1) $$ \begin{align} \varphi ^{(n)}_{\omega} = \varphi _{\sigma ^{n-1}\omega }\circ \varphi _{\sigma ^{n-2}\omega } \circ \cdots \circ \varphi _{\omega} \end{align} $$

with the notation $\varphi _{\omega } = \Phi (1, \omega , \cdot )$ . Conversely, for each measurable map $\varphi : \Omega \times \Sigma \to \Sigma : (\omega , x) \mapsto \varphi _{\omega } (x)$ , the measurable map $(n,\omega , x) \mapsto \varphi _{\omega } ^{(n)}(x)$ given by (1) is an RDS. We call it an RDS induced by $\varphi $ over $\sigma $ and simply denote it by $(\varphi , \sigma )$ .

It is easy to see that if we define a skew-product map $\Theta : \Omega \times \Sigma \to \Omega \times \Sigma $ by $\Theta (\omega , x)=(\sigma \omega , \varphi _{\omega } (x))$ for each $(\omega , x)\in \Omega \times \Sigma $ , then

$$ \begin{align*} \Theta ^n (\omega , x) =(\sigma ^n \omega , \varphi ^{(n)}_{\omega} (x)) \quad \text{for all } n\in {\mathbb N}_0. \end{align*} $$

Rather than work with a single $\Theta $ -invariant measure on the product space $\Omega \times \Sigma $ , we prefer to work with a family of equivariant measures supported on $\Sigma $ fibers, the definition and existence of which we now recall from [Reference Arnold1, §1.4]. A measure $\mu $ on $(\Omega \times \Sigma , \mathcal F\times \mathcal G)$ is said to have marginal $\mathbb {P}$ on $(\Omega , \mathcal {F})$ if $ \mu \circ \pi _{\Omega }^{-1} = {\mathbb P}$ , where $\pi _{\Omega } : \Omega \times \Sigma \to \Omega $ is the projection onto the first coordinate. A probability measure $\mu $ on $(\Omega \times \Sigma , \mathcal F\times \mathcal G)$ is $\Theta $ -invariant and has marginal $\mathbb {P}$ on $(\Omega , \mathcal {F})$ if and only if there is a measurable family of probability measures (a family of probability measures $\{ \mu _{\omega } \} _{\omega \in \Omega }$ on $(\Sigma , \mathcal {G})$ is measurable if the map $\omega \mapsto \mu _{\omega } (A)$ is $(\mathcal F, \mathcal B_{\mathbb R})$ -measurable for each $A\in \mathcal G$ ) $\{ \mu _{\omega } \} _{\omega \in \Omega }$ such that ${\mu (A)= \int _{\Omega } \int _{\Sigma } 1_A(\omega ,x) \mu _{\omega } (\mathrm {d}x) {\mathbb P}(\mathrm {d}\omega )}$ for each $A\in \mathcal F\times \mathcal G$ and so that we have

(2) $$ \begin{align} \mu_{\omega } \circ \varphi _{\omega }^{-1} = \mu_{\sigma \omega } \quad \text{for almost every } \omega \in \Omega. \end{align} $$

Hence, a measurable family of probability measures $\{ \mu _{\omega } \} _{\omega \in \Omega }$ is said to be equivariant for $(\varphi , \sigma )$ if it satisfies (2).

2.2 The GKL theorem

We recall the statement of the GKL theorem from [Reference Baladi7] (although we note that the result first appeared in full generality in [Reference Gouëzel and Liverani28, Reference Gouëzel and Liverani29] and in less generality in [Reference Keller and Liverani33]). Fix an integer $N \ge 1$ and let $E_j$ , $j \in \{0, \ldots , N\}$ , be Banach spaces with $E_j \hookrightarrow E_{j-1}$ for each $j \in \{1, \ldots , N\}$ . For a family of linear operators $\{ A_{\epsilon } \}_{\epsilon \in [-1,1]}$ on these spaces, we consider the following conditions.

1. (GKL1) For all $i \in \{1, \ldots , N\}$ and ${\lvert {\epsilon }\rvert } \le 1$ ,

   $$ \begin{align*} \lVert A_{\epsilon}\rVert_{L(E_i)} \le C. \end{align*} $$

2. (GKL2) There exists $M> 0$ such that $\lVert A_{\epsilon }^n\rVert _{L(E_0)} \le C M^n$ for all ${\lvert {\epsilon }\rvert } \le 1$ and $n \in {\mathbb {N}}$ .

3. (GKL3) There exists $\alpha < M$ such that, for every ${\lvert {\epsilon }\rvert } \le 1$ , $f \in E_1$ and $n \in {\mathbb {N}}$ ,

   $$ \begin{align*} \lVert A_{\epsilon}^n f\rVert_{E_1} \le C\alpha ^n\lVert f\rVert_{E_1} + C M^n \lVert f\rVert_{E_0}. \end{align*} $$

4. (GKL4) For every ${\lvert {\epsilon }\rvert } \leq 1$ ,

   $$ \begin{align*} \lVert A_{\epsilon} - A_0\rVert_{L(E_N, E_{N-1})} \le C {\lvert{\epsilon}\rvert}. \end{align*} $$

If $N \ge 2$ , we have the following additional requirement.

1. (GKL5) There exist linear operators $Q_1, \ldots ,Q_{N-1}$ such that, for all $j \in \{1, \ldots , N-1\}$ and $i \in \{j, \ldots , N\}$ , we have $Q_j(E_i) \subseteq E_{i-j}$ and

   (3) $$ \begin{align} \lVert Q_j\rVert_{L(E_i, E_{i-j})} \le C, \end{align} $$

   and so that, for all ${\lvert {\epsilon }\rvert } \leq 1$ and $j \in \{2, \ldots , N \}$ ,

   (4) $$ \begin{align} \bigg\lVert A_{\epsilon} - A_0 - \sum_{k=1}^{j-1} \epsilon^k Q_k \bigg\rVert_{L(E_N, E_{N-j})} \le C {\lvert{\epsilon}\rvert}^j. \end{align} $$

Theorem 2.1. (The GKL theorem, [Reference Baladi7, Theorem A.4])

Fix an integer $N \ge 1$ and let $E_j$ , $j \in \{0, \ldots , N\}$ , be Banach spaces with $E_j \hookrightarrow E_{j-1}$ for each $j \in \{1, \ldots , N\}$ . Suppose that $\{ A_{\epsilon } \}_{\epsilon \in [-1,1]}$ satisfies (GKL1)–(GKL4) and if $N \ge 2$ , then also (GKL5). For ${z \notin \sigma (A_0 | E_N)}$ , set $R_0(z) = (z - A_0)^{-1}$ and define

(5) $$ \begin{align} S_{\epsilon}^{(N)}(z) = R_0(z) + \sum_{k=1}^{N-1} \epsilon^k \sum_{j=1}^k \sum_{\substack{l_1 + \cdots + l_j = k \\ l_i \ge 1}} R_0(z) Q_{l_1} R_0(z) \cdots R_0(z)Q_{l_j}R_0(z). \end{align} $$

In addition, for any $a> \alpha $ , let

$$ \begin{align*} \eta = \frac{\log(a/\alpha )}{\log(M/\alpha )}, \end{align*} $$

and, for $\delta> 0$ , set

$$ \begin{align*} \mathcal{V}_{\delta, a}(A_0) = \{z \in {\mathbb{C}} : {\lvert{z}\rvert} \ge a \quad\text{and}\quad \operatorname*{\mathrm{dist}}(z, \sigma(A_0 | E_j )) \ge \delta \quad \text{for all } j \in \{1, \ldots, N\} \}. \end{align*} $$

There exists $\epsilon _0> 0$ so that $\mathcal {V}_{\delta , a}(A_0) \cap \sigma (A_{\epsilon } | E_1 ) = \emptyset $ for every ${\lvert {\epsilon }\rvert } \le \epsilon _0$ and so that, for each $z \in \mathcal {V}_{\delta , a}(A_0)$ ,

$$ \begin{align*} \lVert(z- A_{\epsilon})^{-1}\rVert_{L(E_1)} \le C \end{align*} $$

and

$$ \begin{align*} \lVert(z- A_{\epsilon})^{-1} - S_{\epsilon}^{(N)}(z)\rVert_{L(E_N, E_0)} \le C {\lvert{\epsilon}\rvert}^{N-1 + \eta}. \end{align*} $$

3.1 Main result

Given a bounded, $\xi $ -Markov, $\xi $ -mixing operator cocycle $(A,\sigma )$ , we are interested in the question of stability (and differentiability) of the $\xi $ -normalized fixed point v of $\mathbb {A}$ . To this end, we formulate a number of conditions on operator cocycles that are reminiscent of the conditions of the GKL theorem.

Fix an integer $N \ge 1$ and let $E_j$ , $j \in \{0, \ldots , N\}$ , be Banach spaces. Let $\{(A_{\epsilon } , \sigma )\}_{\epsilon \in [-1,1]}$ be a family of operator cocycles on these spaces.

1. (QR0)

   1. (a) The Banach spaces $\{E_j\}_{j \in \{0, \ldots , N\}}$ satisfy $E_j \hookrightarrow E_{j-1}$ for each $j \in \{1, \ldots , N\}$ . Moreover, $E_N$ is separable and $\lVert \cdot \rVert _{E_j}$ -dense in $E_j$ for each $j \in \{0, \ldots , N\}$ (in particular, $E_1$ is separable).

   2. (b) There exists a non-zero functional $\xi \in E_0^*$ such that $(A_{\epsilon } , \sigma )$ is $\xi $ -Markov on $E_j$ for each $\vert \epsilon \vert \leq 1$ , $j \in \{0, \ldots , N\}$ and so that $(A_0 , \sigma )$ is $\xi $ -mixing on $E_j$ for $j \in \{1, N\}$ .

2. (QR1) For all $i \in \{1, \ldots , N\}$ and ${\lvert {\epsilon }\rvert } \le 1$ , we have $\operatorname *{\mathrm {ess\ sup}}_{\omega } \lVert A_{\epsilon }(\omega )\rVert _{L(E_i)} \le C$ .

3. (QR2) There exists $M> 0$ such that $ \operatorname *{\mathrm {ess\ sup}}_{\omega } \lVert A_{\epsilon }^{(n)}(\omega )\rVert _{L(E_0)} \le C M^n$ for all ${\lvert {\epsilon }\rvert } \le 1$ and $n \in {\mathbb {N}}$ .

4. (QR3) There exists $\alpha < M$ such that, for every $f \in E_1$ , ${\lvert {\epsilon }\rvert } \le 1$ and $n \in {\mathbb {N}}$ ,

   $$ \begin{align*} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert A_{\epsilon}^{(n)}(\omega) f\rVert_{E_1} \le C\alpha ^n\lVert f\rVert_{E_1} + C M^n \lVert f\rVert_{E_0}. \end{align*} $$

5. (QR4) For every ${\lvert {\epsilon }\rvert } \leq 1$ ,

   $$ \begin{align*} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert A_{\epsilon}(\omega) - A_0(\omega)\rVert_{L(E_N, E_{N-1})} \le C {\lvert{\epsilon}\rvert}. \end{align*} $$

If $N \ge 2$ , we have the following additional requirement.

1. (QR5) There exists linear operators $Q_1(\omega ), \ldots ,Q_{N-1}(\omega )$ for each $\omega $ such that, for all $j \in \{1, \ldots , N-1\}$ and $i \in \{j, \ldots , N\}$ ,

   $$ \begin{align*} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert Q_j(\omega)\rVert_{L(E_i, E_{i-j})} \le C \end{align*} $$

   and such that, for all ${\lvert {\epsilon }\rvert } \leq 1$ and $j \in \{2, \ldots , N \}$ ,

   $$ \begin{align*} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert A_{\epsilon}(\omega) - A_0(\omega) - \sum_{k=1}^{j-1} \epsilon^k Q_k(\omega) \rVert_{L(E_N, E_{N-j})} \le C {\lvert{\epsilon}\rvert}^j. \end{align*} $$

We need not assume that $Q_1, \ldots , Q_{N-1}$ are measurable (recall that the essential supremum of a (not necessarily measurable) complex-valued function f on $\Omega $ is the infimum of $\sup _{\omega \in \Omega _0} \vert f(\omega )\vert $ over all ${\mathbb P}$ -full measure sets $\Omega _0$ ), which will make our application in §4 simpler.

Our main theorem for this section is the following.

Theorem 3.6. Fix an integer $N \ge 1$ . Let $E_j$ , $j \in \{0, \ldots , N\}$ , be Banach spaces and let $\{ (A_{\epsilon } ,\sigma ) \}_{\epsilon \in [-1,1]}$ be a family of operator cocycles on these spaces. Suppose that $\{ (A_{\epsilon } ,\sigma ) \}_{\epsilon \in [-1,1]}$ satisfies $\mathrm {(QR0)}$ – $\mathrm {(QR4)}$ and, if $N \ge 2$ , then also (QR5). Then there exists $\epsilon _0 \in (0, 1]$ such that $(A_{\epsilon } ,\sigma )$ is $\xi $ -mixing whenever ${\lvert {\epsilon }\rvert } < \epsilon _0$ . Moreover, for each $\epsilon \in (-\epsilon _0 , \epsilon _0)$ , there is a unique $v_{\epsilon } \in L^{\infty } (\Omega , E_1)$ such that $A_{\epsilon } (\omega )v_{\epsilon } (\omega )=v_{\epsilon } (\sigma \omega )$ and $\xi (v_{\epsilon } (\omega ))=1$ almost everywhere and so that

$$ \begin{align*} \sup_{{\lvert{\epsilon}\rvert} < \epsilon_0} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert v_{\epsilon}(\omega)\rVert_{E_1} < \infty. \end{align*} $$

Lastly, there exists $\{v_0^{(k)}\}_{k=1}^{N-1} \subset L^{\infty }(\Omega , E_0) $ such that $\xi (v_0^{(k)}) = 0$ almost surely for each k and so that, for every $\eta \in (0, \log (1/\alpha )/{\log} (M/\alpha ))$ ,

(10) $$ \begin{align} v_{\epsilon} = v_0 + \sum_{k=1}^{N-1} \epsilon^k v_0^{(k)} + O_{\eta}(\epsilon^{N-1+\eta}), \end{align} $$

where $O_{\eta }(\epsilon ^{N-1+\eta })$ is to be understood as an essentially bounded term in $E_0$ that possibly depends on $\eta $ .

Remark 3.9. The claim that ‘there exists $\epsilon _0 \in (0, 1]$ such that $(A_{\epsilon } ,\sigma )$ is $\xi $ -mixing whenever ${\lvert {\epsilon }\rvert } < \epsilon _0$ ’ is exactly the content of [Reference Dragičević and Sedro20, Proposition 6] (as well as being an easy corollary of [Reference Crimmins14, Proposition 3.11]).

In fact, [Reference Dragičević and Sedro20, Proposition 6] and [Reference Crimmins14, Proposition 3.11] tell us that the claim follows from (b) of (QR0), (QR3) and (QR4) with $N=1$ . Furthermore, upon examining these proofs, it is clear that something slightly stronger is true: in the setting of Theorem 3.6, for every $\kappa \in (\rho ,1)$ , there exists $\epsilon _{\kappa }> 0$ such that, for all $\epsilon \in (-\epsilon _{\kappa }, \epsilon _{\kappa })$ ,

(11)

3.2 The proof of Theorem 3.6

Before detailing the proof of Theorem 3.6, we introduce some basic constructs. For each $j \in \{0, \ldots , N\}$ , let

$$ \begin{align*} \mathcal{E}_j = \{ f \in L^{\infty}(\Omega, E_j) : \xi(f) \text{ is almost surely constant} \}. \end{align*} $$

Since $\xi \in E_j^*$ for each $j \in \{0, \ldots , N\}$ we observe that each $\mathcal {E}_j$ is a closed subspace of $L^{\infty }(\Omega , E_j)$ and, therefore, is a Banach space. Moreover, we have $\mathcal {E}_j \hookrightarrow \mathcal {E}_{j-1}$ for $j \in \{1, \ldots , N\}$ . For each $j \in \{1, \ldots , N\}$ , we may consider the lift $\mathbb {A}_{\epsilon ,j}$ of the operator cocycle $( A_{\epsilon } , \sigma )$ on $E_j$ , although we omit the subscript j and just write $\mathbb {A}_{\epsilon }$ , which will be of no consequence.

The beginning of the proof of Theorem 3.6 is straightforward. First, we note that (QR1) implies that $(A_0,\sigma )$ is bounded on $E_j$ for $j \in \{1, N\}$ and so Proposition 3.5 may be applied to characterize the spectrum of $\mathbb {A}_0$ on $\mathcal {E}_1$ and $\mathcal {E}_N$ . Let $\rho $ be the rate of $\xi $ -mixing in (QR0): that is, $(A_0, \sigma )$ is $\xi $ -mixing on $E_j$ with rate $\rho $ for each $j\in \{ 1, N\}$ . Then it follows from (QR0) and Proposition 3.5 that 1 is a simple eigenvalue of $\mathbb {A}_0$ , when considered on either space, and we have

(12) $$ \begin{align} \sigma(\mathbb{A}_0 | \mathcal{E}_j) \setminus \{1\} \subseteq \{ z \in {\mathbb{C}} : {\lvert{z}\rvert} \le \rho \} \end{align} $$

for $j \in \{1, N\}$ . For each $j \in \{1, \ldots , N\}$ , one may use basic functional analysis and the fact that $\mathcal {E}_N \hookrightarrow \mathcal {E}_j \hookrightarrow \mathcal {E}_1$ to deduce that 1 is a simple eigenvalue of $\mathbb {A}_0 : \mathcal {E}_j \to \mathcal {E}_j$ and that (12) holds. As a consequence, we find a $\xi $ -normalized $v_0 \in \mathcal {E}_N$ that is the unique $\xi $ -normalized fixed point of ${\mathbb A}_0 : \mathcal E_j \to \mathcal E_j$ for each $j \in \{1, \ldots , N\}$ .

We now turn to constructing the $\xi $ -normalized fixed points of $\mathbb A_{\epsilon } : \mathcal E_1 \to \mathcal E_1$ . By Remark 3.9, we may find some $\kappa \in (\rho ,1)$ and $\epsilon _0> 0$ such that $(A_{\epsilon } ,\sigma )$ is $\xi $ -mixing on $E_1$ with rate $\kappa $ for every $\epsilon \in (-\epsilon _0, \epsilon _0)$ . We note that each $( A_{\epsilon } ,\sigma )$ is bounded on $E_1$ due to (QR1) and so, by Proposition 3.5, we find that $1$ is a simple eigenvalue of $\mathbb {A}_{\epsilon } : \mathcal {E}_1 \to \mathcal {E}_1$ and that $\sigma (\mathbb {A}_{\epsilon } | \mathcal {E}_1) \setminus \{1\} \subseteq \{z \in {\mathbb {C}} : {\lvert {z}\rvert } \le \kappa \}$ . Thus, $\mathbb A_{\epsilon } : \mathcal E_1 \to \mathcal E_1$ has a unique $\xi $ -normalized fixed point $v_{\epsilon } \in \mathcal {E}_1$ for each $\epsilon \in (-\epsilon _0, \epsilon _0)$ , by Proposition 3.5. Moreover, by virtue of the uniform bound (11), we may strengthen the conclusion of Proposition 3.5: for all n sufficiently large, the family of maps $\{\mathbb {A}_{\epsilon }^n\}_{{\lvert {\epsilon }\rvert } < \epsilon _0}$ uniformly contract the set $\mathcal {X} _1$ from (8). Hence, we deduce the bound

(13) $$ \begin{align} \sup_{{\lvert{\epsilon}\rvert} < \epsilon_0} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert v_{\epsilon}(\omega)\rVert_{E_1} < \infty, \end{align} $$

as required for Theorem 3.6.

Thus, to complete the proof of Theorem 3.6, it suffices to prove (10). It may be easily seen from the proof of Proposition 3.5 that the eigenprojection $\Pi _{\epsilon } \in L(\mathcal {E}_1)$ of ${\mathbb {A}_{\epsilon } : \mathcal {E}_1 \to \mathcal {E}_1}$ onto the eigenspace for $1$ is defined for $f \in \mathcal {E}_1$ and $\epsilon \in (-\epsilon _0, \epsilon _0)$ by

$$ \begin{align*} \Pi_{\epsilon} (f) = \xi(f) v_{\epsilon}. \end{align*} $$

Since each $v_{\epsilon }$ is $\xi $ -normalized, we consequently have

(14) $$ \begin{align} v_{\epsilon} = v_0 + (\Pi_{\epsilon}- \Pi_0)v_0. \end{align} $$

If $\delta \in (0, 1-\kappa )$ , then $D_{\delta } = \{ z \in {\mathbb {C}} : {\lvert {z -1}\rvert } = \delta \} \subseteq {\mathbb {C}} \setminus \sigma (\mathbb {A}_{\epsilon } | \mathcal {E}_1)$ for every ${\epsilon \in (-\epsilon _0, \epsilon _0)}$ . Thus,

(15)

Applying (15) to (14) yields

(16)

The idea is to apply the GKL theorem to the lifts $\{\mathbb {A}_{\epsilon }\}_{\epsilon \in [-1,1]}$ with Banach spaces $\{\mathcal {E}_j\}_{0 \le j \le N}$ and then develop a Taylor expansion in (16). The hypothesis that (QR1)–(QR4) hold for $\{ (A_{\epsilon } ,\sigma )\}_{\epsilon \in [-1, 1]}$ with constants almost surely independent of $\omega $ readily implies that the lifts $\{\mathbb {A}_{\epsilon }\}_{\epsilon \in [-1,1]}$ satisfy (GKL1)–(GKL4) for the spaces $\{\mathcal {E}_j\}_{0 \le j \le N}$ . Hence, in the case where $N = 1$ , we may apply Theorem 2.1 to the lifts $\{{\mathbb A}_{\epsilon } \}_{\epsilon \in [-1, 1]}$ .

However, the case where $N \ge 2$ is more delicate because the measurability of $Q_j$ is not required in Theorem 3.6. Thus, we introduce the following functional space instead of $L^{\infty } (\Omega , E_j)$ , where the objects are defined up to almost everywhere equality but we loosen the measurability requirement. For each $j\in \{0,\ldots ,N\}$ , let $B(\Omega , E_j)$ denote the set of (not necessarily measurable) bounded $E_j$ -valued functions on $\Omega $ equipped with the uniform norm $\lVert \cdot \rVert _{B(\Omega , E_j)}$ that is defined by

$$ \begin{align*} \lVert f\rVert_{B(\Omega , E_j)} = \sup _{\omega \in \Omega} \lVert f(\omega )\rVert_{E_j}, \quad f\in B(\Omega , E_j), \end{align*} $$

and let

$$ \begin{align*} \mathcal{N}_j = \{ f \in B(\Omega, E_j) : f= 0 \text{ almost surely} \}. \end{align*} $$

Then $\mathcal {N}_j$ is a closed subspace of $B(\Omega , E_j)$ and, thus, we can form a quotient space

$$ \begin{align*} I^{\infty}( \Omega , E_j) = B(\Omega, E_j) / \mathcal{N}_j. \end{align*} $$

Since $B(\Omega , E_j)$ is a Banach space, $I^{\infty }( \Omega , E_j) $ is also a Banach space with respect to the quotient norm

$$ \begin{align*} \lVert f\rVert_{I^{\infty}( \Omega , E_j) } = \inf _{h\in {\mathcal N}_j} \lVert g - h\rVert_{B(\Omega , E_j )} , \quad f\in I^{\infty}( \Omega , E_j), \end{align*} $$

where g is a representative of f. As for $L^{\infty }( \Omega , E_j)$ , under the identification of each element of $I^{\infty }( \Omega , E_j)$ with its representative, we have

$$ \begin{align*} \Vert f \Vert _{I^{\infty}( \Omega , E_j) } = \operatorname*{\mathrm{ess\ sup}} _{\omega} \Vert f(\omega )\Vert _{E_j}. \end{align*} $$

Thus, under the identification, we have $\Vert f\Vert _{L^{\infty }(\Omega , E_j)} = \Vert f\Vert _{I^{\infty }(\Omega , E_j)}$ for each $f\in L^{\infty }(\Omega , E_j)$ . In particular, $L^{\infty }(\Omega , E_j)$ isometrically injects into $I^{\infty }(\Omega , E_j)$ . Finally, let

(17) $$ \begin{align} \widetilde{\mathcal{E}}_j = \{ f \in I^{\infty}(\Omega, E_j) : \xi(f) \text{ is almost surely constant}\}. \end{align} $$

We simply write $\Vert f\Vert _{ \widetilde {\mathcal {E}}_j }$ for $\Vert f\Vert _{I^{\infty } (\Omega , E_j)}$ if $f \in \widetilde {\mathcal {E}}_j $ . Repeating the previous argument, one can show (GKL1)–(GKL4) for the lifts $\{\mathbb {A}_{\epsilon }\}_{\epsilon \in [-1,1]}$ with respect to the spaces $\{ \widetilde {\mathcal {E}}_j\}_{0 \le j \le N}$ . We now deduce (GKL5) for the lifted systems.

Proposition 3.10. Assume the setting of Theorem 3.6 with $N \ge 2$ . Then (GKL5) holds with operators $\mathbb {Q}_j$ defined by

$$ \begin{align*} (\mathbb{Q}_jf)(\omega) = Q_j(\sigma^{-1}\omega) f_{\sigma^{-1} \omega}, \end{align*} $$

where $j \in \{1, \ldots , N-1\}$ , $i \in \{j, \ldots , N\}$ and $f \in \widetilde {\mathcal {E}}_i$ .

Proof. It is straightforward to verify the required inequalities in (GKL5) from those in (QR5). Hence, it only remains to show that $\mathbb {Q}_j(\widetilde {\mathcal {E}}_i) \subset \widetilde {\mathcal {E}}_{i-j}$ for each $j\in \{1,\ldots , N-1\}$ and $i\in \{j, \ldots , N\}$ . Let $j\in \{1,\ldots ,N\}$ , $i\in \{j,\ldots ,N-1\}$ and fix $f \in \widetilde {\mathcal {E}}_i$ . That $\mathbb {Q}_jf \in E_{i-j}$ almost everywhere follows immediately from (3), and so, to complete the proof, it is sufficient to show that $\xi (\mathbb {Q}_jf)$ is almost surely constant. Since $E_N$ is $\lVert \cdot \rVert _{E_j}$ -dense in $E_j$ for almost every $\omega $ and each $\eta> 0$ , we may find a $g_{\eta , \omega } \in E_N$ such that $\lVert f_{\sigma ^{-1} \omega } - g_{\eta , \omega }\rVert _{E_j} \le \eta $ . By (QR5), we have, for almost every $\omega $ , that

(18) $$ \begin{align} \begin{aligned} \lvert{\xi(\mathbb{Q}_jf)(\omega)}\rvert &= \lvert{\xi(Q_j(\sigma^{-1}\omega) f_{\sigma^{-1} \omega})}\rvert \\ &\le \lvert{\xi(Q_j(\sigma^{-1}\omega)g_{\eta, \omega})}\rvert + \lvert{\xi(Q_j(\sigma^{-1}\omega)(f_{\sigma^{-1} \omega} - g_{\eta, \omega})}\rvert. \end{aligned} \end{align} $$

By (QR5) and as $\xi \in E_0^*$ , we have $\lim _{\eta \to 0} \lvert {\xi (Q_j(\sigma ^{-1}\omega )(f_{\sigma ^{-1} \omega } - g_{\eta , \omega }))}\rvert = 0$ for almost every $\omega $ . On the other hand, since $g_{\eta , \omega } \in E_N$ by (QR5), again we have

(19) $$ \begin{align} &\xi(Q_j(\sigma^{-1}\omega)g_{\eta, \omega})\nonumber\\ &\quad= \lim_{\epsilon \to 0} \xi\bigg(\epsilon^{-j} (A_{\epsilon}(\sigma^{-1}\omega) - A_0(\sigma^{-1}\omega))g_{\eta, \omega} - \sum_{k=1}^{j-1} \epsilon^{k-j} Q_k(\sigma^{-1}\omega) g_{\eta, \omega}\bigg). \end{align} $$

Therefore, using the fact that $( A_{\epsilon } ,\sigma )$ is $\xi $ -Markov for every $\epsilon \in [-1,1]$ , we have $\xi (Q_1 (\sigma ^{-1}\omega )g_{\eta , \omega }) = 0$ almost surely. By a simple induction on (19), we find that $\xi (Q_j(\sigma ^{-1}\omega ) g_{\eta , \omega }) = 0$ almost surely too. Thus, in (18),

(20) $$ \begin{align} \lvert{\xi(\mathbb{Q}_jf)(\omega)}\rvert \le \limsup_{\eta \to 0} \lvert{\xi(Q_j(\sigma^{-1}\omega)g_{\eta, \omega})}\rvert + \lvert{\xi(Q_j(\sigma^{-1}\omega)(f_{\sigma^{-1} \omega} - g_{\eta, \omega})}\rvert = 0, \end{align} $$

which completes the proof.

By Proposition 3.10, we have (GKL5) for the lifts $\{\mathbb {A}_{\epsilon }\}_{\epsilon \in [-1,1]}$ with the spaces $\{ \widetilde {\mathcal {E}}_j \} _{0\leq j\leq N}$ in the setting of Theorem 3.6 whenever $N \ge 2$ , and so we can apply Theorem 2.1 in this case. As a consequence, we may now finish the proof of Theorem 3.6. Let $\eta \in (0, \log (1/\alpha )/{\log} (M/\alpha ))$ and fix $a \in ( \alpha ,1)$ so that $\eta = \log (a/\alpha )/{\log} (M/\alpha )$ . Recall $\delta $ from (15) and notice that we may take $\delta $ to be as small as we like. Henceforth, we fix $\delta \in (0, 1-a)$ and choose some $\delta _0 \in (0, \min \{\delta , 1 - a - \delta \})$ . Upon recalling the statement of Theorem 2.1 and our earlier characterization of $\sigma (\mathbb {A}_0 | \widetilde {\mathcal {E}}_j)$ for $j \in \{1, \ldots , N\}$ (see the paragraph following (12)),

$$ \begin{align*} D_{\delta} \subseteq \{ z \in {\mathbb{C}} : {\lvert{z}\rvert} \ge s \text{ and } {\lvert{z-1}\rvert} \ge \delta_0 \} \subseteq \mathcal{V}_{\delta_0, a}(\mathbb{A}_0). \end{align*} $$

We now apply Theorem 2.1 to the lifts $\{\mathbb {A}_{\epsilon }\}_{\epsilon \in [-1,1]}$ with Banach spaces $\widetilde {\mathcal {E}}_j$ , ${j \in \{0, \ldots , N\}}$ , to deduce the existence of $\epsilon _{\eta } \in (0, \epsilon _0)$ such that, for every $\epsilon \in (-\epsilon _{\eta }, \epsilon _{\eta })$ , we have $\mathcal {V}_{\delta _0, a}(A_0) \cap \sigma (\mathbb {A}_{\epsilon } | \widetilde {\mathcal {E}}_1) = \emptyset $ and, for each $z \in \mathcal {V}_{\delta _0, a}(\mathbb {A}_0)$ , that

(21) $$ \begin{align} \lVert(z- \mathbb{A}_{\epsilon})^{-1} - \mathbb{S}_{\epsilon}^{(N)}(z)\rVert_{L(\widetilde{\mathcal{E}}_N, \widetilde{\mathcal{E}}_0)} \le C {\lvert{\epsilon}\rvert}^{N-1 + \eta}, \end{align} $$

where $\mathbb {S}_{\epsilon }^{(N)}(z)$ is defined as in (5). With (21) in hand, we may proceed with obtaining (10) via (16). In particular, for $z \in D_{\delta }$ ,

(22)

For each $k \in \{1, \ldots , N-1\}$ , we now define $v_0^{(k)} \in \widetilde {\mathcal E}_0$ by

Furthermore, by the proof of Proposition 3.10, we have $\xi (v_0^{(k)}) = 0$ almost everywhere for all k. By integrating (22) over $D_{\delta }$ and recalling (16), we get

(23)

in $\widetilde {\mathcal E}_0$ . Moreover, since $D_{\delta } \subseteq \mathcal {V}_{\delta _0,a}(\mathbb {A}_0)$ , it follows from (21) that

(24)

Finally, we show that $ v_0^{(k)}$ lies in $\mathcal E_0 \subset L^{\infty } (\Omega , E_0)$ for each $k=1, \ldots ,N-1$ . Recall that $v_{\epsilon } \in \mathcal E_0 $ for every $\epsilon \in (-\epsilon _0,\epsilon _0)$ . Thus, $\epsilon ^{-1}(v_{\epsilon } - v_0)$ belongs to $ \mathcal E_0 $ . Therefore, since $ \mathcal E_0 $ isometrically injects into $\widetilde {\mathcal E}_0 $ (recall the argument above (17)), it follows from the Taylor expansion (23) and (24) that $\{\epsilon ^{-1}(v_{\epsilon } - v_0)\}_{\vert \epsilon \vert <\epsilon _0}$ is a Cauchy sequence in $ \mathcal E_0 $ . Denote its limit by $v^{\prime }_0$ so that $\epsilon ^{-1}(v_{\epsilon } - v_0) - v_0' $ lies in $ \mathcal E_0 $ and $\Vert \epsilon ^{-1}(v_{\epsilon } - v_0) - v_0' \Vert _{ \mathcal E_0 }\to 0$ as $\epsilon \to 0$ . Then, by using again the fact that $ \mathcal E_0 $ isometrically injects into $\widetilde {\mathcal E}_0$ , we deduce that $v^{\prime }_0$ equals the limit of $\{\epsilon ^{-1}(v_{\epsilon } - v_0)\}_{\vert \epsilon \vert <\epsilon _0}$ in $\widetilde {\mathcal E}_0$ . Hence, $v_0' = v_0^{(1)}$ by (23) and (24), which concludes that $v_0^{(1)}$ lies in $ \mathcal E_0 $ . By considering $\epsilon ^{-k}( v_{\epsilon } - v_0 - \sum _{j=1}^{k-1} \epsilon ^{j} v_0^{(j)} )$ instead of $\epsilon ^{-1}(v_{\epsilon } - v_0)$ , we can show via induction that $v_0^{(k)}$ also lies in $ \mathcal E_0 $ for each $k=2, \ldots , N-1$ . This completes the proof of Theorem 3.6 because (23) and (24) hold with $\mathcal E_j$ in place of $\widetilde {\mathcal E}_j $ .

4.1 Equivariant family of measures

Let M be a compact connected ${\mathcal C}^{\infty }$ Riemannian manifold and let m denote the associated Riemannian probability measure on M. Fix a Lebesgue space $(\Omega , \mathcal {F}, \mathbb {P})$ and an invertible, ergodic, ${\mathbb P}$ -preserving map $\sigma : \Omega \to \Omega $ . For some $r \ge 1$ , let $\mathcal {T} : \Omega \to \mathcal {C}^{r+1}(M,M)$ denote a $(\mathcal {F}, \mathcal {B}_{\mathcal {C}^{r+1}(M,M)})$ -measurable map. Recall from §2.1 that the RDS $(\mathcal T, \sigma )$ induced by $\mathcal T$ over $\sigma $ is explicitly written as a measurable map

$$ \begin{align*} {\mathbb N}_0 \times \Omega \times X \ni (n, \omega ,x)\mapsto \mathcal T^{(n)}_{\omega} (x), \quad \mathcal T^{(n)}_{\omega} :=\mathcal T_{\sigma^{n-1} \omega} \circ \cdots \circ \mathcal T _{\omega}, \end{align*} $$

and, since $\sigma $ is invertible, the equivariance of a measurable family of probability measures $\{ \mu _{\omega } \}_{\omega \in \Omega }$ for $(\mathcal T,\sigma )$ is given as

$$ \begin{align*} \mu_{\omega } \circ \mathcal T _{\omega }^{-1} = \mu_{\sigma \omega } \quad \text{for almost every } \omega \in \Omega. \end{align*} $$

We aim to study the regularity of the dependence of $\{ \mu _{\omega } \}_{\omega \in \Omega }$ on the map $\mathcal {T}$ as $\mathcal {T}$ is fiber-wise varied in a uniformly $\mathcal {C}^N$ way for some $N \le r $ . To do this, we shall realize equivariant families of probability measures as fixed points of (the lifts of) certain operator cocycles (linear RDSs) and then apply Theorem 3.6. In particular, we shall consider the Perron–Frobenius operator cocycle associated to the RDS $( \mathcal {T} ,\sigma )$ on an appropriate Banach space. Recall that the Perron–Frobenius operator ${\mathcal {L}}_T$ associated to a non-singular (recall that a measurable map $T : M \to M$ is said to be non-singular (with respect to m) if $m(A) = 0$ implies that $m(T^{-1}(A)) = 0$ ) measurable map $T : M\to M$ is given by

(25) $$ \begin{align} \mathcal L_T f = \frac{\mathrm{d} [(f m) \circ T]}{\mathrm{d}m} \quad \text{for } f\in L^1(M,m), \end{align} $$

where $fm$ is a finite signed measure given by for $A\in \mathcal {B}_M$ and $\mathrm {d}\mu /\mathrm {d}m$ is the Radon–Nikodym derivative of an absolutely continuous finite signed measure $\mu $ . Note that, for each M-valued random variable $\psi $ whose distribution is $fm$ for some density $f\in L^1(M,m)$ , $T(\psi )$ has the distribution $(\mathcal L_T f) m$ (and, thus, $\mathcal L_T$ is also called the transfer operator associated with T). It is routine to verify that

(26)

and that $\mathcal L_T$ is an m-Markov operator, where, in an abuse of notation, we let m denote the linear functional . In addition, ${\mathcal {L}}_T$ is positive: if $f \in L^1(M,m)$ satisfies $f \ge 0$ almost everywhere, then ${\mathcal {L}}_{T} f \ge 0$ almost everywhere.

Let $\mathcal {N}^{r+1}(M,M)$ denote the set of $T \in \mathcal {C}^{r+1}(M,M)$ satisfying $\det D_xT \ne 0$ for all $x\in M$ . Notice that if $T \in \mathcal {N}^{r+1}(M,M)$ , then T is automatically non-singular with respect to m and so ${\mathcal {L}}_T$ is a well-defined operator on $L^1(M,m)$ . Additionally, for such T, we have ${\mathcal {L}}_T \in L(\mathcal {C}^{r}(M))$ with

$$ \begin{align*} ({\mathcal{L}}_T f)(x) = \sum_{T(y) = x} \frac{f(y)}{{\lvert{\det D_y T}\rvert}} \quad \text{for all } f \in \mathcal{C}^r(M). \end{align*} $$

Hence, from a measurable map $\mathcal {T} : \Omega \to \mathcal {N}^{r+1}(M,M)$ , we obtain a map ${\mathcal {L}}_{\mathcal {T}}: \omega \mapsto {\mathcal {L}}_{\mathcal {T}_{\omega }} : \Omega \to L(\mathcal {C}^r(M))$ , which is measurable by virtue of the following proposition (we postpone its proof until Appendix A because it is mundane but technical).

Thus, if we demand that $\mathcal {T} \in \mathcal {N}^{r+1}(M,M)$ almost surely, then $( {\mathcal {L}}_{\mathcal {T}}, \sigma )$ is an m-Markov operator cocycle on $\mathcal {C}^r(M)$ , which we shall call the Perron–Frobenius operator cocycle (on $\mathcal {C}^r(M)$ ) associated to $\mathcal {T}$ . In order to apply the theory of §3, we require that the Perron–Frobenius operator cocycle is bounded and m-mixing. This later condition will entail some mixing hypotheses on our random systems. However, as in the deterministic case, in order to realize the mixing of the RDS in operator theoretic terms, we may be forced to consider the Perron–Frobenius operator cocycle on an alternative Banach space. Specifically, we shall seek Banach spaces $(X, \lVert \cdot \rVert _X)$ satisfying the following conditions.

1. (S1) $\mathcal {C}^r(M)$ is dense in X with $\mathcal {C}^r(M) \hookrightarrow X$ .

2. (S2) The embedding $\mathcal {C}^r(M) \hookrightarrow (\mathcal {C}^{\infty }(M))^*$ given by the map continuously extends to an embedding .

It is clear that any X satisfying (S1) must be separable. Moreover, we note that the functional $\varphi \in (\mathcal {C}^{\infty }(M))^* \mapsto \varphi (1_M)$ is continuous on $(\mathcal {C}^{\infty }(M))^*$ and yields m when pulled back via the embedding $\mathcal {C}^r(M) \hookrightarrow (\mathcal {C}^{\infty }(M))^*$ that is described in (S2). Hence, if (S2) holds so that we have an embedding $X \hookrightarrow (\mathcal {C}^{\infty }(M))^*$ that continuously extends the $\mathcal {C}^r(M) \hookrightarrow (\mathcal {C}^{\infty }(M))^*$ , then m induces a continuous linear functional on X. In particular, we may speak of m-Markov operators in $L(X)$ . The following proposition gives a sufficient condition for an m-Markov operator in $L({\mathcal C}^r(M))$ to be extended to an m-Markov operator in $L(X)$ .

Proposition 4.2. Let $(A, \sigma )$ be a bounded, m-Markov operator cocycle on ${\mathcal C}^r(M)$ and let X be a Banach space satisfying (S1) and (S2). Suppose that

(27) $$ \begin{align} \operatorname*{\mathrm{ess\ sup}}_{\omega} \sup_{\substack{f \in \mathcal{C}^r(M) \\ \lVert f\rVert_X = 1}} \lVert A (\omega )f\rVert_{X} < \infty. \end{align} $$

Then A almost surely extends to a unique, bounded operator on X such that ${\omega \mapsto A(\omega ) : \Omega \to L(X)}$ is strongly measurable. Consequently, $( A,\sigma )$ is a bounded, m-Markov operator cocycle on X such that

(28) $$ \begin{align} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert A (\omega )\rVert_{L(X)} <\infty. \end{align} $$

Proof. It is clear that A almost surely extends to a unique, bounded operator on X and that

$$ \begin{align*} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert A (\omega )\rVert_{L(X)} = \operatorname*{\mathrm{ess\ sup}}_{\omega} \sup_{\substack{f \in \mathcal{C}^r(M) \\ \lVert f\rVert_X = 1}} \lVert A (\omega )f\rVert_{X} < \infty. \end{align*} $$

That A is almost surely m-Markov in $L(X)$ follows straightforwardly from the fact that A is almost surely m-Markov in $L(\mathcal {C}^r(M))$ and that m uniquely extends to a continuous linear functional on X. Hence, it only remains to show that $\omega \mapsto A (\omega )$ is strongly measurable in $L(X)$ . Suppose that $f \in X$ . Then there exists a sequence $\{f_n\}_{n \in {\mathbb {N}}} \subset \mathcal {C}^r(M)$ with limit f in X. For each n, the map $\omega \mapsto A (\omega ) f_n$ is $(\mathcal {F}, \mathcal {B}_{\mathcal {C}^r(M)})$ -measurable and so it must be $(\mathcal {F}, \mathcal {B}_{X})$ measurable too due to (S1). Moreover, for almost every $\omega $ ,

$$ \begin{align*} \lim_{n \to \infty} \lVert A (\omega ) f_n - A (\omega ) f\rVert_X = 0, \end{align*} $$

which is to say that $\omega \mapsto A (\omega ) f$ is the almost everywhere pointwise limit (in X) of $(\mathcal {F}, \mathcal {B}_{X})$ -measurable functions. Hence, $\omega \mapsto A (\omega ) f$ is $(\mathcal {F}, \mathcal {B}_{X})$ -measurable since $(\Omega , \mathcal F, {\mathbb P})$ is a Lebesgue space (in particular, complete). That $\omega \mapsto A (\omega )$ is strongly measurable in $L(X)$ then follows from Lemma 3.1 and the fact that $f \in X$ was arbitrary.

Hence, by Propositions 4.1 and 4.2, if $\mathcal T: \Omega \to {\mathcal N}^{r+1}(M,M)$ is measurable and X satisfies (S1) and (S2), then the Perron–Frobenius operator cocycle $( {\mathcal {L}}_{\mathcal {T}} ,\sigma )$ on ${\mathcal C}^r(M)$ can be extended to a bounded, m-Markov operator cocycle on X. Compare also (28) with (QR1).

The following proposition will help us to describe the relationship between the equivariant family of probability measures for $(\mathcal {T} , \sigma )$ and the fixed point of the lift of a bounded, m-mixing Perron–Frobenius operator cocycle $(\mathcal L_{\mathcal {T} }, \sigma )$ .

Proof. Notice that the set

$$ \begin{align*} \mathcal{D} = \{ f \in L^{\infty}(\Omega,X) : m(f) = 1 \text{ and } f \in X_+ \text{ almost surely}\} \end{align*} $$

is almost surely invariant under $A(\omega )$ since $(A,\sigma )$ is bounded, positive and m-Markov. Hence, we may carry out the construction of h in Proposition 3.5 with $\mathcal {D}$ in place of $\mathcal {X}_1$ to conclude that $h \in X_+$ almost surely. Thus, there exists $\{ f_k\}_{k \in {\mathbb {N}}} \subseteq L^{\infty } (\Omega ,\mathcal {C}^r(M))$ such that $f_k(\omega ) \ge 0$ and for every k and so that $\lim _{k \to \infty } f_k (\omega )= h(\omega )$ in X for almost every $\omega $ . As $X \hookrightarrow (\mathcal {C}^{\infty }(M))^*$ , it follows that $\lim _{k \to \infty } f_k (\omega )= h(\omega )$ in the sense of distributions as well. Thus, for any positive $g \in \mathcal {C}^{\infty }(M)$ ,

(29)

(recall the embedding of ${\mathcal C}^r(M)$ in (S2)). As $f_k(\omega )$ and g are positive, it follows from (29) that $h(\omega )(g) \ge 0$ for every such g. Hence, $h(\omega )$ is a positive distribution for almost every $\omega $ . On the other hand, as is well known, for any positive $f\in ({\mathcal C}^{\infty } (M))^*$ , one can find a positive Radon measure $\mu _f$ such that for every $g\in {\mathcal C}^{\infty } (M)$ . We denote by $\mu _{\omega }$ the positive Radon measure corresponding to $h(\omega )$ .

To see that $\mu _{\omega } $ is a probability measure for almost every $\omega $ , we note that, by (29) and as

for every k,

Finally, $\{ \mu _{\omega } \} _{\omega \in \Omega }$ is a measurable family on the complete probability space $(\Omega , \mathcal F, {\mathbb P})$ because for, any $A\in \mathcal B_M$ , by using (29) again,

for almost every $\omega $ , while, for every k, $\omega \mapsto f_k(\omega ) :\Omega \to {\mathcal C}^r(M)$ is measurable and

is continuous, so that

is measurable too.

Hence, if X satisfies (S1) and (S2) and if the Perron–Frobenius operator cocycle $( {\mathcal {L}}_{\mathcal {T}} , \sigma )$ on X is m-mixing, then we obtain a measurable family of Radon probability measures $\{ \mu _{\omega } \} _{\omega \in \Omega }$ such that

is in $L^{\infty } (\Omega , X)$ . Furthermore, $\{ \mu _{\omega } \} _{\omega \in \Omega }$ is equivariant because it follows from (29) that, for any $A\in \mathcal B_M$ and almost every $\omega $ ,

Due to (26), the continuity of $\mathcal L_{T_{\omega }} : X\to X$ and the fact that h is the fixed point of the lift of $(\mathcal L_{\mathcal T}, \sigma )$ , this coincides with

4.2 The conditions (QR4) and (QR5)

In this section, we discuss a sufficient condition for a family of Perron–Frobenius operator cocycles $\{ (\mathcal L_{\mathcal T_{\epsilon } } ,\sigma )\} _{\epsilon \in [-1,1]}$ to satisfy (QR4) and (QR5). We emphasize that these conditions hold rather independently of how the underlying random dynamics $(\mathcal T_{\epsilon }, \sigma )$ behave (see Proposition 4.5 for a precise statement), so we treat (QR4) and (QR5) here as a final preparation before specializing to our applications. For simplicity, throughout this section, we assume that M is a d-dimensional torus $\mathbb T^d$ . One may straightforwardly remove this assumption by considering a partition of unity. (Refer to, e.g., [Reference Baladi7, Reference Gouëzel and Liverani28]; see also Appendix A).

Notice that (QR4) and (QR5) are conditions for a single iteration $\mathcal L_{T_{\epsilon ,\omega }}$ (not for $\mathcal L_{T_{\epsilon , \sigma ^{n-1} \omega }} \circ \cdots \circ \mathcal L_{T_{\epsilon , \omega }}$ , $n\in {\mathbb N}$ ), and so clear observations may be found in the non-random setting. Fix $r\geq 1$ and $1\leq s \leq r$ , and consider $T\in {\mathcal C}^N([-1,1], {\mathcal C}^{r+1} (\mathbb T^d ,\mathbb T^d))$ . Let ${1\leq N\leq s}$ be an integer and let $E_j$ , $j \in \{0, \ldots , N\}$ , be Banach spaces with $E_j \hookrightarrow E_{j-1}$ for each $j \in \{1, \ldots , N\}$ satisfying the following conditions.

1. (P1) The condition (S1) holds with $E_j$ in place of X for each $j \in \{0, \ldots , N\}$ .

2. (P2) The condition (S2) holds with $E_j$ in place of X for each $j \in \{0, \ldots , N\}$ .

3. (P3) There are constants $C>0$ and $0\leq \rho \leq r-N$ such that

   $$ \begin{align*} \Vert uf \Vert _{E_j} \leq C\Vert u \Vert _{{\mathcal C}^{\rho +j}} \Vert f\Vert _{E_j} \quad \text{for each }u, f\in {\mathcal C}^r(\mathbb T^d)\quad \text{and}\quad j\in \{0,\ldots ,N\}. \end{align*} $$

4. (P4) There is a constant $C>0$ such that

   $$ \begin{align*} &\bigg\Vert \frac{\partial}{\partial x_l } f\bigg\Vert _{E_{j-1}}\\ &\quad\leq C\Vert f\Vert _{E_j} \quad \text{for each } f\in {\mathcal C}^r(\mathbb T^d), l\in \{1,\ldots , d\} \quad\text{and}\quad j\in \{ 1, \ldots , N\}. \end{align*} $$

Observe that all conditions (P1)–(P4) are not for the operators $\mathcal L_{T_{\epsilon } }$ , $\epsilon \in [-1,1]$ , with ${T_{\epsilon } := T(\epsilon )}$ , but for the spaces $E_j$ , $j\in \{0,\ldots ,N\}$ , so the following proposition is quite useful in our applications. Note that if

(30) $$ \begin{align} \lVert\mathcal L_{T_{\epsilon} }f\rVert_{E_j} < C_{\epsilon} \Vert f\Vert _{E_j}\quad \text{for each }f \in \mathcal{C}^r(\mathbb T^d), j \in \{0, \ldots, N\}\quad\text{and}\quad \vert \epsilon \vert \leq 1, \end{align} $$

then it follows from Proposition 4.2 that $\mathcal L_{T_{\epsilon }}$ is a bounded operator on $E_j$ for each ${j \in \{0, \ldots , N\}}$ and $\vert \epsilon \vert \leq 1$ .

Before starting the proof of Proposition 4.5, we discuss a consequence of Proposition 4.5 with respect to the conditions (QR4) and (QR5). Let $\{(\mathcal T_{\epsilon } , \sigma )\}_{\epsilon \in [-1,1]}$ be a family of RDSs such that, for almost every $\omega $ , the map $\epsilon \mapsto T_{\epsilon , \omega }:= \mathcal T_{\epsilon } (\omega )$ is in ${\mathcal C}^N([-1,1], {\mathcal C}^{r+1}(\mathbb T^d ,\mathbb T^d))$ . Let $E_j$ , $j \in \{0, \ldots , N\}$ , be Banach spaces with $E_j \hookrightarrow E_{j-1}$ for each $j \in \{1, \ldots , N\}$ satisfying (P1)–(P4). We suppose the following.

1. (P5) $T_{\epsilon ,\omega } \in {\mathcal N}^{r+1}(\mathbb T^d ,\mathbb T^d)$ for each $\epsilon \in [-1,1]$ and almost every $\omega $ . Furthermore, (27) holds with $E_j$ and $\mathcal L_{T_{\epsilon , \omega }}$ in place of X and $A(\omega )$ for every $j\in \{ 0,\ldots , N\}$ and $\vert \epsilon \vert \leq 1$ .

Then it follows, from Proposition 4.2, that the Perron–Frobenius operator cocyles $(\mathcal L_{\mathcal T_{\epsilon }}, \sigma )$ , $\epsilon \in [-1,1]$ , can be extended to bounded operator cocycles on each $E_j$ and that (QR1) holds for these operator cocycles by virtue of (28).

For each $j\in \{ 0, \ldots , N\}$ , $i\in \{ j,\ldots , N\}$ and almost every $\omega $ , it follows from Proposition 4.5 that we can define $Q_j(\omega ) : E_i \to E_{i-j}$ by

$$ \begin{align*} Q_j (\omega ) f = \frac{1}{j!} \bigg(\frac{d^j}{d\epsilon ^j} \mathcal L_{T_{\epsilon ,\omega } } f \bigg) _{\epsilon =0} \quad \text{for } f\in E_i. \end{align*} $$

By the definition, it is straightforward to see that, for all $\epsilon \in [-1,1]$ and $2\leq j\leq N$ ,

$$ \begin{align*} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert\mathcal L_{T_{\epsilon ,\omega}} - \mathcal L_{T_{0 ,\omega}} \rVert_{L(E_N, E_{N-1})} \le C {\lvert{\epsilon}\rvert} \end{align*} $$

and

$$ \begin{align*} \operatorname*{\mathrm{ess\ sup}}_{\omega} \lVert\mathcal L_{T_{\epsilon ,\omega}} - \mathcal L_{T_{0 ,\omega}} - \sum_{k=1}^{j-1} \epsilon^k Q_k(\omega) \rVert_{L(E_N, E_{N-j})} \le C {\lvert{\epsilon}\rvert}^j. \end{align*} $$

To summarize the above argument, we conclude the following corollary.

We now return to the proof of Proposition 4.5.

Proof of Proposition 4.5

Fix $1\leq \sigma \leq N$ , $1\leq \rho \leq r$ , $f\in {\mathcal C}^{\sigma }([-1,1] , {\mathcal C}^{\rho } (\mathbb T^d))$ , ${g\in {\mathcal C}^{\rho } (\mathbb T^d)}$ and $1\leq l\leq d$ for the time being (notice that this f is different from f in the statement of Proposition 4.5 in the sense that this f depends on $\epsilon \in [-1,1]$ ). We simply denote ${d^a}/{d\epsilon ^a} f \in {\mathcal C}^{\sigma -a}([-1,1] , {\mathcal C}^{\rho } (\mathbb T^d))$ by $f^{(a)}$ for each integer $a\in [0, \sigma ]$ . We also simply denote by $\partial _l g$ the partial derivative of g with respect to the lth coordinate and let $\partial ^{\alpha } = \partial ^{\alpha _1}_1 \cdots \partial ^{\alpha _d}_d$ and $\vert \alpha \vert = \alpha _1 +\cdots + \alpha _d$ for each multi-index $\alpha =(\alpha _1 ,\ldots , \alpha _d) \in {\mathbb N}_0^d$ . Then, for each $ \epsilon \in [-1,1]$ and $x\in \mathbb T^d$ ,

$$ \begin{align*} \partial _{l} (f ^{(1)}_{\epsilon} )(x)= \partial _{\epsilon} \partial _{l} \tilde f (\epsilon , x) = \partial _{l}\partial _{\epsilon} \tilde f (\epsilon , x), \end{align*} $$

where $\tilde f: [-1,1] \times \mathbb T^d \to \mathbb T^d$ is given by $\tilde f (\epsilon , x) = f_{\epsilon } (x)$ . (Since $f^{(1)} \in {\mathcal C}^{0}([-1,1] , {\mathcal C}^1 (\mathbb T^d))$ , it is straightforward to see that the first equality holds and that $(\epsilon ,x) \mapsto \partial _{l} (f ^{(1)}_{\epsilon } )(x) $ is continuous. The second equality also immediately follows from these observations together with the Schwarz–Clairaut theorem on equality of mixed partials.) In particular,

(31) $$ \begin{align} f ^{(1)} =(\epsilon \mapsto \partial _{\epsilon} \tilde f (\epsilon , \cdot ) )\quad \text{in } {\mathcal C}^{\sigma -1 }([-1,1], {\mathcal C}^{\rho }(\mathbb T^d)). \end{align} $$

Furthermore, it is also straightforward to see that the map $\epsilon \mapsto \partial _l f_ \epsilon $ is in $ {\mathcal C}^{\sigma } ([-1,1], {\mathcal C}^{\rho -1}(\mathbb T^d))$ , which we denote by $\partial _l f$ as a slight abuse of notation, and that

(32) $$ \begin{align} (\partial _{l}f )^{(1)}= \partial _{l} (f^{(1)}) \quad \text{in } {\mathcal C}^{\sigma -1}([-1,1] , {\mathcal C}^{\rho -1}(\mathbb T^d)), \end{align} $$

(33) $$ \begin{align} (\det DT ) ^{(1)} = \det DT ^{(1)} \quad \text{in } {\mathcal C}^{N-1}([-1,1] , {\mathcal C}^{r-1}(\mathbb T^d)). \end{align} $$

Moreover, we denote by $T_{(l)}\in {\mathcal C}^N([-1,1], {\mathcal C}^{r+1}(\mathbb T^d))$ the map $\epsilon \mapsto (x\mapsto T_{(l), \epsilon } (x))$ , where $ T_{(l), \epsilon } (x)$ is the lth coordinate of $T_{\epsilon } (x) \in \mathbb T^d$ (under the identification of $\mathbb T^d$ with $\mathbb R^d$ ). Finally, we define a map $\mathbf Lf: [-1,1] \to {\mathcal C}^{\rho }(\mathbb T^d)$ by

$$ \begin{align*} (\mathbf Lf )_ \epsilon = \mathcal L_{T_ \epsilon } f_{\epsilon} \quad \text{for } \epsilon \in [-1,1], \end{align*} $$

which is well defined by virtue of (30). The following is the key lemma for the proof of Proposition 4.5.

Lemma 4.7. For each $f\in {\mathcal C}^{\sigma } ([-1,1], {\mathcal C}^{\rho } (\mathbb T^d))$ with $\sigma \in [1,N]$ and $\rho \in [1,r]$ , $(\mathbf L f) ^{(1)}$ exists in ${\mathcal C}^{\sigma -1} ([-1,1], {\mathcal C}^{\rho -1}(\mathbb T^d))$ and is of the form

(34) $$ \begin{align} (\mathbf L f) ^{(1)}= \mathbf L \bigg(\sum _{\vert \alpha \vert \leq 1} J_{0,\alpha } \cdot \partial ^{\alpha} f + \sum _{\vert \alpha \vert \leq 1} J_{1, \alpha } \cdot \partial ^{\alpha} f^{(1)} \bigg), \end{align} $$

where $J_{k ,\alpha } $ is in ${\mathcal C}^{N-1}([-1,1] ,{\mathcal C}^{r-1} (\mathbb T^d))$ is a polynomial function of $\partial ^{\beta } T_{(l )}$ , $\partial ^{\beta } T_{(l )}^{(1)}$ ( $1\leq l \leq d$ , $\vert \beta \vert \leq 2$ ) and $( \det DT)^{-1}$ for each $k=0,1$ and multi-index $\alpha $ with $\vert \alpha \vert \leq 1$ .

Proof. Observe that $ \det D_x T_{\epsilon }>0$ for all $\vert \epsilon \vert \leq 1$ and $x\in \mathbb T^d$ or $\det D_x T_{\epsilon } <0$ for all $\vert \epsilon \vert \leq 1$ and $x\in \mathbb T^d$ because $T_{\epsilon } \in {\mathcal N}^{r+1}(\mathbb T^d,\mathbb T^d)$ for each ${\vert \epsilon \vert \leq 1}$ and $\epsilon \mapsto T_{\epsilon }$ is continuous. We consider only the former case because the other case is similar. Also, we show (34) only around $\epsilon =0$ to keep our notation simple (the general case can be literally treated). First, we note that there is $\epsilon _0>0$ , $B\in {\mathbb N}$ , a finite covering $\{ U_{\unicode{x3bb} } \} _{\unicode{x3bb} \in \Lambda }$ of $\mathbb T^d$ and ${\mathcal C}^{r+1}$ maps $(T _{\epsilon } )^{-1}_{\unicode{x3bb} ,b} : U_{\unicode{x3bb} } \to (T _{\epsilon } )^{-1}_{\unicode{x3bb} ,b} (U_{\unicode{x3bb} } ) $ for $\vert \epsilon \vert <\epsilon _0$ , $\unicode{x3bb} \in \Lambda $ and $b\in \{ 1,\ldots , B\}$ such that, for each $\vert \epsilon \vert <\epsilon _0$ , $\unicode{x3bb} \in \Lambda $ , $b\in \{ 1,\ldots , B\}$ and $g\in {\mathcal C}^r(\mathbb T^d)$ ,

(35) $$ \begin{align} T _{\epsilon} \circ (T _{\epsilon} )^{-1}_{\unicode{x3bb} ,b} (x) = x\quad \text{on } U_{\unicode{x3bb}} \end{align} $$

and

(36) $$ \begin{align} \mathcal M_1(\epsilon , g)(x) := \sum _{T_{\epsilon} (y) =x} g(y) = \sum _{b=1}^B g\circ (T _{\epsilon} )^{-1}_{\unicode{x3bb} ,b} (x) \quad \text{on } U_{\unicode{x3bb}} \end{align} $$

(because $T_0 \in {\mathcal N}^{r+1}(\mathbb T^d,\mathbb T^d)$ and $\mathbb T^d$ is compact; see Appendix A for details). Note also that if we define $\mathcal M_2: [-1,1]\times {\mathcal C}^r(\mathbb T^d) \to {\mathcal C}^r(\mathbb T^d)$ by

$$ \begin{align*} \mathcal M_2 (\epsilon , g) = \frac{g}{ \det DT_{\epsilon} } \quad \text{for } \epsilon \in [-1,1], g\in {\mathcal C}^r(\mathbb T^d), \end{align*} $$

then, for each $f\in {\mathcal C}^{\sigma } ([-1,1],{\mathcal C}^r(\mathbb T^d))$ and $\vert \epsilon \vert <\epsilon _0$ ,

$$ \begin{align*} (\mathbf L f)_{\epsilon} =\mathcal M_1(\epsilon , \mathcal M_2(\epsilon , f_{\epsilon} )). \end{align*} $$

Notice that both $\mathcal M_1$ and $\mathcal M_2$ are linear with respect to $g\in {\mathcal C}^r(\mathbb T^d)$ . Hence, it follows from the chain rule for (Fréchet) derivatives that

(37) $$ \begin{align} (\mathbf L f)_{\epsilon} ^{(1)} = \partial _{\epsilon} \mathcal M_1(\epsilon , \mathcal M_2(\epsilon , f_{\epsilon} )) + \mathcal M_1(\epsilon , \partial _{\epsilon} \mathcal M_2(\epsilon , f_{\epsilon} ) + \mathcal M_2(\epsilon , f_{\epsilon} ^{(1)} )) \end{align} $$

if the derivatives exist, where $\partial _{\epsilon } ={\partial }/{\partial \epsilon }$ .

Now we calculate $ \partial _{\epsilon } \mathcal M_1$ and $ \partial _{\epsilon } \mathcal M_2$ . First, we show that

(38) $$ \begin{align} \partial _{\epsilon} \mathcal M_1(\epsilon , g) =\mathcal M_1\bigg( \epsilon , -\sum _{l =1}^d \partial _{l} g \cdot \frac{\sum _{k=1}^d (\mathrm{adj} ( DT_{\epsilon})) _{l , k} \cdot T_{(k),\epsilon } ^{(1)} }{\det D T_{\epsilon} } \bigg), \end{align} $$

where $\mathrm {adj} (A)$ is the adjugate matrix (i.e., the transpose of the cofactor matrix) of a matrix A. By (31) (with $\epsilon \mapsto \mathcal M_1 (\epsilon , g)$ in place of f), (36) and the chain rule for derivatives,

(39) $$ \begin{align} \partial _{\epsilon} \mathcal M_1 (\epsilon , g)(x) = \partial _{\epsilon} \widetilde{ \mathcal M}_{1,g} (\epsilon , x) = \sum _{b=1}^B \sum _{l =1}^d \partial _{l} g \circ (T _{\epsilon} )^{-1} _{b,\unicode{x3bb}}(x) \cdot \partial _{\epsilon} ((T _{\epsilon} )^{-1}_{b,\unicode{x3bb}})_{(l)} (x), \quad x\in U_{\unicode{x3bb}} , \end{align} $$

where $((T _{\epsilon } )^{-1}_{b,\unicode{x3bb} } )_{(l)}(x)$ is the lth coordinate of $(T _{\epsilon } )^{-1}_{b,\unicode{x3bb} } (x)$ and $\widetilde { \mathcal M}_{1,g} (\epsilon , x) = \mathcal M _1(\epsilon , g)(x)$ . On the other hand, by differentiating the lth coordinate of (35) for $1\leq l \leq d$ , we get

$$ \begin{align*} T_{(\ell ) ,\epsilon }^{(1)} (y) + \sum _{k =1}^d \partial _{k } T_{(\ell ),\epsilon }(y) \cdot \partial _{\epsilon} ((T _{\epsilon} )^{-1}_{b,\unicode{x3bb}} )_{(k)} (x)=0, \quad y=(T _{\epsilon} )^{-1}_{b,\unicode{x3bb}} (x), \; x\in U_{\unicode{x3bb}}. \end{align*} $$

In the matrix form (under the identification of $\mathbb T^d$ with $\mathbb R^d$ ), this can be written as

$$ \begin{align*} T^{(1)}_{\epsilon} (y) + D_y T_{\epsilon } [ \partial _{\epsilon} (T _{\epsilon} )^{-1}_{b,\unicode{x3bb}}(x) ] =0, \quad y =(T _{\epsilon} )^{-1}_{b,\unicode{x3bb}} (x), \; x\in U_{\unicode{x3bb}}, \end{align*} $$

where we see $ T^{(1)}_{\epsilon } (y)$ and $ \partial _{\epsilon } (T _{\epsilon } )^{-1}_{b,\unicode{x3bb} } (x)$ as column vectors. Thus, since $A^{-1} = (\det A)^{-1} \mathrm {adj}(A)$ for any invertible matrix A,

(40) $$ \begin{align} \partial _{\epsilon} (T _{\epsilon} )^{-1}_{b,\unicode{x3bb}}(x) =- (\det D_yT_{\epsilon} )^{-1} \mathrm{adj}(D_yT_{\epsilon} ) [ T^{(1)}_{\epsilon} (y) ], \quad y =(T _{\epsilon} )^{-1}_{b,\unicode{x3bb}} (x), \; x\in U_{\unicode{x3bb}}. \end{align} $$

Equation (38) immediately follows from (39) and (40). Furthermore, by the quotient rule for derivatives and (33),

(41) $$ \begin{align} \partial _{\epsilon} \mathcal M_2 (\epsilon , g) =-\frac{g \cdot \det D T^{(1)}_{\epsilon } }{(\det D T_{\epsilon } )^2} \end{align} $$

and

(42) $$ \begin{align} \partial _l (\mathcal M_2 (\epsilon , g) )= \frac{\partial _l g}{ \det DT_{\epsilon} } - \frac{g\cdot \partial _l(\det DT_{\epsilon})}{ (\det DT_{\epsilon})^2 }. \end{align} $$

The conclusion immediately follows from (37), (38), (41) and (42).

Now we complete the proof of Proposition 4.5. We first consider the case when ${f\in {\mathcal C}^r(\mathbb T^d)}$ . We will show by induction that, for each $1\leq k \leq j$ , $(\mathbf L f)^{(k)}$ exists and is of the form

(43) $$ \begin{align} (\mathbf L f)^{(k)} = \mathbf L \bigg(\sum _{\vert \alpha \vert \leq k} \widehat J_{k,\alpha } \cdot \partial ^{\alpha} f \bigg), \end{align} $$

where $\widehat J_{k,\alpha }$ is in ${\mathcal C}^{N-k}([-1,1],{\mathcal C}^{r-k} (\mathbb T^d))$ is a polynomial function of $\partial ^{\beta } T_{(l )}^{(k')}$ ( $1\leq l \leq d$ , $0\leq k'\leq k$ , $\vert \beta \vert \leq k+1$ ) and $( \det DT)^{-1}$ for each multi-index $\alpha $ with $\vert \alpha \vert \leq k$ . Equation (43) for $k=1$ is an immediate consequence of Lemma 4.7 (notice that f in Lemma 4.7 depended on $\epsilon $ while f here does not). Suppose that $k\geq 2$ and (43) holds with $k-1$ instead of k. Then, by Lemma 4.7,

$$ \begin{align*} (\mathbf L f)^{(k )} &= \mathbf L \bigg(\sum _{\vert \alpha \vert \leq 1} J_{ 0,\alpha } \cdot \partial ^{\alpha} \bigg(\sum _{\vert \beta \vert \leq k-1 } \widehat J_{k-1 ,\beta } \cdot \partial ^{\beta} f \bigg)\\ &\quad+ \sum _{\vert \alpha \vert \leq 1} J_{1, \alpha } \cdot \partial ^{\alpha} \bigg(\sum _{\vert \beta \vert \leq k -1} \widehat J_{k-1 ,\beta }^{(1)} \cdot \partial ^{\beta} f \bigg) \bigg). \end{align*} $$

Therefore, (43) immediately follows from (32) and (33). Furthermore, exists as an element of ${\mathcal C}^{0}([-1,1], E_{i-j})$ by (P3) and the fact that $s-j\geq 0$ .

Next, we consider the general case, i.e., the case when $f\in E_i$ . By (P1), one can find $\{ f_n\} _{n\geq 1} \subset {\mathcal C}^r(\mathbb T^d)$ such that $\Vert f -f_n\Vert _{E_i} \to 0$ as $n\to \infty $ . By the result in the previous paragraph, $ (\mathbf L f_n)^{(k)}$ exists as an element of a Banach space ${\mathcal C}^0([-1,1], E_{i-k})$ for all ${1\leq k\leq j}$ . On the other hand, it follows from (P3), (P4), (30) and (43) that

$$ \begin{align*} \sup _{\epsilon \in [-1,1]}\Vert (\mathbf L f_n)^{(k)}_{\epsilon} - (\mathbf L f_m)^{(k)}_{\epsilon}\Vert _{E_{i-k}} \leq C\sum _{\vert \alpha \vert \leq k} \sup _{\epsilon \in [-1,1]}\Vert \widehat J_{k,\alpha ,\epsilon }\Vert _{{\mathcal C}^{r-k}} \Vert f_n - f_m\Vert _{E_i} \to 0 \end{align*} $$

as $n, m\to \infty $ . In particular, $\lim _{n\to \infty } (\mathbf Lf_n)^{(j)} $ exists. In a similar manner, we can show that $\mathbf L$ is a bounded operator from ${\mathcal C}^0([-1,1], E_i)$ to ${\mathcal C}^j([-1,1], E_{i-j})$ , so that $\lim _{n\to \infty } (\mathbf Lf_n)^{(j)} =(\mathbf L f)^{(j)}$ in ${\mathcal C}^0([-1,1], E_{i-j})$ . In conclusion, $\mathbf L f : \epsilon \mapsto \mathcal L_{T_{\epsilon } } f$ is in ${\mathcal C}^j([-1,1],E_{i-j})$ .

4.3 Random Anosov maps

Let M be a compact, connected $\mathcal {C}^{\infty }$ Riemannian manifold with dimension d. In this section, we consider RDSs consisting of Anosov maps lying in a small $\mathcal {C}^{r+1}(M,M)$ -neighborhood of a fixed, topologically transitive Anosov diffeomorphism $T \in \mathcal {C}^{r+1}(M,M)$ for some $r \ge 1$ . The setting we consider is very similar to that of [Reference Dragičević and Sedro20, §4]; however, we obtain more general conclusions than those of [Reference Dragičević and Sedro20]. For the remainder of this section, we fix a topologically transitive Anosov diffeomorphism $T \in \mathcal {C}^{r+1}(M,M)$ . Recall that $(\Omega , \mathcal {F}, \mathbb {P})$ is a Lebesgue space and that $\sigma : \Omega \to \Omega $ is a measurably invertible, ergodic, measure-preserving map. For every $\eta> 0$ , we define

$$ \begin{align*} \mathcal{O}_{\eta}(T) = \{ S \in \mathcal{C}^{r+1}(M,M) : d_{\mathcal{C}^{r+1}}(S,T) < \eta \}. \end{align*} $$

Recall that if $\eta $ is sufficiently small, then $\mathcal {O}_{\eta }(T) \subset \mathcal {N}^{r+1}(M,M) $ and every $S \in \mathcal {O}_{\eta }(T)$ is an Anosov diffeomorphism. A map $\mathcal {T} : \Omega \to \mathcal {C}^{r+1}(M,M)$ will be said to be measurable if it is $(\mathcal {F}, \mathcal {B}_{\mathcal {C}^{r+1}(M,M)})$ -measurable.

We consider RDSs induced by measurable maps $\mathcal {T} : \Omega \to \mathcal {O}_{\eta }(T)$ for some small, fixed $\eta $ , over $\sigma $ . Our main result for this section concerns the stability properties of the equivariant family of probability measures associated to such systems. We will formulate our result in the setting developed by Gouëzel and Liverani in [Reference Gouëzel and Liverani28]. In particular, in [Reference Gouëzel and Liverani28] it is shown that, when a topologically transitive Anosov map is smoothly perturbed, the Sinai–Ruelle–Bowen measure varies with similar regularity in certain anisotropic Banach spaces.

A small technical comment is required before proceeding: in [Reference Gouëzel and Liverani28] the usual metric on M is replaced by an adapted metric for T (which will also be adapted for $S \in \mathcal {O}_{\eta }(T)$ provided that $\eta $ is sufficiently small); we shall do the same here. We denote by m the Riemannian probability measure induced by the adapted metric on M. For each $q \geq 0$ , $p \in {\mathbb {N}} _0$ with $p \le r$ , one obtains a space $B_{p,q}(T)$ by taking the completion of $\mathcal {C}^r(M)$ with respect to anisotropic norms $\lVert \cdot \rVert _{p,q}$ as defined in [Reference Gouëzel and Liverani28, §3] (actually, the norms in [Reference Gouëzel and Liverani28, §3] are defined on the real Banach space $\mathcal {C}^r(M,{\mathbb {R}})$ . Here, we consider the complexification, which is of no consequence). Since the map T is fixed, we will just write $B_{p,q}$ in place of $B_{p,q}(T)$ . Our main result for this section is the following theorem.

We will use Theorem 3.6 to prove Theorem 4.8, with the help of Propositions 4.2 and 4.4 and of Corollary 4.6. Therefore, we should check (P1)–(P5) and (QR0)–(QR3) for appropriate Banach spaces. We start with the basic properties of the $B_{p,q}$ spaces from [Reference Gouëzel and Liverani28].

1. (1) By the definition of $\Vert \cdot \Vert _{p,q}$ , it is straightforward to see that $\Vert \partial _l f \Vert _{p,q} \leq \Vert f\Vert _{p+1,q-1}$ for each $f\in {\mathcal C}^{r}(M)$ and $1\leq l\leq d$ . Furthermore, $B_{p+1,q-1} \hookrightarrow B_{p,q}$ .

2. (2) [Reference Gouëzel and Liverani28, Lemma 2.1] If $p+q <r$ , then the unit ball in $B_{p+1,q-1}$ is relatively compact in $B_{p,q}$ .

3. (3) [Reference Gouëzel and Liverani28, Lemma 3.2] $\Vert uf \Vert _{p,q} \leq C\Vert u\Vert _{{\mathcal C}^{p+q}} \Vert f\Vert _{p,q}$ for each $u \in {\mathcal C}^{p+q}(M)$ and $f\in {\mathcal C}^{r}(M)$ . In particular, if $p+q <r$ , then $\mathcal {C}^r(M) \hookrightarrow B_{p,q}$ (see also [Reference Gouëzel and Liverani28, Remark 4.3]).

4. (4) [Reference Gouëzel and Liverani28, Proposition 4.1] We have $B_{p,q} \hookrightarrow ({\mathcal C}^q(M) )^*$ . Specifically, for each $h \in \mathcal {C}^r(M)$ , one obtains a distribution $\tilde {h} \in ({\mathcal C}^q(M) )^*$ defined by . The map $h \mapsto \tilde {h}$ continuously extends from $\mathcal {C}^r(M)$ to $B_{p,q}$ and yields the required inclusion.

We also remark that there exist injections $B_{p,q} \to B_{p-1, q}$ and $B_{p,q} \to B_{p,q'}$ for $q'> q$ due to [Reference Gouëzel and Liverani28, Remark 4.2]. By the fourth item of the above list, the functional on $\mathcal {C}^r(M)$ extends to a continuous functional on $B_{p,q}$ , which we shall also denote by m. The following result summarizes some facts from [Reference Conze and Raugi13, Reference Gouëzel and Liverani28] pertaining to the boundedness and mixing of the Perron–Frobenius operator associated to maps in $\mathcal {O}_{\eta }(T)$ for small $\eta $ . We refer the reader to [Reference Gouëzel and Liverani28, Lemma 2.2] and the discussion at the beginning of [Reference Gouëzel and Liverani28, §7] for the first and second items and to [Reference Conze and Raugi13, Proposition 2.10] for the third item (see also [Reference Dragičević, Froyland, González-Tokman and Vaienti19, §3]).

Fix $q \geq 0$ and $p \in {\mathbb N}$ with $p+q < r-N$ . Let $E_{j} = B_{p+ j, q + N-j}$ , $j \in \{0, \ldots , N\}$ . Then the conditions (P1)–(P4) on these Banach spaces immediately follow from the above list (recall that each $E_j$ is the completion of ${\mathcal C}^r(M)$ with respect to $\Vert \cdot \Vert _{E_j}$ ). Furthermore, fix a bounded subset $\mathcal K$ of $\mathcal {C}^N([-1,1],\mathcal {C}^{r+1}(M,M))$ and let $\{ \mathcal {T}_{\epsilon } : \Omega \to \mathcal {O}_{\eta _0}(T)\}_{\epsilon \in [-1,1]}$ be a family of measurable maps such that $\epsilon \mapsto \mathcal {T}_{\epsilon }(\omega )$ lies in $\mathcal K$ for almost every $\omega $ . Then, by virtue of Proposition 4.2, the first part of Proposition 4.9 and the above list, Perron–Frobenius operator cocycles $\{ (\mathcal L_{\mathcal T_{\epsilon } } , \sigma )\} _{\epsilon \in [-1,1]}$ associated with the random dynamics $\{ (\mathcal T_{\epsilon } ,\sigma )\} _{\epsilon \in [-1,1]}$ are precisely defined on these Banach spaces and they satisfy (P5) and (QR0) with $\xi =m$ (see the remark following (S2)) except the m-mixing property. In fact, (QR2), (QR3) and the mixing of $\{ ( {\mathcal {L}}_{\mathcal {T}_{\epsilon } }, \sigma )\}_{\epsilon \in [-1,1]}$ on $E_j$ for $j\in \{ 1, N\}$ are consequences of each item of Proposition 4.9, respectively. By Corollary 4.6, (QR1), (QR4) and (QR5) also hold for $\{ ( {\mathcal {L}}_{\mathcal {T_{\epsilon } }}, \sigma )\}_{\epsilon \in [-1,1]}$ on $E_j$ (see also [Reference Gouëzel and Liverani28, Lemma 7.1] and [Reference Gouëzel and Liverani28, §9]).

By Proposition 4.4 (and the remark following it), for each $\epsilon \in [-1,1]$ , $\mathcal {T} _{\epsilon }$ has an equivariant measurable family of Radon probability measures $\{ \mu ^{\mathcal {T} _{\epsilon } } _{\omega } \} _{\omega \in \Omega }$ and ${h_{\mathcal T_{\epsilon } } \in L^{\infty }(\Omega , E_N) = L^{\infty }(\Omega , B_{p+N-1,q})}$ such that $h_{\mathcal T_{\epsilon } }(\omega )(g) = g \mathrm {d}\mu ^{\mathcal {T} _{\epsilon }} _{\omega }$ for each $g\in {\mathcal C}^{\infty } (M)$ and almost every $\omega $ . Furthermore, we apply Theorem 3.6 to deduce the claim that ${\epsilon \mapsto h_{\mathcal T_{\epsilon } }(\omega )}$ is in $\mathcal {C}^{N-1}([-1,1], E_0) =\mathcal {C}^{N-1}([-1,1], B_{p-1,q+N})$ for almost every $\omega $ , which completes the proof of Theorem 3.6.

4.4 Random U(1) extensions of expanding maps

In this section, we will apply Theorem 3.6 to quenched linear response problems for random U(1) extensions of expanding maps. Let $\mathcal U $ be the set of ${\mathcal C} ^{\infty }$ endomorphisms $T: \mathbb T^2 \to \mathbb T^2$ on the torus $\mathbb {T} ^2=\mathbb R^2/ \mathbb Z^2$ of the form

(44) $$ \begin{align} T:\binom{x}{s}\mapsto \left( \! \! \begin{array}{cc} E(x) & \\ s+\tau(x) & \!\!\! \mod\! 1\end{array} \! \! \right)\hspace{-1pt}, \end{align} $$

where $E: \mathbb S^1\to \mathbb S^1$ is a ${\mathcal C}^{\infty }$ orientation-preserving endomorphism on the circle $\mathbb S^1 =\mathbb R/ \mathbb Z$ and $\tau : \mathbb S^1 \rightarrow \mathbb {R}$ is a ${\mathcal C}^{\infty }$ function (T is called the U(1) extension of E over $\tau $ ). U(1) extensions of expanding maps can be seen as toy models of (piecewise) hyperbolic flows such as geodesic flows on manifolds with negative curvature or dispersive billiard flows (via suspension flows of hyperbolic maps; see [Reference Hasselblatt31, Reference Parry and Pollicott40]), and have been intensively studied by several authors (see, e.g., [Reference Dolgopyat15, Reference Faure21, Reference Nakano, Tsujii and Wittsten38, Reference Nakano and Wittsten39]). When we want to emphasize the dependence of E and $\tau $ in (44) on T, we write them as $E_T$ and $\tau _T$ . Fix $T\in \mathcal U$ and assume that E is an expanding map on $\mathbb S^1$ in the sense that $\min _{x\in \mathbb S^1} E'(x)>1$ . Let r be a positive integer. For every $\eta>0$ , we define

$$ \begin{align*} \mathcal O_{\eta} (T) =\{ S \in \mathcal U \mid d_{\mathcal{C}^{r+1 }}(S,T) < \eta \}. \end{align*} $$

Note that $\mathcal U\subset {\mathcal N}^{r+1}(\mathbb T^2,\mathbb T^2)$ and that, if $\eta $ is sufficiently small, then $E_S$ is an expanding map for every $S \in \mathcal {O}_{\eta }(T)$ .

Recall that $(\Omega , \mathcal F, {\mathbb P})$ is a Lebesgue space and that $\sigma :\Omega \to \Omega $ is a measurably invertible, ergodic, ${\mathbb P}$ -preserving map on $(\Omega , \mathcal F, {\mathbb P})$ . When $\tau _T(x) =\alpha $ for any $x\in \mathbb S^1$ with some constant $\alpha $ , then obviously T does not admit any mixing physical measure because the rotation $s\mapsto s+({1}/{2\pi })\alpha $ mod one has no mixing physical measure. However, it is known that if $\tau $ satisfies a generic condition, called the partial captivity condition, then T admits a unique absolutely continuous invariant probability measure for which correlation functions of T decay exponentially fast (in particular, T is mixing). The partial captivity condition was first introduced by Faure [Reference Faure21] and was proved to be generic in [Reference Nakano, Tsujii and Wittsten38]. Furthermore, it was shown in [Reference Nakano and Wittsten39, Theorem 1.6] that, if T satisfies the partial captivity condition, then there is an $\eta _0>0$ and an $m_0\in {\mathbb N}$ , only depending on T (see the comment above Proposition 4.11 for more a precise choice of $\eta _0$ and $m_0$ ), such that, if $r\geq m_0$ , then, for any measurable map $\mathcal T:\Omega \to \mathcal O_{\eta _0}(T)$ , the RDS $(\mathcal T, \sigma )$ induced by $\mathcal T$ over $\sigma $ admits a unique equivariant measurable family of absolutely continuous probability measures $\{ \mu ^{\mathcal T}_{\omega } \}_{\omega \in \Omega }$ such that the Radon–Nikodym derivative of $\mu ^{\mathcal T}_{ \omega }$ is in the usual Sobolev space $H^{r}(\mathbb T^2)$ of regularity r for ${\mathbb P}$ -almost every $\omega $ and that quenched correlation functions of $(\mathcal T,\sigma )$ for $\{ \mu ^{\mathcal T}_{\omega } \}_{\omega \in \Omega }$ decay exponentially fast.

Assume that T satisfies the transversality condition, and fix such an $\eta _0>0$ and an $m_0\in ~{\mathbb N}$ . Assume also that $r\geq m_0+1$ . The main result in this section is the following theorem.

We recall the basic properties of the Sobolev spaces $H^m(\mathbb T^2)$ with regularity $m\in {\mathbb N}_ 0$ . Recall that $\Vert f\Vert _{H^m} ^2= \sum _{\vert \alpha \vert \leq m} \Vert \partial ^{\alpha } f \Vert _{L^2}^2$ .

1. (1) By the definition of $\Vert \cdot \Vert _{H^m}$ , it is straightforward to see that $\Vert \partial _l f \Vert _{H^m} \leq \Vert f\Vert _{H^{m+1}}$ for each $f\in {\mathcal C}^{m+1}(\mathbb T^2)$ and $1\leq l\leq d$ and that $\Vert uf \Vert _{H^m} \leq C\Vert u\Vert _{{\mathcal C}^{m}} \Vert f\Vert _{H^m}$ for each $u, f \in {\mathcal C}^{m}(\mathbb T^2)$ .

2. (2) By Kondrachov’s embedding theorem, $H^{m+1}(\mathbb T^2) \hookrightarrow H^m(\mathbb T^2)$ and the unit ball in $H^{m+1}(\mathbb T^2)$ is relatively compact in $H^m(\mathbb T^2)$ .

3. (3) ${\mathcal C}^{m'}(\mathbb T^2)$ is dense in $H^{m}(\mathbb T^2)$ for each $m'\geq m$ because ${\mathcal C}^{m'}(\mathbb T^2)\subset H^{m}(\mathbb T^2)\subset {\mathcal C}^{m-1}(\mathbb T^2)$ by Sobolev’s embedding theorem.

4. (4) By the Cauchy–Schwarz inequality, we have $H^m(\mathbb T^2) \hookrightarrow ({\mathcal C}^0(\mathbb T^2) )^*$ by $h\mapsto \tilde h$ given by for $g\in {\mathcal C}^0(\mathbb T^2)$ .

Let $\unicode{x3bb} _0:=(\inf _{S\in \mathcal O_{\eta _0}} \min _{x\in \mathbb S^1} E_{S}'(x) )^{-1}$ , which is less than one by taking $\eta _0$ small, if necessary. Fix $\unicode{x3bb} \in ( \unicode{x3bb} _0^{1/2} , 1)$ . Let $m_0$ be a sufficiently large integer given in [Reference Nakano and Wittsten39, Theorem 1.5]. Let $N\leq r-m_0$ be a positive integer. By taking $\eta _0$ small, if necessary, we assume that $(\inf _{S\in \mathcal O_{\eta _0}} \min _{x\in \mathbb S^1} E_{S}'(x))^{-1}<\unicode{x3bb} $ . Fix a family of measurable maps ${\{ \mathcal {T}_{\epsilon } : \Omega \to \mathcal {O}_{\eta _0}(T)\}_{\epsilon \in [-1,1]}}$ such that there is a bounded subset $\mathcal K$ of $\mathcal {C}^N([-1,1],\mathcal {C}^{r+1} (\mathbb T^2,\mathbb T^2))$ satisfying that $\epsilon \mapsto \mathcal {T}_{\epsilon }(\omega )$ lies in $\mathcal K$ for ${\mathbb P}$ -almost every $\omega \in \Omega $ . Let $(\mathcal L_{\mathcal T_{\epsilon } },\sigma )$ be the Perron–Probenius cocycle induced by $(\mathcal T_{\epsilon } , \sigma )$ . Then it follows from [Reference Nakano and Wittsten39, §4] that $\mathcal L_{\mathcal T_{\epsilon }}$ almost surely extends to a unique, bounded operator on $H^m(\mathbb T^2)$ such that $\omega \mapsto \mathcal L_{\mathcal T_{\epsilon }}(\omega ) : \Omega \to L(H^m(\mathbb T^2))$ is strongly measurable for each $\epsilon \in [-1,1]$ . The following estimates were proved in [Reference Chung, Nakano and Wittsten12, §§2.3 and 2.4].

We now prove Theorem 4.10. Let $m=r-N$ . For $j \in \{0, \ldots , N\}$ , set ${E_{j} = H^{m+j}(\mathbb T^2)}$ . Then, in the same manner as in the proof of Theorem 4.8, we can apply Theorem 3.6, with the help of Propositions 4.2 and 4.4 and of Corollary 4.6, to deduce the claim that $\epsilon \mapsto \mu ^{\mathcal {T}_{\epsilon }}_{\omega }$ is in $\mathcal {C}^{N-1}([-1,1], E_0)=\mathcal {C}^{N-1}([-1,1], H^{r-N}(\mathbb T^2))$ for almost every $\omega $ , which completes the proof of Theorem 4.10.