# On the linearization of infinite-dimensional random dynamical systems

Lucas Backes  Departamento de Matemática, Universidade Federal do Rio Grande
do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, RS, Brazil.
<EMAIL_ADDRESS>and  Davor Dragičević  Department of Mathematics,
University of Rijeka, Croatia<EMAIL_ADDRESS>
###### Abstract.

We present a new version of the Grobman-Hartman’s linearization theorem for
random dynamics. Our result holds for infinite dimensional systems whose
linear part is not necessarily invertible. In addition, by adding some
restrictions on the non-linear perturbations, we don’t require for the linear
part to be nonuniformly hyperbolic in the sense of Pesin but rather (besides
requiring the existence of stable and unstable directions) allow for the
existence of a third (central) direction on which we don’t prescribe any
behaviour for the dynamics. Moreover, under some additional nonuniform growth
condition, we prove that the conjugacies given by the linearization procedure
are Hölder continuous when restricted to bounded subsets of the space.

###### Key words and phrases:

random dynamical systems; linearization; Hölder continuity

###### 2020 Mathematics Subject Classification:

37H15

Dedicated to the memory of Vlado Dragičević

##  1\. Introduction

The classical Grobman-Hartman theorem represents one of the cornerstones of
the modern dynamical systems theory. It asserts that if $x_{0}$ is a
hyperbolic fixed point of a $C^{1}$-diffeomorphism
$f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$, then $f$ is on a neighborhood of
$x_{0}$ topologically conjugated to $Df(x_{0})$. The original references for
this results are the papers of Grobman [16] and Hartman [17]. This result was
extended to the case of Banach spaces independently by Palis [21] and Pugh
[23], who also simplified the original arguments of Grobman and Hartman. It is
well-known that even if $f$ is $C^{\infty}$, the conjugacy can fail to be even
locally Lipschitz. Indeed, it turns out that in general the conjugacy is only
locally Hölder continuous [24]. The first versions of the Grobman-Hartman
theorem for nonautonomous dynamics were established by Palmer [22] in the case
of continuous time and by Aulbach and Wanner [2] in the case of discrete time.

In the context of random dynamical systems, the first version of the Grobman-
Hartman theorem was proved by Wanner [25]. It represents an important
contribution to the study of the qualitative behaviour of random dynamical
systems which are used as models of many phenomena with some degree of noise
(including stochastic differential equations). For a detailed exposition of
the theory of random dynamical systems we refer to [1]. For certain extensions
of the original approach developed by Wanner, we refer to [8, 11, 12, 26] and
references therein. In [9], Barreira and Valls formulated sufficient
conditions under which the conjugacies in the random version of the Grobman-
Hartman theorem are locally Hölder continuous. The higher regularity of
conjugacies was discussed in the important works of Li and Lu [18] and Lu,
Zhang and Zhang [20].

The goal of the present paper is to make further contributions to the
linearization theory of random dynamical systems. In sharp contrast to the
existing results in the literature, our first main result (see Theorem 1)
gives a random version of the Grobman-Hartman theorem such that the linear
part of our dynamics is not necessarily invertible (although the associated
perturbed dynamics needs to be invertible). In addition, we don’t require for
the linear part to be nonuniformly hyperbolic in the sense of Pesin but rather
(besides requiring the existence of stable and unstable directions) allow for
the existence of a third (central) direction on which we don’t prescribe any
behaviour for the linear part. On the other hand, we stress that our result
doesn’t allow any type of perturbations along this central direction.
Consequently, to the best of our knowledge, our Theorem 1 represents the first
version of the Grobman-Hartman theorem in the setting when the linear part of
the dynamics can have zero Lyapunov exponents (assuming that a suitable
version of the multiplicative ergodic theorem can be applied to it).

In our second main result (see Theorem 2), we formulate sufficient conditions
under which the conjugacies given by Theorem 1 are Hölder continuous. When
compared with the main result in [9], our Theorem 2 is more general and
sharper since for instance, we deal with the infinite-dimensional dynamics
where a multiplicative ergodic theorem is not always available and, moreover,
we don’t require for the linear part to be hyperbolic.

Our techniques build on our recent work with K. Palmer [7] (see also [5]),
where we obtained similar results in the context of nonautonomous dynamics.
However, we stress that the results in the present paper require a nontrivial
adjustments of the ideas from [7] that now need to be combined with certain
ergodic theory tools.

##  2\. Linearization of random dynamical systems

Let $X=(X,|\cdot|)$ be an arbitrary Banach space and let $\mathcal{B}(X)$
denote the space of all bounded linear operators on $X$. The norm on
$\mathcal{B}(X)$ will be denoted by $\lVert\cdot\rVert$. Furthermore, let
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let
$\sigma\colon\Omega\to\Omega$ be an invertible $\mathbb{P}$-preserving
measurable transformation. For the sake of simplicity of the presentation we
will assume that $\mathbb{P}$ is ergodic (see Remark 6 for the discussion
regarding the general case).

We recall that a measurable map $K\colon\Omega\to(0,\infty)$ is said to be
_tempered_ if:

$\lim_{n\to\pm\infty}\frac{1}{n}\log K(\sigma^{n}(\omega))=0,\quad\text{for
$\mathbb{P}$-a.e. $\omega\in\Omega$.}$ (1)

Let $A\colon\Omega\to\mathcal{B}(X)$ be a strongly measurable map. We recall
that this means that the map $\omega\mapsto A(\omega)x$ is measurable for each
$x\in X$. Let
$\mathcal{A}\colon\Omega\times\\{0,1,2,\ldots\\}\to\mathcal{B}(X)$ be the
_linear cocycle_ generated by $A$. We recall that $\mathcal{A}$ is given by

$\mathcal{A}(\omega,n)=\begin{cases}\text{\rm Id}&\text{if $n=0$;}\\\
A(\sigma^{n-1}(\omega))\cdots A(\omega)&\text{if $n>0$,}\end{cases}$

for $\omega\in\Omega$ and $n\in\\{0,1,\ldots\\}$. Here, Id denotes the
identity operator on $X$.

We now introduce some assumptions related to $\mathcal{A}$. Namely, we will
assume the following:

  * •

there exist three families of projections
$(\Pi^{i}(\omega))_{\omega\in\Omega}$, $i\in\\{1,2,3\\}$ on $X$ such that

$\text{\rm Id}=\Pi^{1}(\omega)+\Pi^{2}(\omega)+\Pi^{3}(\omega),\quad\text{for
$\mathbb{P}$-a.e. $\omega\in\Omega$.}$

In addition, for each $i\in\\{1,2,3\\}$, the map $\omega\to\Pi^{i}(\omega)$ is
strongly measurable;

  * •

for $i\in\\{1,2,3\\}$ and $\mathbb{P}$-a.e. $\omega\in\Omega$,

$A(\omega)\Pi^{i}(\omega)=\Pi^{i}(\sigma(\omega))A(\omega);$ (2)

  * •

for $i\in\\{2,3\\}$ and for $\mathbb{P}$-a.e. $\omega\in\Omega$,

$A(\omega)\rvert_{\operatorname{Im}\Pi^{i}(\omega)}\colon\operatorname{Im}\Pi^{i}(\omega)\to\operatorname{Im}\Pi^{i}(\sigma(\omega))\quad\text{is
invertible;}$ (3)

  * •

there exist $\lambda>0$ and a tempered random variable
$K\colon\Omega\to[1,\infty)$ such that for $\mathbb{P}$-a.e. $\omega\in\Omega$
and $n\geq 0$

$\lVert\mathcal{A}(\omega,n)\Pi^{1}(\omega)\rVert\leq K(\omega)e^{-\lambda n}$
(4)

and

$\lVert\mathcal{A}(\omega,-n)\Pi^{3}(\omega)\rVert\leq K(\omega)e^{-\lambda
n},$ (5)

where

$\mathcal{A}(\omega,-n):=\bigg{(}\mathcal{A}(\sigma^{-n}(\omega),n)\rvert_{\operatorname{Im}\Pi^{3}(\sigma^{-n}(\omega))}\bigg{)}^{-1}\colon\operatorname{Im}\Pi^{3}(\omega)\to\operatorname{Im}\Pi^{3}(\sigma^{-n}(\omega)).$

######  Remark 1.

We would like to point out that the above assumptions are natural in the
context of the ergodic theory. Let $X=\mathbb{R}^{d}$ and assume that

$\int_{\Omega}\log^{+}\|A(\omega)\|\,d\mathbb{P}(\omega)<+\infty.$

It follows from the version of the multplicative ergodic theorem (MET)
established in [14] that there exist numbers

$-\infty\leq\lambda_{1}<\lambda_{2}<\ldots<\lambda_{k}$

and for $\mathbb{P}$-a.e. $\omega\in\Omega$ a measurable decomposition

$X=E_{1}(\omega)\oplus E_{2}(\omega)\oplus\ldots\oplus E_{k}(\omega)$

such that

$A(\omega)E_{i}(\omega)\subset E_{i}(\sigma(\omega))\text{(with equality if
$\lambda_{i}>-\infty$)}$

and

$\lim_{n\to\infty}\frac{1}{n}\log|\mathcal{A}(\omega,n)v|=\lambda_{i},\quad\text{for
$v\in E_{i}(\omega)$, $v\neq 0$ and $i\in\\{1,\ldots,k\\}$.}$

Set

$E^{s}(\omega):=\bigoplus_{i;\lambda_{i}<0}E_{i}(\omega),\quad
E^{u}:=\bigoplus_{i;\lambda_{i}>0}E_{i}(\omega),$

and in the case when $\lambda_{j}=0$ for some $j$, let
$E^{c}(\omega):=E_{j}(\omega)$ (if such $j$ doesn’t exist, set
$E^{c}(\omega)=\\{0\\}$).

$E^{s}(\omega)\oplus E^{c}(\omega)\oplus E^{u}(\omega)=X,\quad\text{for
$\mathbb{P}$-a.e. $\omega\in\Omega$.}$ (6)

Let $\Pi^{1}(\omega)\colon X\to E^{s}(\omega)$, $\Pi^{2}(\omega)\colon X\to
E^{c}(\omega)$ and $\Pi^{3}(\omega)\colon X\to E^{u}(\omega)$ be projections
associated to (6). Clearly, (2) and (3) holds. Finally, it follows from [13,
Theorem 2] that there exist $\lambda>0$ and a tempered random variable
$K\colon\Omega\to[1,+\infty)$ such that (4) and (5) holds.

We remark that under some additional assumptions an analogous statement can be
formulated in the infinite-dimensional setting (by using [4, Proposition
3.7.]) relying on the infinite-dimensional versions of MET (see [10, 15, 19]
and references therein).

Furthermore, we assume that $(f_{\omega})_{\omega\in\Omega}$ and
$(g_{\omega})_{\omega\in\Omega}$ are two families of continuous maps
$f_{\omega}\colon X\to X$, $g_{\omega}\colon X\to X$ such that:

  * •

for every $x\in X$ and $\mathbb{P}$-a.e. $\omega\in\Omega$,
$f_{\omega}(x),g_{\omega}(x)\in
E^{s,u}(\omega):=\operatorname{Ker}\Pi^{2}(\omega)$;

  * •

for every $x\in X$, $\omega\mapsto f_{\omega}(x)$ and $\omega\mapsto
g_{\omega}(x)$ are measurable;

  * •

there exists a tempered measurable map $D\colon\Omega\to[1,\infty)$ such that

$\max\\{||f_{\omega}||_{\infty},||g_{\omega}||_{\infty}\\}\leq D(\omega),$ (7)

where for any $h\colon X\to X$, we set

$\lVert h\rVert_{\infty}:=\sup_{x\in X}|h(x)|;$

  * •

there exist $c>0$ such that for $\mathbb{P}$-a.e. $\omega\in\Omega$ and
$x,y\in X$,

$|f_{\omega}(x)-f_{\omega}(y)|\leq\frac{c}{K(\sigma(\omega))}|x-y|$ (8)

and

$|g_{\omega}(x)-g_{\omega}(y)|\leq\frac{c}{K(\sigma(\omega))}|x-y|;$ (9)

  * •

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and $y\in
E^{c}(\omega):=\operatorname{Im}\Pi^{2}(\omega)$,

$x\mapsto A(\omega)x+f_{\omega}(x+y)$ (10)

and

$x\mapsto A(\omega)x+g_{\omega}(x+y)$

are invertible maps from $E^{s,u}(\omega)$ onto $E^{s,u}(\sigma(\omega))$.

######  Remark 2.

We observe that if $A(\omega)$ is invertible and
$\|A(\omega)^{-1}\|\cdot\text{Lip}(f_{\omega})<1$ where
$\text{Lip}(f_{\omega})$ stands for the Lipschitz constant of $f_{\omega}$,
then $A(\omega)+f_{\omega}$ is a homeomorphism. Consequently, if $A(\omega)$
is invertible for $\mathbb{P}$-a.e. $\omega\in\Omega$,
$a=\operatorname{esssup}_{\omega\in\Omega}\|A(\omega)^{-1}\|<+\infty$ and
$ac<1$, then $A(\omega)+f_{\omega}$ is a homeomorphism for $\mathbb{P}$-a.e
$\omega\in\Omega$ and, in particular, the last hypothesis above is satisfied
(see [3, p. 433]).

######  Remark 3.

  1. 1.

In the case when $K$ is a constant random variable, we note that the
requirements (8) and (9) mean that $f_{\omega}$ and $g_{\omega}$ are Lipschitz
with a uniform (independent on $\omega$) Lipschitz constant. We stress that
even in this setting, our Theorems 1 and 2 are new results.

  2. 2.

Let us now describe a procedure for the construction of maps $f_{\omega}$
satisfying (8) in the general case. The discussion is essentially taken from
[6, Section 5.2] but we include for the sake of completeness. Take an
arbitrary $\rho>0$. By [1, Proposition 4.3.3], there exists a random variable
$D\colon\Omega\to(0,+\infty)$ such that

$K(\omega)\leq D(\omega)\quad\text{and}\quad D(\sigma^{n}(\omega))\leq
D(\omega)e^{\rho|n|},$ (11)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and $n\in\mathbb{Z}$. Choose $T>0$
sufficiently large so that $\mathbb{P}(A)>0$, where

$A:=\\{\omega\in\Omega:D(\omega)\leq T\\}.$

For $n\in\mathbb{N}$, set

$A_{n}:=\sigma^{-n}(A)\setminus\cup_{k=0}^{n-1}\sigma^{-k}(A).$

Letting $A_{0}=A$, we observe that $A_{n}\cap A_{m}=\emptyset$ for $n\neq m$.
Moreover, due to the ergodicity of $(\Omega,\mathcal{F},\mathbb{P},\sigma)$,
we have that $\mathbb{P}(\cup_{n=0}^{\infty}A_{n})=1$. For $n\geq 0$ and
$\omega\in A_{n}$, let $f_{\omega}$ be such that

$|f_{\omega}(x)-f_{\omega}(y)|\leq\frac{c}{T}e^{-\rho|n-1|}|x-y|,\quad\text{for
$x,y\in X$.}$ (12)

We now claim that (8) holds. Take $\omega\in A_{n}$ for $n\geq 1$. Choose
$\omega^{\prime}\in A$ such that $\omega=\sigma^{-n}(\omega^{\prime})$. By
(11), we have that

$K(\sigma(\omega))\leq
D(\sigma(\omega))=D(\sigma^{-(n-1)}(\omega^{\prime}))\leq
e^{\rho|n-1|}D(\omega^{\prime})\leq Te^{\rho|n-1|},$

and thus

$\frac{c}{T}e^{-\rho|n-1|}\leq\frac{c}{K(\sigma(\omega))}.$

We conclude that (8) follows from (12). One can argue in the same manner for
$\omega\in A$.

######  Remark 4.

We note that the condition in the spirit of (8) appeared in some of the
earlier versions of the Grobman-Hartman theorem for random dynamical systems
(see for example [8, eq.(21)]). Furthermore, it is analogous to the condition
(see [27, eq.(2.2)]) imposed in the roughness theorem for tempered exponential
dichotomies established in [27].

We are now in a position to formulate our first result.

######  Theorem 1.

For $c>0$ sufficiently small, there exists a family of homeomorphisms
$(H_{\omega})_{\omega\in\Omega}$, $H_{\omega}\colon X\to X$ such that:

  * •

for $\mathbb{P}$-a.e. $\omega\in\Omega$,

$H_{\sigma(\omega)}\circ(A(\omega)+f_{\omega})=(A(\omega)+g_{\omega})\circ
H_{\omega};$ (13)

  * •

for $x\in X$, $\omega\mapsto H_{\omega}(x)$ is measurable;

  * •

there exists a tempered random variable $T\colon\Omega\to(0,\infty)$ such that

$\lVert H_{\omega}-\text{\rm Id}\rVert_{\infty}\leq T(\omega),\quad\text{for
$\mathbb{P}$-a.e. $\omega\in\Omega$;}$

  * •

$H_{\omega}(x)-x\in E^{s,u}(\omega)$ for $\mathbb{P}$-a.e. $\omega\in\Omega$
and every $x\in X$.

Moreover, the family $(H_{\omega})_{\omega\in\Omega}$ satisfying the previous
properties is essentially unique. More precisely, if
$(H_{\omega}^{\prime})_{\omega\in\Omega}$ is another family that satisfies
these properties, then $H_{\omega}=H_{\omega}^{\prime}$ for $\mathbb{P}$-a.e.
$\omega\in\Omega$.

###### Proof.

For $\omega\in\Omega$, we consider two systems:

$\begin{split}x(n+1)&=A(\sigma^{n}(\omega))x(n)+f_{\sigma^{n}(\omega)}(x(n)+y(n))\\\
y(n+1)&=A(\sigma^{n}(\omega))y(n),\end{split}$ (14)

and

$\begin{split}x(n+1)&=A(\sigma^{n}(\omega))x(n)+g_{\sigma^{n}(\omega)}(x(n)+y(n))\\\
y(n+1)&=A(\sigma^{n}(\omega))y(n).\end{split}$ (15)

For $(\xi,\eta)\in E^{s,u}(\omega)\times E^{c}(\omega)$, let
$n\mapsto(x_{1}(n,\omega,\xi,\eta),y(n,\omega,\eta))$ denote the solution of
(14) such that $(x_{1}(0,\omega,\xi,\eta),y(0,\omega,\eta))=(\xi,\eta)$.
Similarly, let $n\mapsto(x_{2}(n,\omega,\xi,\eta),y(n,\omega,\eta))$ denote
the solution of (15) satisfying
$(x_{2}(0,\omega,\xi,\eta),y(0,\omega,\eta))=(\xi,\eta)$.

It follows from our assumptions and [1, Proposition 4.3.3] that there exists a
tempered random variable $C\colon\Omega\to(0,\infty)$ such that for
$\mathbb{P}$-a.e. $\omega\in\Omega$ and $n\in\mathbb{Z}$,

$K(\sigma(\omega))D(\omega)\leq C(\omega)\quad\text{and}\quad
C(\sigma^{n}(\omega))\leq C(\omega)e^{\frac{\lambda}{2}|n|}.$ (16)

Let $Y_{\infty}$ denote the space of all maps $h\colon\Omega\times X\to X$
with the property that:

  * •

for $\mathbb{P}$-a.e. $\omega\in\Omega$, $h(\omega,\cdot)\colon X\to X$ is
continuous;

  * •

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and $x\in X$, $h(\omega,x)\in
E^{s,u}(\omega)$;

  * •

for $x\in X$, $\omega\mapsto h(\omega,x)$ is measurable;

  * •

$||h||_{\infty}^{\prime}:=\operatorname{esssup}_{\omega\in\Omega}(C(\omega)^{-1}||h(\omega,\cdot)||_{\infty})<\infty.$

It is straightforward to verify that
$(Y_{\infty},||\cdot||_{\infty}^{\prime})$ is a Banach space. For
$\omega\in\Omega$ and $n\in\mathbb{Z}$, set

$\mathcal{G}(\omega,n):=\begin{cases}\mathcal{A}(\omega,n)\Pi^{1}(\omega)&\text{for
$n\geq 0$;}\\\ -\mathcal{A}(\omega,n)\Pi^{3}(\omega)&\text{for
$n<0$.}\end{cases}$

For $h\in Y_{\infty}$, let

$(\mathcal{T}h)(\omega,x):=\sum_{n\in\mathbb{Z}}\mathcal{G}(\sigma^{-n}(\omega),n)p(n,\omega,x),$

where

$\begin{split}p(n,\omega,x)&=g_{\sigma^{-(n+1)}(\omega)}(x_{1}(-(n+1),\omega,\xi,\eta)+y(-(n+1),\omega,\eta)\\\
&\phantom{=}+h(\sigma^{-(n+1)}(\omega),x_{1}(-(n+1),\omega,\xi,\eta)+y(-(n+1),\omega,\eta)))\\\
&\phantom{=}-f_{\sigma^{-(n+1)}(\omega)}(x_{1}(-(n+1),\omega,\xi,\eta)+y(-(n+1),\omega,\eta)),\end{split}$
(17)

for $n\in\mathbb{Z}$, $\omega\in\Omega$ and $x\in X$, where $\xi=(\text{\rm
Id}-\Pi^{2}(\omega))x$, $\eta=x-\xi$.

By (4), (5),(7) and (16), we have that

$\begin{split}|(\mathcal{T}h)(\omega,x)|&\leq\sum_{n\in\mathbb{Z}}\lVert\mathcal{G}(\sigma^{-n}(\omega),n)\rVert\cdot(\lVert
g_{\sigma^{-(n+1)}(\omega)}\rVert_{\infty}+\lVert
f_{\sigma^{-(n+1)}(\omega)}\rVert_{\infty})\\\ &\leq
2\sum_{n=0}^{\infty}K(\sigma^{-n}(\omega))e^{-\lambda
n}D(\sigma^{-(n+1)}(\omega))\\\
&\phantom{\leq}+2\sum_{n=1}^{\infty}K(\sigma^{n}(\omega))e^{-\lambda
n}D(\sigma^{n-1}(\omega))\\\ &\leq 2\sum_{n=0}^{\infty}e^{-\lambda
n}C(\sigma^{-(n+1)}(\omega))+2\sum_{n=1}^{\infty}e^{-\lambda
n}C(\sigma^{n-1}(\omega))\\\ &\leq
2e^{\frac{\lambda}{2}}C(\omega)\bigg{(}\sum_{n=0}^{\infty}e^{-\frac{\lambda}{2}n}+\sum_{n=1}^{\infty}e^{-\frac{\lambda}{2}n}\bigg{)}\\\
&\leq
2e^{\frac{\lambda}{2}}\frac{1+e^{-\frac{\lambda}{2}}}{1-e^{-\frac{\lambda}{2}}}C(\omega),\end{split}$
(18)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and $x\in X$. Hence,
$||\mathcal{T}h||_{\infty}^{\prime}<\infty$.

We now claim that for $\mathbb{P}$-a.e. $\omega\in\Omega$,
$x\mapsto\mathcal{T}h(\omega,x)$ is a continuous map. We begin by observing
that due to the continuity of maps $f_{\omega}$, we have that

$(\xi,\eta)\mapsto(x_{1}(n,\omega,\xi,\eta),y(n,\omega,\eta))\quad\text{is
continuous,}$ (19)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and $n\in\mathbb{Z}$. Fix now an
arbitrary $\omega\in\Omega$ for which (4), (5), (16) and (19) hold, and such
that $h(\sigma^{n}\omega,\cdot)$ is continuous for each $n\in\mathbb{Z}$. We
note that the set of $\omega$’s satisfying these properties is a full measure
set. Take $x\in X$ and $\varepsilon>0$. Choose $N>0$ such that

$\sum_{|n|>N}\lVert\mathcal{G}(\sigma^{-n}(\omega),n)\rVert\cdot(\lVert
g_{\sigma^{-(n+1)}(\omega)}\rVert_{\infty}+\lVert
f_{\sigma^{-(n+1)}(\omega)}\rVert_{\infty})<\frac{\varepsilon}{3}.$

We note that the existence of $N$ follows from the estimates in (18).
Furthemore, by using (19) together with the continuity of
$f_{\sigma^{n}(\omega)}$, $g_{\sigma^{n}(\omega)}$ and
$h(\sigma^{n}(\omega),\cdot)$, we conclude that

$\sum_{|n|\leq
N}\|\mathcal{G}(\sigma^{-n}(\omega),n)\rVert\cdot|p(n,\omega,x)-p(n,\omega,x^{\prime})|<\frac{\varepsilon}{3},$

whenever $|x-x^{\prime}|$ is sufficiently small. The last two estimates easily
imply that

$|(\mathcal{T}h)(\omega,x)-(\mathcal{T}h)(\omega,x^{\prime})|<\varepsilon,$

whenever $|x-x^{\prime}|$ is sufficiently small. Hence,
$x\mapsto\mathcal{T}h(\omega,x)$ is a continuous map and thus $\mathcal{T}h\in
Y_{\infty}$.

Moreover, for $h_{1},h_{2}\in Y_{\infty}$, we have using (9) that

$\begin{split}&|(\mathcal{T}h_{1})(\omega,x)-(\mathcal{T}h_{2})(\omega,x)|\\\
&\leq\sum_{n\in\mathbb{Z}}\frac{c}{K(\sigma^{-n}(\omega))}\lVert\mathcal{G}(\sigma^{-n}(\omega),n)\rVert\cdot\lVert
h_{1}(\sigma^{-(n+1)}(\omega),\cdot)-h_{2}(\sigma^{-(n+1)}(\omega),\cdot)\rVert_{\infty}\\\
&\leq c\sum_{n=0}^{\infty}e^{-\lambda n}C(\sigma^{-(n+1)}(\omega))\lVert
h_{1}-h_{2}\rVert_{\infty}^{\prime}+c\sum_{n=1}^{\infty}e^{-\lambda
n}C(\sigma^{n-1}(\omega))\lVert h_{1}-h_{2}\rVert_{\infty}^{\prime}\\\ &\leq
ce^{\frac{\lambda}{2}}\frac{1+e^{-\frac{\lambda}{2}}}{1-e^{-\frac{\lambda}{2}}}C(\omega)\lVert
h_{1}-h_{2}\rVert_{\infty}^{\prime},\end{split}$

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and $x\in X$. Therefore,

$\lVert\mathcal{T}h_{1}-\mathcal{T}h_{2}\rVert_{\infty}^{\prime}\leq
ce^{\frac{\lambda}{2}}\frac{1+e^{-\frac{\lambda}{2}}}{1-e^{-\frac{\lambda}{2}}}\lVert
h_{1}-h_{2}\rVert_{\infty}^{\prime},$

for $h_{1},h_{2}\in Y_{\infty}$. If $c>0$ is such that

$ce^{\frac{\lambda}{2}}\frac{1+e^{-\frac{\lambda}{2}}}{1-e^{-\frac{\lambda}{2}}}<1,$

we conclude that $h\mapsto\mathcal{T}h$ is a contraction on
$(Y_{\infty},\|\cdot\|_{\infty}^{\prime})$. Hence, $h\mapsto\mathcal{T}h$ has
a unique fixed point $h\in Y_{\infty}$. Thus,

$h(\omega,x)=\sum_{n\in\mathbb{Z}}\mathcal{G}(\sigma^{-n}(\omega),n)p(n,\omega,x),$

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and $x\in X$. Hence, we have that

$\begin{split}&h(\sigma(\omega),x_{1}(1,\omega,\xi,\eta)+y(1,\omega,\eta))\\\
&=\sum_{n\in\mathbb{Z}}\mathcal{G}(\sigma^{-(n-1)}(\omega),n)\bigg{(}g_{\sigma^{-n}(\omega)}(x_{1}(-n,\omega,\xi,\eta)+y(-n,\omega,\eta)\\\
&\phantom{=}+h(\sigma^{-n}(\omega),x_{1}(-n,\omega,\xi,\eta)+y(-n,\omega,\eta)))\bigg{)}\\\
&\phantom{=}-\sum_{n\in\mathbb{Z}}\mathcal{G}(\sigma^{-(n-1)}(\omega),n)f_{\sigma^{-n}(\omega)}(x_{1}(-n,\omega,\xi,\eta)+y(-n,\omega,\eta))\\\
&=\Pi^{1}(\sigma(\omega))g_{\omega}(\xi+\eta+h(\omega,\xi+\eta))\\\
&\phantom{=}+A(\omega)\sum_{n=0}^{\infty}\mathcal{G}(\sigma^{-n}(\omega),n)\bigg{(}g_{\sigma^{-(n+1)}(\omega)}(x_{1}(-(n+1),\omega,\xi,\eta)\\\
&\phantom{=}+y(-(n+1),\omega,\eta)+h(\sigma^{-(n+1)}(\omega),x_{1}(-(n+1),\omega,\xi,\eta)+y(-(n+1),\omega,\eta)))\bigg{)}\\\
&\phantom{=}-\Pi^{1}(\sigma(\omega))f_{\omega}(\xi+\eta)\\\
&\phantom{=}-A(\omega)\sum_{n=0}^{\infty}\mathcal{G}(\sigma^{-n}(\omega),n)f_{\sigma^{-(n+1)}(\omega)}(x_{1}(-(n+1),\omega,\xi,\eta)+y(-(n+1),\omega,\eta))\\\
&\phantom{=}+\Pi^{3}(\sigma(\omega))g_{\omega}(\xi+\eta+h(\omega,\xi+\eta))\\\
&\phantom{=}+A(\omega)\sum_{n=-\infty}^{-1}\mathcal{G}(\sigma^{-n}(\omega),n)\bigg{(}g_{\sigma^{-(n+1)}(\omega)}(x_{1}(-(n+1),\omega,\xi,\eta)\\\
&\phantom{=}+y(-(n+1),\omega,\eta)+h(\sigma^{-(n+1)}(\omega),x_{1}(-(n+1),\omega,\xi,\eta)+y(-(n+1),\omega,\eta)))\bigg{)}\\\
&\phantom{=}-\Pi^{3}(\sigma(\omega))f_{\omega}(\xi+\eta)\\\
&\phantom{=}-A(\omega)\sum_{n=-\infty}^{-1}\mathcal{G}(\sigma^{-n}(\omega),n)f_{\sigma^{-(n+1)}(\omega)}(x_{1}(-(n+1),\omega,\xi,\eta)+y(-(n+1),\omega,\eta))\\\
&=A(\omega)h(\omega,x)+g_{\omega}(\xi+\eta+h(\omega,\xi+\eta))-f_{\omega}(\xi+\eta),\end{split}$

and thus

$h(\sigma(\omega),A(\omega)x+f_{\omega}(x))=A(\omega)h(\omega,x)+g_{\omega}(x+h(\omega,x))-f_{\omega}(x),$
(20)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and $x\in X$.

For $\omega\in\Omega$, we define $H_{\omega}\colon X\to X$ by

$H_{\omega}(x)=x+h(\omega,x),\quad x\in X.$

By (20), we have that (13) holds.

In analogous manner, for $\bar{h}\in Y_{\infty}$ we define
$\mathcal{T}^{\prime}\bar{h}$ by

$(\mathcal{T}^{\prime}\bar{h})(\omega,x)=\sum_{n\in\mathbb{Z}}\mathcal{G}(\sigma^{-n}(\omega),n)r(n,\omega,x),$

where

$\begin{split}r(n,\omega,x)&=f_{\sigma^{-(n+1)}(\omega)}(x_{2}(-(n+1),\omega,\xi,\eta)+y(-(n+1),\omega,\eta)\\\
&\phantom{=}+\bar{h}(\sigma^{-(n+1)}(\omega),x_{2}(-(n+1),\omega,\xi,\eta)))\\\
&\phantom{=}-g_{\sigma^{-(n+1)}(\omega)}(x_{2}(-(n+1),\omega,\xi,\eta)),\end{split}$

for $n\in\mathbb{Z}$, $\omega\in\Omega$ and $x\in X$, where $\xi=(\text{\rm
Id}-\Pi^{2}(\omega))x$, $\eta=x-\xi$. By arguing in a same manner as for
$\mathcal{T}$, we have that (for $c>0$ sufficiently small)
$\bar{h}\mapsto\mathcal{T}^{\prime}\bar{h}$ is a contraction and thus it has a
unique fixed point $\bar{h}\in Y_{\infty}$. For $\omega\in\Omega$, we define
$\bar{H}_{\omega}\colon X\to X$ by

$\bar{H}_{\omega}(x)=x+\bar{h}(\omega,x),\quad x\in X.$

Then, we have that

$\bar{H}_{\sigma(\omega)}\circ(A(\omega)+g_{\omega})=(A(\omega)+f_{\omega})\circ\bar{H}_{\omega}.$
(21)

In order to complete the proof of our theorem we need the following auxiliary
result.

######  Lemma 1.

Let $(Q_{\omega})_{\omega\in\Omega}$ be a family of continuous maps
$Q_{\omega}\colon X\to X$ such that:

  * •

for $\mathbb{P}$-a.e. $\omega\in\Omega$,

$Q_{\sigma(\omega)}\circ(A(\omega)+f_{\omega})=(A(\omega)+g_{\omega})\circ
Q_{\omega};$ (22)

  * •

for $x\in X$, $\omega\mapsto Q_{\omega}(x)$ is measurable;

  * •

there exists a tempered random variable $T\colon\Omega\to(0,\infty)$ such that

$\lVert Q_{\omega}-\text{\rm Id}\rVert_{\infty}\leq T(\omega),\quad\text{for
$\mathbb{P}$-a.e. $\omega\in\Omega$;}$ (23)

  * •

$Q_{\omega}(x)-x\in E^{s,u}(\omega)$ for $\mathbb{P}$-a.e. $\omega\in\Omega$
and every $x\in X$.

Then, $q(\omega,x):=Q_{\omega}(x)-x$ is a fixed point of $\mathcal{T}$. In
particular, the family of maps $(Q_{\omega})_{\omega\in\Omega}$ with the above
properties is essentially unique.

###### Proof of the lemma.

Given $\omega\in\Omega$ and $x\in X$, let us consider $\xi=(\text{\rm
Id}-\Pi^{2}(\omega))x$, $\eta=x-\xi$ and
$(x_{1}(n,\omega,\xi,\eta),y(n,\omega,\eta))_{n\in\mathbb{Z}}$ as before. We
start observing that condition (22) implies that

$\begin{split}q(\sigma(\omega),A(\omega)x+f_{\omega}(x))&=Q_{\sigma(\omega)}(A(\omega)x+f_{\omega}(x))-\left(A(\omega)x+f_{\omega}(x)\right)\\\
&=\left(A(\omega)+g_{\omega}\right)(Q_{\omega}(x))-A(\omega)x-f_{\omega}(x)\\\
&=A(\omega)q(\omega,x)+g_{\omega}(x+q(\omega,x))-f_{\omega}(x).\end{split}$

Thus, recalling that
$A(\omega)x+f_{\omega}(x)=x_{1}(1,\omega,\xi,\eta)+y(1,\omega,\eta)$ we can
rewrite the previous equality as

$\begin{split}&q(\sigma(\omega),x_{1}(1,\omega,\xi,\eta)+y(1,\omega,\eta))\\\
&=A(\omega)q(\omega,x_{1}(0,\omega,\xi,\eta)+y(0,\omega,\eta))\\\
&+g_{\omega}(x_{1}(0,\omega,\xi,\eta)+y(0,\omega,\eta)+q(\omega,x_{1}(0,\omega,\xi,\eta)+y(0,\omega,\eta)))\\\
&-f_{\omega}(x_{1}(0,\omega,\xi,\eta)+y(0,\omega,\eta)),\end{split}$

which implies that

$\begin{split}&q(\omega,x)\\\
&=A(\sigma^{-1}(\omega))q(\sigma^{-1}(\omega),x_{1}(-1,\omega,\xi,\eta)+y(-1,\omega,\eta))\\\
&+g_{\sigma^{-1}(\omega)}(x_{1}(-1,\omega,\xi,\eta)+y(-1,\omega,\eta)+q(\sigma^{-1}(\omega),x_{1}(-1,\omega,\xi,\eta)+y(-1,\omega,\eta)))\\\
&-f_{\sigma^{-1}(\omega)}(x_{1}(-1,\omega,\xi,\eta)+y(-1,\omega,\eta)).\end{split}$

Iterating this formula we conclude that for every $k>0$,

$\begin{split}q(\omega,x)&=\mathcal{A}(\sigma^{-k}(\omega),k)q(\sigma^{-k}(\omega),x_{1}(-k,\omega,\xi,\eta)+y(-k,\omega,\eta))\\\
&+\sum_{n=0}^{k-1}\mathcal{A}(\sigma^{-n}(\omega),n)p(n,\omega,x)\\\
\end{split}$

where $p(n,\omega,x)$ is as defined in (17) with $h=q$. Thus, using property
(2) we conclude that

$\begin{split}\Pi^{1}(\omega)q(\omega,x)&=\mathcal{A}(\sigma^{-k}(\omega),k)\Pi^{1}(\sigma^{-k}(\omega))q(\sigma^{-k}(\omega),x_{1}(-k,\omega,\xi,\eta)+y(-k,\omega,\eta))\\\
&+\sum_{n=0}^{k-1}\mathcal{A}(\sigma^{-n}(\omega),n)\Pi^{1}(\sigma^{-n}(\omega))p(n,\omega,x)\\\
&=\mathcal{G}(\sigma^{-k}(\omega),k)q(\sigma^{-k}(\omega),x_{1}(-k,\omega,\xi,\eta)+y(-k,\omega,\eta))\\\
&+\sum_{n=0}^{k-1}\mathcal{G}(\sigma^{-n}(\omega),n)p(n,\omega,x)\\\
\end{split}$

for every $k>0$. It follows from (23) that for $\mathbb{P}$-a.e.
$\omega\in\Omega$ and every $k>0$,

$|q(\sigma^{-k}(\omega),x_{1}(-k,\omega,\xi,\eta)+y(-k,\omega,\eta))|\leq
T(\sigma^{-k}(\omega)).$

Combining this observation with (1) and (4) we conclude that

$|\mathcal{G}(\sigma^{-k}(\omega),k)q(\sigma^{-k}(\omega),x_{1}(-k,\omega,\xi,\eta)+y(-k,\omega,\eta)|\xrightarrow{k\to+\infty}0$

for $\mathbb{P}$-a.e. $\omega\in\Omega$. Consequently,

$\begin{split}\Pi^{1}(\omega)q(\omega,x)&=\sum_{n=0}^{+\infty}\mathcal{G}(\sigma^{-n}(\omega),n)p(n,\omega,x).\\\
\end{split}$

Similarly, one can prove that

$\begin{split}\Pi^{3}(\omega)q(\omega,x)&=\sum_{n=-\infty}^{-1}\mathcal{G}(\sigma^{-n}(\omega),n)p(n,\omega,x).\\\
\end{split}$

Therefore, since $q(\omega,x)=Q_{\omega}(x)-x\in E^{s,u}(\omega)$ for
$\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x\in X$, we conclude that

$\begin{split}q(\omega,x)&=\Pi^{1}(\omega)q(\omega,x)+\Pi^{3}(\omega)q(\omega,x)\\\
&=\sum_{n\in\mathbb{Z}}\mathcal{G}(\sigma^{-n}(\omega),n)p(n,\omega,x)\\\
&=(\mathcal{T}q)(\omega,x),\end{split}$

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x\in X$. Therefore, $q$ is
a fixed point for $\mathcal{T}$ as claimed. The uniqueness property follows
easily from previous observations and from the fact that $\mathcal{T}$ is a
contraction. The proof of the lemma is completed. ∎

Our objective now is to prove that $H_{\omega}\circ\bar{H}_{\omega}=\text{\rm
Id}$ and $\bar{H}_{\omega}\circ H_{\omega}=\text{\rm Id}$. We begin by
observing that (13) and (21) readily imply that

$\bar{H}_{\sigma(\omega)}\circ
H_{\sigma(\omega)}\circ\left(A(\omega)+f_{\omega}\right)=\left(A(\omega)+f_{\omega}\right)\circ\bar{H}_{\omega}\circ
H_{\omega},$

for $\mathbb{P}$-a.e $\omega\in\Omega$. Moreover, note that

$(\bar{H}_{\omega}\circ H_{\omega})(x)-x=(\bar{H}_{\omega}\circ
H_{\omega})(x)-H_{\omega}(x)+H_{\omega}(x)-x,$

and thus by using the properties of $\bar{H}$ and $H$ we conclude that
$\bar{H}_{\omega}\circ H_{\omega}(x)-x\in E^{s,u}(\omega)$ for
$\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x\in X$. Moreover, there exists
a tempered random variable $\tilde{T}\colon\Omega\to(0,\infty)$ such that

$\lVert\bar{H}_{\omega}\circ H_{\omega}-\text{\rm
Id}\rVert_{\infty}\leq\tilde{T}(\omega),\quad\text{for $\mathbb{P}$-a.e.
$\omega\in\Omega$.}$

Finally, since $H_{\omega}$ and $\bar{H}_{\omega}$ are strongly measurable we
have that $\omega\to(\bar{H}_{\omega}\circ H_{\omega})(x)$ is measurable for
every $x\in X$. Thus, it follows from the uniqueness given in Lemma 1 (applied
to the case when $g_{\omega}=f_{\omega}$) that $\bar{H}_{\omega}\circ
H_{\omega}=\text{\rm Id}$ for $\mathbb{P}$-a.e. $\omega\in\Omega$. By changing
the roles of $f_{\omega}$ and $g_{\omega}$ and $H_{\omega}$ and
$\bar{H}_{\omega}$ we also conclude that
$H_{\omega}\circ\bar{H}_{\omega}=\text{\rm Id}$, for $\mathbb{P}$-a.e.
$\omega\in\Omega$. In particular, $H_{\omega}$ is a homeomorphism for
$\mathbb{P}$-a.e. $\omega\in\Omega$. The proof of the theorem is completed. ∎

##  3\. Hölder linearization of random dynamical systems

In this section we show that under some additional assumptions the conjugacies
obtained in Theorem 1 are Hölder continuous. We retain all the notation and
the assumptions introduced in the previous section. Moreover, we assume that
$A(\omega)$ is invertible for $\mathbb{P}$-a.e $\omega\in\Omega$. In this
case, for $\mathbb{P}$-a.e $\omega\in\Omega$, we consider

$\mathcal{A}(\omega,n)=\begin{cases}A(\sigma^{n-1}(\omega))\cdots
A(\omega)&\text{if $n>0$,}\\\ \text{\rm Id}&\text{if $n=0$;}\\\
A(\sigma^{-|n|}(\omega))^{-1}\cdots A(\sigma^{-1}(\omega))^{-1}&\text{if
$n<0$.}\end{cases}$

Finally, suppose that there exists $\rho>0$ and a tempered random variable
$Z\colon\Omega\to[1,+\infty)$ such that

$\|\mathcal{A}(\omega,n)\|\leq Z(\omega)e^{\rho|n|},$ (24)

for $\mathbb{P}\text{-a.e. }\omega\in\Omega\text{ and }n\in\mathbb{Z}$. We may
assume without loss of generality that $\rho\geq\lambda$ (where $\lambda>0$ is
given in (4) and (5)) and that

$K(\omega)\leq Z(\omega),\quad\text{for $\mathbb{P}$-a.e. $\omega\in\Omega$.}$
(25)

Set $\alpha_{0}:=\lambda/\rho\in(0,1]$.

######  Remark 5.

Observe that condition (24) is satisfied, for instance, whenever

$\int\log\|A(\omega)\|d\mathbb{P}<+\infty\text{ and
}\int\log\|A(\omega)^{-1}\|d\mathbb{P}<+\infty.$

Observe also that it follows from (4) and (5) that

$\|\Pi^{2}(\omega)\|\leq M(\omega)\text{ for }\mathbb{P}\text{-a.e.
}\omega\in\Omega,$ (26)

where

$M(\omega)=1+2K(\omega),\quad\omega\in\Omega.$

######  Theorem 2.

Let $\alpha\in(0,\alpha_{0})$. There exists a tempered random variable
$d:\Omega\to(0,+\infty)$ such that, if

$|f_{\omega}(x)-f_{\omega}(z)|\leq d(\omega)|x-z|$ (27)

and

$|g_{\omega}(x)-g_{\omega}(z)|\leq d(\omega)|x-z|$ (28)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$, then the
conjugacies $H_{\omega}$ and the associated inverses $H_{\omega}^{-1}$ given
by Theorem 1 are $\alpha$-Hölder continuous when restricted to bounded subsets
of $X$ for $\mathbb{P}$-a.e $\omega\in\Omega$. More precisely, given a bounded
subset $\tilde{X}\subset X$, there exists a constant $T>0$ depending only on
$\tilde{X}$ such that

$|H_{\omega}(x)-H_{\omega}(z)|\leq T|x-z|^{\alpha}$

and

$|H_{\omega}^{-1}(x)-H_{\omega}^{-1}(z)|\leq T|x-z|^{\alpha}$

for $\mathbb{P}$-a.e $\omega\in\Omega$ and every $x,z\in\tilde{X}$.

###### Proof.

Let $c\in(0,1)$ be a sufficiently small constant as in (8) and (9) so that
Theorem 1 holds. We are going to build the desired map
$d:\Omega\to(0,+\infty)$ with the property that

$d(\omega)\leq\frac{c}{Z(\sigma(\omega))}$ (29)

for $\mathbb{P}$-a.e. $\omega\in\Omega$. In particular, due to (25), whenever
(27) and (28) are satisfied the same is true for (8) and (9).

Let $\varepsilon>0$ be such that $\alpha\rho+10\varepsilon-\lambda<0$. It
follows from our assumptions and [1, Proposition 4.3.3] that there exists a
tempered random variable $N\colon\Omega\to[1,\infty)$ such that for
$\mathbb{P}$-a.e. $\omega\in\Omega$ and $n\in\mathbb{Z}$,

$\frac{1}{N(\omega)}\leq
Z(\omega)(2K(\omega)+M(\omega))+Z(\sigma(\omega))+2D(\omega)\leq N(\omega)$
(30)

and

$N(\omega)e^{-\varepsilon|n|}\leq N(\sigma^{n}(\omega))\leq
N(\omega)e^{\varepsilon|n|}.$ (31)

Thus, using (7), (27) and (30),

$\begin{split}|f_{\omega}(x)-f_{\omega}(z)|&=|f_{\omega}(x)-f_{\omega}(z)|^{1-\alpha}|f_{\omega}(x)-f_{\omega}(z)|^{\alpha}\\\
&\leq\left(2D(\omega)\right)^{1-\alpha}d(\omega)^{\alpha}|x-z|^{\alpha}\\\
&\leq N(\omega)d(\omega)^{\alpha}|x-z|^{\alpha}\end{split}$ (32)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$. Similarly, using
(7), (28) and (30),

$\begin{split}|g_{\omega}(x)-g_{\omega}(z)|\leq
N(\omega)d(\omega)^{\alpha}|x-z|^{\alpha}\end{split}$ (33)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$.

Let $Y_{\infty}$ be as in the proof of Theorem 1. Furthermore, let
$Y^{\alpha}\subset Y_{\infty}$ consist of all $h\in Y_{\infty}$ with the
property that

$|h(\omega,x)-h(\omega,z)|\leq|x-z|^{\alpha},$ (34)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$. We claim that
whenever $c>0$ is sufficiently small, there exists $d:\Omega\to(0,+\infty)$
such that if (27) and (28) are satisfied then $\mathcal{T}(Y^{\alpha})\subset
Y^{\alpha}$, where $\mathcal{T}$ is as in the proof of Theorem 1. Indeed,
given $h\in Y^{\alpha}$, for $x,z\in X$ and $\mathbb{P}$-a.e $\omega\in\Omega$
we have that

$\begin{split}|(\mathcal{T}h)(\omega,x)-(\mathcal{T}h)(\omega,z)|&=\left|\sum_{n\in\mathbb{Z}}\mathcal{G}(\sigma^{-n}(\omega),n)\left(p(n,\omega,x)-p(n,\omega,z)\right)\right|\\\
&\leq\sum_{n\in\mathbb{Z}}\|\mathcal{G}(\sigma^{-n}(\omega),n)\|\left|p(n,\omega,x)-p(n,\omega,z)\right|\\\
&\leq\sum_{n\in\mathbb{Z}}K(\sigma^{-n}(\omega))e^{-\lambda|n|}\left|p(n,\omega,x)-p(n,\omega,z)\right|\\\
&\leq
N(\omega)\sum_{n\in\mathbb{Z}}e^{(-\lambda+\varepsilon)|n|}\left|p(n,\omega,x)-p(n,\omega,z)\right|,\end{split}$
(35)

where $p(n,\omega,x)$ and $p(n,\omega,z)$ are as in (17). Observe that we have
used that $K(\omega)\leq N(\omega)$ for $\mathbb{P}$-a.e. $\omega\in\Omega$
which follows from (30).

Our objective now is to estimate the size of
$\left|p(n,\omega,x)-p(n,\omega,z)\right|$. In order to do it we are going to
need several lemmas. Let us consider $\xi_{x}=(\text{\rm
Id}-\Pi^{2}(\omega))x$, $\eta_{x}=x-\xi_{x}$, $\xi_{z}=(\text{\rm
Id}-\Pi^{2}(\omega))z$ and $\eta_{z}=z-\xi_{z}$. Observe that, although all
these parameters depend on $\omega$, there will be no confusion in omitting
this from the notation itself.

######  Lemma 2.

For $\mathbb{P}$-a.e $\omega\in\Omega$ and every $x,z\in X$ and
$n\in\mathbb{Z}$,

$\begin{split}\left|y(n,\omega,\eta_{x})-y(n,\omega,\eta_{z})\right|&\leq
N(\omega)e^{\rho|n|}|x-z|.\end{split}$ (36)

###### Proof of the lemma.

Since $A(\omega)$ is invertible for $\mathbb{P}$-a.e $\omega\in\Omega$, it
follows from (14) that $y(n,\omega,\eta)=\mathcal{A}(\omega,n)\eta$ for
$\mathbb{P}$-a.e $\omega\in\Omega$ and every $\eta\in E^{c}(\omega)$. Thus,
using (24) and (26) we get that

$\begin{split}\left|y(n,\omega,\eta_{x})-y(n,\omega,\eta_{z})\right|&=\left|\mathcal{A}(\omega,n)\Pi^{2}(\omega)(x-z)\right|\\\
&\leq M(\omega)Z(\omega)e^{\rho|n|}|x-z|,\end{split}$

which together with (30) yields the desired result. ∎

######  Lemma 3.

For $\mathbb{P}$-a.e $\omega\in\Omega$ and every $x,z\in X$ and $n>0$,

$\begin{split}\left|x_{1}(n,\omega,\xi_{x},\eta_{x})-x_{1}(n,\omega,\xi_{z},\eta_{z})\right|&\leq
nN(\omega)\left(e^{\rho}+c\right)^{n}|x-z|.\\\ \end{split}$ (37)

Similarly, whenever $c>0$ is small enough,

$\begin{split}\left|x_{1}(n,\omega,\xi_{x},\eta_{x})-x_{1}(n,\omega,\xi_{z},\eta_{z})\right|&\leq|n|N(\omega)\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)^{|n|+1}|x-z|\end{split}$
(38)

for $\mathbb{P}$-a.e $\omega\in\Omega$ and every $x,z\in X$ and $n<0$.

###### Proof of the lemma.

We first introduce some auxiliary notation. For $\mathbb{P}$-a.e.
$\omega\in\Omega$ and $x\in X$, set

$|x|_{\omega}:=\sup_{n\in\mathbb{Z}}(|\mathcal{A}(\omega,n)x|e^{-\rho|n|}).$

It is easy to verify that

$|x|\leq|x|_{\omega}\leq Z(\omega)|x|,$ (39)

and

$|\mathcal{A}(\omega,n)x|_{\sigma^{n}(\omega)}\leq e^{\rho|n|}|x|_{\omega},$
(40)

for $\mathbb{P}$-a.e. $\omega\in\Omega$, $n\in\mathbb{Z}$ and $x\in X$. We
observe that it follows from (27), (29) and (39) that

$|f_{\omega}(x)-f_{\omega}(y)|_{\sigma(\omega)}\leq c|x-y|,\quad\text{for
$\mathbb{P}$-a.e. $\omega\in\Omega$ and $x,y\in X$.}$ (41)

We will now establish (37). Using (24), (40) and (41), we obtain that for
$\mathbb{P}$-a.e. $\omega\in\Omega$ and $n\geq 0$,

$\begin{split}&\left|x_{1}(n+1,\omega,\xi_{x},\eta_{x})-x_{1}(n+1,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{n+1}(\omega)}\\\
&\leq\left|A(\sigma^{n}(\omega))\left(x_{1}(n,\omega,\xi_{x},\eta_{x})-x_{1}(n,\omega,\xi_{z},\eta_{z})\right)\right|_{\sigma^{n+1}(\omega)}\\\
&+\left|f_{\sigma^{n}(\omega)}(x_{1}(n,\omega,\xi_{x},\eta_{x})+y(n,\omega,\eta_{x}))-f_{\sigma^{n}(\omega)}(x_{1}(n,\omega,\xi_{z},\eta_{z})+y(n,\omega,\eta_{z}))\right|_{\sigma^{n+1}(\omega)}\\\
&\leq
e^{\rho}\left|x_{1}(n,\omega,\xi_{x},\eta_{x})-x_{1}(n,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{n}(\omega)}\\\
&\phantom{\leq}+c\left|x_{1}(n,\omega,\xi_{x},\eta_{x})+y(n,\omega,\eta_{x}))-x_{1}(n,\omega,\xi_{z},\eta_{z})-y(n,\omega,\eta_{z})\right|\\\
&\leq
e^{\rho}\left|x_{1}(n,\omega,\xi_{x},\eta_{x})-x_{1}(n,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{n}(\omega)}\\\
&\phantom{\leq}+c\left|x_{1}(n,\omega,\xi_{x},\eta_{x})-x_{1}(n,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{n}(\omega)}+c\left|y(n,\omega,\eta_{x}))-y(n,\omega,\eta_{z})\right|\\\
&=(e^{\rho}+c)\left|x_{1}(n,\omega,\xi_{x},\eta_{x})-x_{1}(n,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{n}(\omega)}+c|\mathcal{A}(\omega,n)\eta_{x}-\mathcal{A}(\omega,n)\eta_{z}|\\\
&\leq(e^{\rho}+c)\left|x_{1}(n,\omega,\xi_{x},\eta_{x})-x_{1}(n,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{n}(\omega)}+Z(\omega)ce^{\rho
n}|\eta_{x}-\eta_{z}|.\end{split}$

Proceeding recursively and using (39), we conclude that

$\begin{split}&\left|x_{1}(n+1,\omega,\xi_{x},\eta_{x})-x_{1}(n+1,\omega,\xi_{z},\eta_{z})\right|\\\
&\leq\left|x_{1}(n+1,\omega,\xi_{x},\eta_{x})-x_{1}(n+1,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{n+1}(\omega)}\\\
&\leq(e^{\rho}+c)^{n+1}\left|x_{1}(0,\omega,\xi_{x},\eta_{x})-x_{1}(0,\omega,\xi_{z},\eta_{z})\right|_{\omega}\\\
&\phantom{\leq}+cZ(\omega)|\eta_{x}-\eta_{z}|\sum_{j=0}^{n}(e^{\rho}+c)^{j}e^{\rho(n-j)}\\\
&\leq(e^{\rho}+c)^{n+1}\left|\xi_{x}-\xi_{z}\right|_{\omega}+cZ(\omega)|\eta_{x}-\eta_{z}|\sum_{j=0}^{n}(e^{\rho}+c)^{n}\\\
&\leq(n+1)(e^{\rho}+c)^{n+1}Z(\omega)\left(\left|\xi_{x}-\xi_{z}\right|+|\eta_{x}-\eta_{z}|\right)\\\
&\leq(n+1)(e^{\rho}+c)^{n+1}Z(\omega)\left(2K(\omega)+M(\omega)\right)\left|x-z\right|,\end{split}$

which implies (37).

We now prove (38). Given $\omega\in\Omega$ and $\eta\in E^{c}(\omega)$ let us
consider $F_{\omega}^{\eta}:E^{s,u}(\omega)\to E^{s,u}(\sigma(\omega))$
defined by

$F_{\omega}^{\eta}(\xi)=A(\omega)\xi+f_{\omega}(\xi+\eta).$ (42)

It follows from our hypothesis (10) that $F_{\omega}^{\eta}$ is invertible for
$\mathbb{P}$-a.e. $\omega\in\Omega$. Moreover, one can easily check that its
inverse is given by

$\left(F_{\omega}^{\eta}\right)^{-1}(\xi)=A(\omega)^{-1}\xi-A(\omega)^{-1}\left(f_{\omega}\left(\left(F_{\omega}^{\eta}\right)^{-1}(\xi)+\eta\right)\right).$
(43)

Consequently, for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every
$\eta,\theta\in E^{c}(\omega)$ and $\xi,\zeta\in E^{s,u}(\sigma(\omega))$, we
have that

$\begin{split}&\left|\left(F_{\omega}^{\eta}\right)^{-1}(\xi)-\left(F_{\omega}^{\theta}\right)^{-1}(\zeta)\right|_{\omega}\\\
&\leq\left|A(\omega)^{-1}\left(\xi-\zeta\right)\right|_{\omega}\\\
&\phantom{\leq}+\left|A(\omega)^{-1}\left(f_{\omega}\left(\left(F_{\omega}^{\eta}\right)^{-1}(\xi)+\eta\right)-f_{\omega}\left(\left(F_{\omega}^{\theta}\right)^{-1}(\zeta)+\theta\right)\right)\right|_{\omega}\\\
&\leq
e^{\rho}|\xi-\zeta|_{\sigma(\omega)}+e^{\rho}\left|f_{\omega}\left(\left(F_{\omega}^{\eta}\right)^{-1}(\xi)+\eta\right)-f_{\omega}\left(\left(F_{\omega}^{\theta}\right)^{-1}(\zeta)+\theta\right)\right|_{\sigma(\omega)}\\\
&\leq
e^{\rho}|\xi-\zeta|_{\sigma(\omega)}+ce^{\rho}\left(\left|\left(F_{\omega}^{\eta}\right)^{-1}(\xi)-\left(F_{\omega}^{\theta}\right)^{-1}(\zeta)\right|+\left|\eta-\theta\right|\right)\\\
&\leq
e^{\rho}|\xi-\zeta|_{\sigma(\omega)}+ce^{\rho}\left(\left|\left(F_{\omega}^{\eta}\right)^{-1}(\xi)-\left(F_{\omega}^{\theta}\right)^{-1}(\zeta)\right|_{\omega}+\left|\eta-\theta\right|\right)\\\
&\leq
e^{\rho}\left(|\xi-\zeta|_{\sigma(\omega)}+\left|\eta-\theta\right|\right)+ce^{\rho}\left|\left(F_{\omega}^{\eta}\right)^{-1}(\xi)-\left(F_{\omega}^{\theta}\right)^{-1}(\zeta)\right|_{\omega}.\end{split}$

Thus, for $c>0$ sufficiently small we get that

$\begin{split}&\left|\left(F_{\omega}^{\eta}\right)^{-1}(\xi)-\left(F_{\omega}^{\theta}\right)^{-1}(\zeta)\right|_{\omega}\leq\frac{e^{\rho}}{1-ce^{\rho}}\left(|\xi-\zeta|_{\sigma(\omega)}+\left|\eta-\theta\right|\right).\end{split}$
(44)

Now, recalling (14) and (42) it follows that for every $n>0$

$x_{1}(-n,\omega,\xi_{x},\eta_{x})=\left(F_{\sigma^{-n}(\omega)}^{\mathcal{A}(\omega,-n)\eta_{x}}\right)^{-1}\left(x_{1}(-(n-1),\omega,\xi_{x},\eta_{x})\right).$

By applying (44), we obtain that

$\begin{split}&\left|x_{1}(-n,\omega,\xi_{x},\eta_{x})-x_{1}(-n,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{-n}(\omega)}\\\
&\leq\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)\left|x_{1}(-(n-1),\omega,\xi_{x},\eta_{x})-x_{1}(-(n-1),\omega,\xi_{z},\eta_{z})\right|_{\sigma^{-(n-1)}(\omega)}\\\
&+\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)\left|\mathcal{A}(\omega,-n)\eta_{x}-\mathcal{A}(\omega,-n)\eta_{z}\right|\\\
&\leq\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)\left|x_{1}(-(n-1),\omega,\xi_{x},\eta_{x})-x_{1}(-(n-1),\omega,\xi_{z},\eta_{z})\right|_{\sigma^{-(n-1)}(\omega)}\\\
&+Z(\omega)\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)e^{\rho
n}\left|\eta_{x}-\eta_{z}\right|.\\\ \end{split}$

By proceeding recursively, we conclude that

$\begin{split}&\left|x_{1}(-n,\omega,\xi_{x},\eta_{x})-x_{1}(-n,\omega,\xi_{z},\eta_{z})\right|\\\
&\leq\left|x_{1}(-n,\omega,\xi_{x},\eta_{x})-x_{1}(-n,\omega,\xi_{z},\eta_{z})\right|_{\sigma^{-n}(\omega)}\\\
&\leq\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)^{n}\left|x_{1}(0,\omega,\xi_{x},\eta_{x})-x_{1}(0,\omega,\xi_{z},\eta_{z})\right|_{\omega}\\\
&\phantom{\leq}+Z(\omega)\left|\eta_{x}-\eta_{z}\right|\sum_{j=0}^{n-1}\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)^{j+1}e^{\rho(n-j)}\\\
&\leq\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)^{n}\left|\xi_{x}-\xi_{z}\right|_{\omega}+Z(\omega)\left|\eta_{x}-\eta_{z}\right|\sum_{j=0}^{n-1}\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)^{n+1}\\\
&\leq
n\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)^{n+1}Z(\omega)\left(\left|\xi_{x}-\xi_{z}\right|+\left|\eta_{x}-\eta_{z}\right|\right)\\\
&\leq
nZ(\omega)\left(2K(\omega)+M(\omega)\right)\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)^{n+1}\left|x-z\right|\\\
&\leq
nN(\omega)\left(\frac{e^{\rho}}{1-ce^{\rho}}\right)^{n+1}\left|x-z\right|,\\\
\end{split}$

which proves our claim (38). The proof of the lemma is completed.

∎

In order to simplify notation, let us consider

$\tilde{x}_{1}(n,\omega,x):=x_{1}(n,\omega,\xi_{x},\eta_{x})+y(n,\omega,\eta_{x})$

and

$\tilde{x}_{1}(n,\omega,z):=x_{1}(n,\omega,\xi_{z},\eta_{z})+y(n,\omega,\eta_{z}).$

Thus, combining (36), (37) and (38) we get that

$|\tilde{x}_{1}(n,\omega,x)-\tilde{x}_{1}(n,\omega,z)|\leq
2(|n|+1)N(\omega)\left(\frac{e^{\rho}+c}{1-ce^{\rho}}\right)^{|n|+1}|x-z|$
(45)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$ and
$n\in\mathbb{Z}$. Then, using (31) and (32) we conclude that

$\begin{split}&\left|f_{\sigma^{-(n+1)}(\omega)}(\tilde{x}_{1}(-(n+1),\omega,x))-f_{\sigma^{-(n+1)}(\omega)}(\tilde{x}_{1}(-(n+1),\omega,z))\right|\\\
&\leq
N(\sigma^{-(n+1)}(\omega))d(\sigma^{-(n+1)}(\omega))^{\alpha}\left|\tilde{x}_{1}(-(n+1),\omega,x)-\tilde{x}_{1}(-(n+1),\omega,z)\right|^{\alpha}\\\
&\leq
2(|n|+2)N(\omega)N(\sigma^{-(n+1)}(\omega))d(\sigma^{-(n+1)}(\omega))^{\alpha}\left(\frac{e^{\rho}+c}{1-ce^{\rho}}\right)^{(|n|+2)\alpha}|x-z|^{\alpha}\\\
&\leq
2(|n|+2)N(\omega)^{2}d(\sigma^{-(n+1)}(\omega))^{\alpha}e^{\varepsilon(|n|+1)}\left(\frac{e^{\rho}+c}{1-ce^{\rho}}\right)^{(|n|+2)\alpha}|x-z|^{\alpha},\end{split}$
(46)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$ and
$n\in\mathbb{Z}$.

Similarly, assuming without loss of generality that $d(\omega)<1$ and using
(28), (31), (33) and (34) we get that, for $\mathbb{P}$-a.e. $\omega\in\Omega$
and every $u,v\in X$,

$\begin{split}&\left|g_{\sigma^{-(n+1)}(\omega)}\left(u+h(\sigma^{-(n+1)}(\omega),u)\right)-g_{\sigma^{-(n+1)}(\omega)}\left(v+h(\sigma^{-(n+1)}(\omega),v)\right)\right|\\\
&\leq
N(\sigma^{-(n+1)}(\omega))d(\sigma^{-(n+1)}(\omega))^{\alpha}\min\Big{\\{}\left|u-v\right|+\left|h(\sigma^{-(n+1)}(\omega),u)-h(\sigma^{-(n+1)}(\omega),v)\right|,\\\
&\phantom{=}\Big{[}\left|u-v\right|+\left|h(\sigma^{-(n+1)}(\omega),u)-h(\sigma^{-(n+1)}(\omega),v)\right|\Big{]}^{\alpha}\Big{\\}}\\\
&\leq
N(\sigma^{-(n+1)}(\omega))d(\sigma^{-(n+1)}(\omega))^{\alpha}\min\Big{\\{}\left|u-v\right|+\left|u-v\right|^{\alpha},\Big{[}\left|u-v\right|+\left|u-v\right|^{\alpha}\Big{]}^{\alpha}\Big{\\}}\\\
&\leq
N(\sigma^{-(n+1)}(\omega))d(\sigma^{-(n+1)}(\omega))^{\alpha}\begin{cases}2^{\alpha}\left|u-v\right|^{\alpha}&\text{if
$\left|u-v\right|>1$ (using the right one)}\\\
2\left|u-v\right|^{\alpha}&\text{if $\left|u-v\right|\leq 1$ (using the left
one)}\end{cases}\\\ &\leq
2N(\sigma^{-(n+1)}(\omega))d(\sigma^{-(n+1)}(\omega))^{\alpha}|u-v|^{\alpha}\\\
&\leq
2N(\omega)d(\sigma^{-(n+1)}(\omega))^{\alpha}e^{\varepsilon(|n|+1)}|u-v|^{\alpha}.\end{split}$

Taking $u=\tilde{x}_{1}(-(n+1),\omega,x)$ and
$v=\tilde{x}_{1}(-(n+1),\omega,z)$ and applying (45) we conclude that

$\begin{split}&\left|g_{\sigma^{-(n+1)}(\omega)}\left(\tilde{x}_{1}(-(n+1),\omega,x)+h(\sigma^{-(n+1)}(\omega),\tilde{x}_{1}(-(n+1),\omega,x))\right)\right.\\\
&\left.-g_{\sigma^{-(n+1)}(\omega)}\left(\tilde{x}_{1}(-(n+1),\omega,z)+h(\sigma^{-(n+1)}(\omega),\tilde{x}_{1}(-(n+1),\omega,z))\right)\right|\\\
&\leq
4(|n|+2)N(\omega)^{2}d(\sigma^{-(n+1)}(\omega))^{\alpha}e^{\varepsilon(|n|+1)}\left(\frac{e^{\rho}+c}{1-ce^{\rho}}\right)^{(|n|+2)\alpha}|x-z|^{\alpha},\end{split}$
(47)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$ and
$n\in\mathbb{Z}$.

Thus, recalling the definition of $p(n,\omega,x)$ and $p(n,\omega,z)$ in (17)
and using (46) and (47), it follows that

$\begin{split}&\left|p(n,\omega,x)-p(n,\omega,z)\right|\\\ &\leq
6(|n|+2)N(\omega)^{2}d(\sigma^{-(n+1)}(\omega))^{\alpha}e^{\varepsilon(|n|+1)}\left(\frac{e^{\rho}+c}{1-ce^{\rho}}\right)^{(|n|+2)\alpha}|x-z|^{\alpha},\end{split}$
(48)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$ and
$n\in\mathbb{Z}$. Consequently, plugging (48) into (35) yields

$\begin{split}|(\mathcal{T}h)(\omega,x)-(\mathcal{T}h)(\omega,z)|&\leq
6N(\omega)^{3}B^{\prime}|x-z|^{\alpha}\end{split}$ (49)

for $\mathbb{P}$-a.e. $\omega\in\Omega$ and every $x,z\in X$ and
$n\in\mathbb{Z}$, where

$B^{\prime}:=\sum_{n\in\mathbb{Z}}\left((|n|+2)e^{-\lambda|n|}e^{2\varepsilon(|n|+1)}\left(\frac{e^{\rho}+c}{1-ce^{\rho}}\right)^{(|n|+2)\alpha}d(\sigma^{-(n+1)}(\omega))^{\alpha}\right).$

Now, using the fact that $\alpha\rho+10\varepsilon-\lambda<0$ we get that if
$c$ is sufficiently small then (mind the term $5\varepsilon$)

$B:=\sum_{n\in\mathbb{Z}}\left((|n|+2)e^{-\lambda|n|}e^{5\varepsilon(|n|+1)}\left(\frac{e^{\rho}+c}{1-ce^{\rho}}\right)^{(|n|+2)\alpha}\right)<+\infty.$

Set

$d(\omega)=\left(\frac{c}{12BN(\omega)^{3}}\right)^{1/\alpha},\quad\omega\in\Omega.$
(50)

It is easy to see (using (30)) that (29) is satisfied. Furthermore, since $N$
is tempered, we have that $d$ is also tempered. Moreover, using the first
inequality in (31) it follows that

$\begin{split}d(\sigma^{-(n+1)(\omega)})^{\alpha}&=\frac{c}{12BN(\sigma^{-(n+1)}(\omega))^{3}}\\\
&\leq\frac{ce^{3\varepsilon(|n|+1)}}{12BN(\omega)^{3}}\\\
&=d(\omega)^{\alpha}e^{3\varepsilon(|n|+1)}.\end{split}$

Inserting this into (49), we find that

$\begin{split}&|(\mathcal{T}h)(\omega,x)-(\mathcal{T}h)(\omega,z)|\\\ &\leq
6N(\omega)^{3}d(\omega)^{\alpha}\sum_{n\in\mathbb{Z}}\left((|n|+2)e^{-\lambda|n|}e^{5\varepsilon(|n|+1)}\left(\frac{e^{\rho}+c}{1-ce^{\rho}}\right)^{(|n|+2)\alpha}\right)|x-z|^{\alpha}\\\
&=6N(\omega)^{3}d(\omega)^{\alpha}B|x-z|^{\alpha}\\\
&=\frac{c}{2}|x-z|^{\alpha}\leq|x-z|^{\alpha},\end{split}$

for $\mathbb{P}$-a.e $\omega\in\Omega$ and every $x,z\in X$. Hence,
$\mathcal{T}(Y^{\alpha})\subset Y^{\alpha}$ as claimed. Thus, since
$Y^{\alpha}$ is a closed subspace of
$(Y_{\infty},\|\cdot\|^{\prime}_{\infty})$ and $\mathcal{T}:Y_{\infty}\to
Y_{\infty}$ is a contraction, it follows that the unique fixed point $h$ of
$\mathcal{T}$ is in $Y^{\alpha}$. Therefore, since the conjugacy given by
Theorem 1 is of the form $H_{\omega}(x)=x+h(\omega,x)$, it follows that

$\begin{split}|H_{\omega}(x)-H_{\omega}(z)|&=|x+h(\omega,x)-z-h(\omega,z)|\\\
&\leq\left(|x-z|^{1-\alpha}+1\right)|x-z|^{\alpha}\end{split}$

for $\mathbb{P}$-a.e $\omega\in\Omega$ and every $x,z\in X$. Now, given a
bounded subset $\tilde{X}\subset X$ there exists a constant $\tilde{T}>1$
depending only on $\tilde{X}$ such that $|x-z|\leq\tilde{T}$ for every
$x,z\in\tilde{X}$. Consequently, taking $T:=\tilde{T}+1$ we get that

$\begin{split}|H_{\omega}(x)-H_{\omega}(z)|\leq T|x-z|^{\alpha}\end{split}$

for $\mathbb{P}$-a.e $\omega\in\Omega$ and every $x,z\in\tilde{X}$. The fact
that $H_{\omega}^{-1}$ satisfies a similar property follows by changing the
roles of $f_{\omega}$ and $g_{\omega}$ in the previous argument. The proof of
the theorem is complete. ∎

######  Remark 6.

Theorems 1 and 2 were stated and proved in the case when the base dynamics
$(\Omega,\sigma,\mathbb{P})$ is ergodic. Nevertheless, similar versions with
generalized hypothesis can be obtained in the non-ergodic setting simply by
combining our results with the Ergodic Decomposition Theorem: if $\mathbb{P}$
is non-ergodic then we can decompose it into ergodic components, each of which
“living” in disjoint subsets of $\Omega$, and apply our results to each of
these components. Then, by “gluing” together each of the respective maps we
get the desired result. The main differences in the statements of the results
in the non-ergodic case with respect to the ergodic one appear in the constant
$c$ in Theorem 1 and in the Hölder exponent in Theorem 2: both will become
$\sigma$-invariant maps $c,\alpha:\Omega\to(0,1)$ since these quantities will
depend on the ergodic component of $\mathbb{P}$. For instance, instead of
getting a constant Hölder exponent that holds for $\mathbb{P}$-a.e
$\omega\in\Omega$, we will get a $\sigma$-invariant map
$\alpha:\Omega\to(0,1)$ such that each $H_{\omega}$ and $H_{\omega}^{-1}$ is
$\alpha(\omega)$-Hölder continuous for $\mathbb{P}$-a.e $\omega\in\Omega$. As
already mentioned, in the non-ergodic setting our hypothesis can be relaxed
accordingly. For instance, in conditions (4), (5) and (24) the constants
$\lambda$ and $\rho$ can be replaced by $\sigma$-invariant maps
$\lambda,\rho:\Omega\to(0,+\infty)$.

##  4\. Acknowledgements

We would like to thank the referee for useful comments that helped us improve
our paper. L.B. was partially supported by a CNPq-Brazil PQ fellowship under
Grant No. 306484/2018-8. D.D. was supported in part by Croatian Science
Foundation under the project IP-2019-04-1239 and by the University of Rijeka
under the projects uniri-prirod-18-9 and uniri-pr-prirod-19-16.

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