#  Quenched limit theorems for random U(1) extensions of expanding maps

Yong Moo Chung  Department of Applied Mathematics, Hiroshima University,
Higashi-Hiroshima, 739-8527, JAPAN<EMAIL_ADDRESS>,  Yushi
Nakano  Department of Mathematics, Tokai University, 4-1-1 Kitakaname,
Hiratuka, Kanagawa, 259-1292, JAPAN<EMAIL_ADDRESS>and  Jens
Wittsten  Centre for Mathematical Sciences, Lund University, Box 118, SE221 00
Lund, Sweden, and Department of Engineering, University of Borås, SE-501 90
Borås, Sweden<EMAIL_ADDRESS>
###### Abstract.

The Lyapunov spectra of random U(1) extensions of expanding maps on the torus
were investigated in our previous work [NW2015]. Using the result, we extend
the recent spectral approach for quenched limit theorems for expanding maps
[DFGV2018] and hyperbolic maps [DFGV2019] to our partially hyperbolic
dynamics. Quenched central limit theorems, large deviations principles and
local central limit theorems for random U(1) extensions of expanding maps on
the torus are proved via corresponding theorems for abstract random dynamical
systems.

###### Key words and phrases:

Quenched limit theorems, transfer operator, U(1) extensions of expanding maps

##  1\. Introduction

This paper concerns quenched limit theorems for random dynamical systems on a
compact smooth Riemannian manifold $M$. Given a measure-preserving mapping
$\sigma:\Omega\to\Omega$ on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$, a _random dynamical system_ (abbreviated
RDS henceforth) on $M$ over $\sigma$ is given as a measurable map
$\mathcal{T}:\mathbb{N}_{0}\times\Omega\times M\to M$ satisfying
$\mathcal{T}(0,\omega,\cdot)=\mathrm{id}_{M}$ and
$\mathcal{T}(n+m,\omega,x)=\mathcal{T}(n,\sigma^{m}\omega,\mathcal{T}(m,\omega,x))$
for each $n,m\in\mathbb{N}_{0}$, $\omega\in\Omega$ and $x\in M$, with
$\mathbb{N}_{0}=\\{0,1,\ldots\\}$. Here $\sigma\omega$ denotes the value
$\sigma(\omega)$, and $\sigma$ is called a _driving system_. A standard
reference for random dynamical systems is the monograph by Arnold [Arnold].
Here we merely recall that if we denote $\mathcal{T}(n,\omega,\cdot)$ and
$\mathcal{T}(1,\omega,\cdot)$ by $T^{(n)}_{\omega}$ and $T_{\omega}$,
respectively, then we have

(1.1)
$\mathcal{T}(n,\omega,\cdot)=T^{(n)}_{\omega}=T_{\sigma^{n-1}\omega}\circ
T_{\sigma^{n-2}\omega}\circ\cdots\circ T_{\omega}.$

Conversely, given a measurable map $T:\Omega\times M\to M:(\omega,x)\mapsto
T_{\omega}(x)$, the map $\mathcal{T}:\mathbb{N}_{0}\times\Omega\times M\to M$
defined by (1.1) is an RDS over $\sigma$. We call it the _RDS induced by $T$
over $\sigma$_.

Given a measurable function $g:\Omega\times M\to\mathbb{R}:(\omega,x)\mapsto
g_{\omega}(x)$, we consider the _quenched_ (i.e. noisewise) Birkhoff sum
$(S_{n}g)_{\omega}:M\to\mathbb{C}$ of $g$, given by

(1.2) $(S_{n}g)_{\omega}=\sum_{j=0}^{n-1}g_{\sigma^{j}\omega}\circ
T^{(j)}_{\omega}\quad(n\geq 1,\;\omega\in\Omega).$

Our ultimate interest is the $\mathbb{P}$-almost sure asymptotic behavior of
such Birkhoff sums as $n\to\infty$. Results on such behavior are known as
quenched limit theorems (for the random process $\\{g_{\sigma^{n}\omega}\circ
T^{(n)}_{\omega}\\}_{n\geq 0}$ in the random environment
$\\{\sigma^{n}\omega\\}_{n\geq 0}$). Conversely, results on the asymptotic
behavior of the expectation $\mathbb{E}_{\mathbb{P}}[S_{n}g]$ are referred to
as _annealed_ (i.e. averaged) limit theorems. For the history of annealed and
quenched limit theorems for random dynamical systems, we refer to [ANV2015,
DFGV2018, DFGV2019].

In [ANV2015], Aimino et al. gave a comprehensive study of annealed limit
theorems for abundant random dynamical systems (including random piecewise
expanding maps) via a Nagaev-Guivarc’h type perturbative spectral approach for
random dynamical systems (see §1.4 for a brief description of the approach).
This was extended by Dragičević et al. [DFGV2018] to quenched schemes by using
the newly-developed theory of _Lyapunov spectrum_ of random operators (see
e.g. [FLQ2013, GQ2014] for a presentation of this theory) and the detailed
analysis of the _leading_ eigenvalue of the associated transfer operator. This
was recently further extended to random piecewise hyperbolic maps [DFGV2019].
In this paper we extend the technology in [DFGV2018] to a class of partially
expanding maps with a neutral direction, namely, to U(1) extensions of
expanding maps on the circle.

###  1.1. Random U(1) extensions of expanding maps on $\mathbb{S}^{1}$

Let $e_{0}:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a
${\mathscr{C}}^{\infty}$ orientation-preserving diffeomorphism on the circle
$\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$. Let $k\geq 2$ be an integer and set
$E_{0}(x)=ke_{0}(x)$ mod 1. Let $\tau_{0}:\mathbb{S}^{1}\rightarrow\mathbb{R}$
be a ${\mathscr{C}}^{\infty}$ function. We consider the skew product
$T_{0}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ of class $\mathscr{C}^{\infty}$ given
by

(1.3) $T_{0}:\binom{x}{s}\mapsto\left(\\!\\!\begin{array}[]{cc}E_{0}(x)&\\\
s+\tau_{0}(x)&\\!\\!\mod 1\end{array}\\!\\!\right)$

on the torus $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$. The map $E_{0}$
is assumed to be an expanding map on $\mathbb{S}^{1}$ in the sense that
$\min_{x}E_{0}^{\prime}(x)>1$, and we then say that $T_{0}$ is the U(1)
extension of the expanding map $E_{0}$ over $\tau_{0}$. This dynamical system
can be seen as a toy model of (piecewise) hyperbolic flows such as geodesic
flows on negatively curved manifolds or dispersive billiard flows (via
suspension flows of hyperbolic maps; see [hasselblatt2017, PP1990]), and has
been intensively studied by several authors (see e.g. [Dolgopyat2002, Faure,
FaureTsujii2, NW2015, NTW2016, BE2017, FW2017]).

######  Remark.

We recall that if $\tau_{0}$ _is cohomologous to a constant function_ (i.e.
there is a smooth function $u$ such that $\tau_{0}-u\circ E_{0}+u$ is a
constant function), then $T_{0}$ is not even weakly mixing ([Tsujii]; see also
[BE2017]) which implies that the corresponding transfer operator does not have
a spectral gap (and several limit theorems thus fail to hold). This would
prevent us from applying the Nagaev-Guivarc’h spectral method. Therefore, in
this paper we shall always assume that the function $\tau_{0}$ is not
cohomologous to a constant function, which is known to be a generic condition
[Tsujii]. We do so by imposing the slightly stronger condition that $T_{0}$ is
partially captive which is still a generic condition on $\tau_{0}$ once
$E_{0}$ is fixed [NTW2016]. A precise description of the partial captivity
condition is given in Appendix A.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let
${\mathscr{C}}^{\infty}(\mathbb{T}^{2},\mathbb{T}^{2})$ be the space of smooth
endomorphisms on $\mathbb{T}^{2}$, endowed with the Borel $\sigma$-algebra.
Given a noise level $\epsilon\geq 0$, we let $T\equiv
T(\epsilon):\Omega\rightarrow{\mathscr{C}}^{\infty}(\mathbb{T}^{2},\mathbb{T}^{2}):\omega\mapsto
T_{\omega}$ be a measurable map such that $T_{\omega}$ is for
$\mathbb{P}$-almost every $\omega\in\Omega$ of the form

(1.4)
$T_{\omega}:\binom{x}{s}\mapsto\left(\\!\\!\begin{array}[]{cc}E_{\omega}(x)&\\!\\!\\\
s+\tau_{\omega}(x)&\\!\\!\mod 1\end{array}\\!\\!\right),$

and $\omega\mapsto T_{\omega}(z)$ is a measurable mapping from $\Omega$ to
$\mathbb{T}^{2}$ for each $z=(x,s)\in\mathbb{T}^{2}$. Here
$E_{\omega}:\mathbb{S}^{1}\to\mathbb{S}^{1}$ is given by
$E_{\omega}(x)=ke_{\omega}(x)$ mod 1 where $k\geq 2$ is the same integer as
before while $e_{\omega}:\mathbb{S}^{1}\to\mathbb{S}^{1}$ is a
${\mathscr{C}}^{\infty}$ diffeomorphism and
$\tau_{\omega}:\mathbb{S}^{1}\to\mathbb{R}$ is a ${\mathscr{C}}^{\infty}$
function, $\mathbb{P}$-almost surely. We also assume that

(1.5)
$\operatorname*{ess\,sup}_{\omega}d_{{\mathscr{C}}^{\infty}}(T_{\omega},T_{0})\leq\epsilon,$

where $T_{0}$ is the partially expanding map given by (1.3) and
$d_{{\mathscr{C}}^{\infty}}$ is some choice of metric on
${\mathscr{C}}^{\infty}(\mathbb{T}^{2},\mathbb{T}^{2})$. Note that
$E_{\omega}$ is an expanding map for any sufficiently small $\epsilon>0$ and
$\mathbb{P}$-almost every $\omega\in\Omega$ (see Remark 2.2 for details), and
that when $\epsilon=0$, $T_{\omega}=T_{0}$ for $\mathbb{P}$-almost every
$\omega\in\Omega$.

Let $\sigma:\Omega\to\Omega$ be a measurably invertible, ergodic, measure-
preserving map on $(\Omega,\mathcal{F},\mathbb{P})$. Then $(\omega,z)\mapsto
T_{\omega}(z)$ is a measurable map from $\Omega\times\mathbb{T}^{2}$ to
$\mathbb{T}^{2}$ (see [CV77]*Lemma 3.14), so the map $\mathcal{T}$ induced by
$T$ over $\sigma$ (recall (1.1)) is an RDS on $\mathbb{T}^{2}$. The purpose of
this paper is to obtain quenched limit theorems for $\mathcal{T}$ as special
cases of corresponding results for abstract random dynamical systems
satisfying certain hypotheses inspired by the conditions placed on concrete
Banach spaces in [DFGV2018, DFGV2019].

###  1.2. Abstract setting

Let $(M,\mathcal{G})$ be a measurable space, where $\mathcal{G}$ is a
$\sigma$-algebra of $M$. We consider a topological vector space $\mathscr{D}$
consisting of complex-valued functions on $M$, and a map $T:M\to M$ preserving
$\mathscr{D}$ (i.e. $u\circ T\in\mathscr{D}$ for any $u\in\mathscr{D}$). Then,
we define the (Perron-Frobenius) _transfer operator_ $\mathcal{L}_{T}$ of $T$
on the (continuous) dual space $\mathscr{D}^{\prime}$ of $\mathscr{D}$ by

(1.6) $\langle\mathcal{L}_{T}\varphi,u\rangle=\langle\varphi,u\circ
T\rangle,\quad\varphi\in\mathscr{D}^{\prime},\quad u\in\mathscr{D},$

where $\langle\phantom{i},\phantom{i}\rangle$ is the dual pairing. When $T$ is
a smooth map on a compact smooth Riemannian manifold $M$ equipped with the
normalized Lebesgue measure $\mathrm{Leb}_{M}$, and $\mathscr{D}=L^{1}(M)$, it
follows from the canonical form $L^{1}(M)\ni
u\mapsto\langle\varphi,u\rangle=\int\varphi u\,d\mathrm{Leb}_{M}$ of
$\varphi\in L^{\infty}(M)$ as an element of $\left(L^{1}(M)\right)^{\prime}$
that (1.6) is equivalent to $\int\mathcal{L}_{T}\varphi\cdot
u\,d\mathrm{Leb}_{M}=\int\varphi\cdot u\circ T\,d\mathrm{Leb}_{M}$ for each
$\varphi\in L^{\infty}(M)$ and $u\in L^{1}(M).$ Hence, if $\det DT(x)\neq 0$
Lebesgue almost everywhere, then by a change of variables one can get

(1.7) $\mathcal{L}_{T}\varphi(x)=\sum_{T(y)=x}\frac{\varphi(y)}{|\det DT(y)|}$

for each $\varphi\in L^{\infty}(M)$ and Lebesgue almost every $x\in M$. Due to
the relation (1.6), one may expect that several statistical properties of $T$
directly follow from properties of the spectrum of $\mathcal{L}$. Standard
references for transfer operators are the monographs of Baladi [Baladi00,
Baladibook2].

In some literature, the transfer operator of $T$ is defined as the bounded
operator on a space of functions on a smooth manifold $M$ given by (1.7).
However, in the last two decades it has been clearly realized that it is
important to investigate the spectrum of the transfer operator on Banach
spaces of _distributions_ if one hopes to obtain information on the Sinai-
Ruelle-Bowen measures of the dynamics, which are the most relevant measures in
smooth dynamical systems theory. We thus employ relation (1.6) for the
definition of transfer operators and interpret it in the distributional sense
whenever appropriate.

Let $(\Omega,\mathcal{F},\mathbb{P})$ and $\sigma$ be as in Subsection 1.1.
Let $\mathscr{M}$ be a measurable space consisting of maps preserving
$\mathscr{D}$, and $T:\Omega\to\mathscr{M}:\omega\mapsto T_{\omega}$ a
measurable map such that $(\omega,x)\mapsto T_{\omega}(x)$ is a measurable map
from $\Omega\times M$ to $M$. We simply write $\mathcal{L}_{\omega}$ for the
transfer operator
$\mathcal{L}_{T_{\omega}}:\mathscr{D}^{\prime}\to\mathscr{D}^{\prime}$ of
$T_{\omega}$.

######  Definition 1.1.

We say that a Banach space $\mathscr{B}\subset\mathscr{D}^{\prime}$ with a
norm $\|\cdot\|$ is _$T$ -admissible_ if $\mathcal{L}_{\omega}$ is bounded on
$\mathscr{B}$ for $\mathbb{P}$-almost every $\omega\in\Omega$, and there
exists a constant $C_{0}>0$ such that

$\mathrm{(A1)}$ $\|\mathcal{L}_{\omega}\varphi\|\leq
C_{0}\|\varphi\|,\quad\varphi\in\mathscr{B},\quad\text{ $\mathbb{P}$-almost
every $\omega$.}$

Let $\mathcal{Z}(\mathscr{D})$ be a linear subspace of $\mathscr{D}$ given by

$\mathcal{Z}(\mathscr{D})=\\{u\in\mathscr{D}\mid uv\in\mathscr{D}\;\text{for
any $v\in\mathscr{D}$}\\}.$

For each $u\in\mathcal{Z}(\mathscr{D})$ and $\varphi\in\mathscr{D}^{\prime}$,
define $u\varphi\in\mathscr{D}^{\prime}$ via duality by

(1.8) $\langle u\varphi,v\rangle=\langle\varphi,uv\rangle,\quad
v\in\mathscr{D}.$

Let $1_{A}$ be the indicator function of $A\subset M$.

######  Definition 1.2.

We say that a vector space $\mathscr{E}$ with a seminorm
$\|\cdot\|_{\mathscr{E}}$ is _associated with $(\mathscr{B},\|\cdot\|)$_ if
$\mathscr{E}\subset\mathcal{Z}(\mathscr{D})$, $1_{M}\in\mathscr{E}$ and there
exists a constant $C_{1}>0$ such that

$\mathrm{(A2)}$
$\displaystyle\max\left\\{\langle\varphi,u\rangle,\|u\varphi\|\right\\}$
$\displaystyle\leq
C_{1}\|\varphi\|\|u\|_{\mathscr{E}},\quad\varphi\in\mathscr{B},\
u\in\mathscr{E},$ $\displaystyle\|uv\|_{\mathscr{E}}$ $\displaystyle\leq
C_{1}\|u\|_{\mathscr{E}}\|v\|_{\mathscr{E}},\quad u,v\in\mathscr{E}.$

We call each element in $\mathscr{E}$ an _observable_.

We now fix a seminormed vector space $(\mathscr{E},\|\cdot\|_{\mathscr{E}})$
associated with a $T$-admissible Banach space $(\mathscr{B},\|\cdot\|)$. We
define a cocycle $(n,\omega,\varphi)\mapsto\mathcal{L}_{\omega}^{(n)}\varphi$
by

$\mathcal{L}_{\omega}^{(n)}=\mathcal{L}_{\sigma^{n-1}\omega}\circ\mathcal{L}_{\sigma^{n-2}\omega}\circ\cdots\circ\mathcal{L}_{\omega},\quad
n\geq 1,$

and $\mathcal{L}_{\omega}^{(0)}=\mathrm{id}_{\mathscr{B}}$ for each
$\omega\in\Omega$, which is called the _transfer operator cocycle_ (of the RDS
$\mathcal{T}$ induced by $T$ over $\sigma$), and denoted by
$(\mathcal{L},\sigma)$ for simplicity.

For each topological space $\mathscr{V}$, we denote by
$\mathcal{B}_{\mathscr{V}}$ the Borel $\sigma$-algebra of $\mathscr{V}$. In
particular, $\mathcal{B}_{L(\mathscr{B})}$ is the Borel $\sigma$-algebra
generated by the norm topology on $L(\mathscr{B})$, where $L(\mathscr{B})$ is
the space of all bounded operators on $\mathscr{B}$ equipped with the operator
norm. When $\mathscr{B}$ is separable, we also consider the $\sigma$-algebra
$\mathcal{S}_{L(\mathscr{B})}$ generated by the strong operator topology on
$L(\mathscr{B})$, see [GQ2014, Appendix A] for its definition and basic
properties. Here we merely recall that a map $\mathcal{A}:\Omega\to
L(\mathscr{B})$ is $(\mathcal{F},\mathcal{S}_{L(\mathscr{B})})$-measurable if
and only if $\mathcal{A}$ is strongly measurable, that is, for any
$\varphi\in\mathscr{B}$, the map $\omega\mapsto\mathcal{A}(\omega)(\varphi)$
is $(\mathcal{F},\mathcal{B}_{\mathscr{B}})$-measurable. Furthermore, as
pointed out in [GQ2014] (see also [BY93, Bogenschutz00]), the
$(\mathcal{F},\mathcal{B}_{L(\mathscr{B})})$-measurability of $\mathcal{L}$ is
a strong requirement (at least stronger than the
$(\mathcal{F},\mathcal{S}_{L(\mathscr{B})})$-measurability). Indeed,
[DFGV2018, DFGV2019] ensured the
$(\mathcal{F},\mathcal{B}_{L(\mathscr{B})})$-measurability of $\mathcal{L}$ by
assuming that $\\{T_{\omega}\mid\omega\in\Omega\\}$ is at most countable. The
main hypotheses which we shall place on an abstract RDS in this setting are
the following two spectral conditions.

######  Definition 1.3 (Uniform spectral gap condition).

We say that $T$ satisfies the uniform spectral gap condition (UG) with respect
to $\mathscr{B}$ and $\mathscr{E}$ if the following holds: There exists a
unique $h\in L^{\infty}(\Omega,\mathscr{B})$ and a measurable family of
probability measures $\\{\mu_{\omega}\\}_{\omega\in\Omega}$ on $M$ such that

$\mathcal{L}_{\omega}h_{\omega}=h_{\sigma\omega},\quad\langle
h_{\omega},u\rangle=\int
u\,d\mu_{\omega},\quad\operatorname*{ess\,inf}_{\omega\in\Omega}\|h_{\omega}\|\geq
1$

for $\mathbb{P}$-almost every $\omega\in\Omega$ and any $u\in\mathscr{E}$.
Furthermore, there is a constant $C_{2}>0$ and $\rho\in(0,1)$ such that if
$\varphi\in\mathscr{B}$ satisfies $\langle\varphi,1_{M}\rangle=0$, then

(1.9)
$\operatorname*{ess\,sup}_{\omega\in\Omega}\|\mathcal{L}_{\omega}^{(n)}\varphi\|\leq
C_{2}\rho^{n}\|\varphi\|\quad\text{for all $n\in\mathbb{N}$.}$

######  Definition 1.4 (Lasota-Yorke inequality condition).

We say that $T$ satisfies the Lasota-Yorke inequality condition (LY) with
respect to $\mathscr{B}\subset\mathscr{B}_{+}$ if the following holds: either

  1. $\mathrm{(i)}$

$(\Omega,\mathcal{F},\mathbb{P})$ is a Polish space, $\sigma$ is a
homeomorphism, and $\mathcal{L}$ is
$(\mathcal{F},\mathcal{B}_{L(\mathscr{B})})$-measurable, or

  2. $\mathrm{(ii)}$

$(\Omega,\mathcal{F},\mathbb{P})$ is a Lebesgue-Rokhlin probability space,
$\mathscr{B}$ is separable, and $\mathcal{L}$ is
$(\mathcal{F},\mathcal{S}_{L(\mathscr{B})})$-measurable.

Furthermore, $(\mathscr{B}_{+},|\cdot|)$ is a Banach space with
$\mathscr{B}\subset\mathscr{B}_{+}\subset\mathscr{D}^{\prime}$, and it is
required that there exists a positive integer $n_{0}$, and random variables
$\alpha,\beta$ with values in $\mathbb{R}_{+}=\\{x>0\\}$ such that the
inclusion $(\mathscr{B},\|\cdot\|)\hookrightarrow(\mathscr{B}_{+},|\cdot|)$ is
compact, $\mathcal{L}$ is $\mathbb{P}$-almost surely bounded on
$\mathscr{B}_{+}$, and

(1.10)
$\|\mathcal{L}_{\omega}^{(n_{0})}\varphi\|\leq\alpha_{\omega}\|\varphi\|+\beta_{\omega}|\varphi|,\quad\varphi\in\mathscr{B},\quad\text{$\mathbb{P}$-almost
every $\omega\in\Omega$},$

with $\mathbb{E}_{\mathbb{P}}\left[\alpha\right]<1$ and
$\mathbb{E}_{\mathbb{P}}\left[\beta\right]<\infty.$

######  Remark.

The expectation $\mathbb{E}_{\mathbb{P}}[u]$ of $u$ with respect to a
probability measure $\mathbb{P}$ is $\mathbb{E}_{\mathbb{P}}[u]=\int
u\,d\mathbb{P}$. It follows from (1.6) and (UG) that

(1.11) $\mathbb{E}_{\mu_{\omega}}[u\circ
T_{\omega}]=\mathbb{E}_{\mu_{\sigma\omega}}[u]$

for any $u\in\mathscr{E}$. Moreover, for several random dynamics including
random U(1) extensions of expanding maps in Subsection 1.1, one can also show
that $\\{\mu_{\omega}\\}_{\omega\in\Omega}$ is $T$-invariant, i.e.

$\mu_{\omega}(T_{\omega}^{-1}A)=\mu_{\sigma\omega}(A)$

for $\mathbb{P}$-almost every $\omega\in\Omega$ and any $A\in\mathcal{G}$ (or
equivalently, (1.11) with $\\{1_{A}\mid A\in\mathcal{G}\\}$ in place of
$\mathscr{E}$), although in this paper we will not use this property (use
(1.11) instead) to prove quenched limit theorems. Another remark on (UG) is
that for any positive $h_{\omega}\in(\mathscr{C}^{0}(M))^{\prime}$ with
$\langle h_{\omega},1_{M}\rangle=1$, there exists a unique probability measure
$\mu_{\omega}$ such that $\langle h_{\omega},u\rangle=\int u\,d\mu_{\omega}$
by the Riesz-Markov-Kakutani representation theorem, and that
$\operatorname*{ess\,inf}_{\omega\in\Omega}\|h_{\omega}\|\geq 1$ if
$\|\cdot\|\geq\|\cdot\|_{L^{1}}$.

The setting of (i) in (LY) is required in order to apply the Froyland-Lloyd-
Quas version of multiplicative ergodic theorem ([FLQ2013]), while the setting
of (ii) comes from the González-Tokman-Quas version of multiplicative ergodic
theorem ([GQ2014]), see Subsection 1.4. We also remark that it follows from
[FLQ2013, Remark 13] that when $(\Omega,\mathcal{F},\mathbb{P})$ is a Polish
space, the $(\mathcal{F},\mathcal{B}_{L(\mathscr{B})})$-measurability of
$\mathcal{L}$ is equivalent to the $\mathbb{P}$-continuity of $\mathcal{L}$
(i.e. there is a countable collection of Borel sets
$\\{A_{n}\\}_{n\in\mathbb{N}}$ such that $\bigcup_{n\in\mathbb{N}}A_{n}$ has
$\mathbb{P}$-full measure and the restriction of $\mathcal{L}$ on each $A_{n}$
is continuous), which is exactly the condition used in [DFGV2018, DFGV2019].

The abstract setting considered here is quite natural. Dynamical systems
satisfying conditions (UG) and (LY) can be found in [DFGV2018, DFGV2019]:

  * •

For random piecewise expanding maps considered in [DFGV2018], the conditions
can be verified with
$\|\cdot\|_{\mathscr{E}}=\|\cdot\|_{L^{1}}+\mathrm{var}(\cdot)+\|\cdot\|_{L^{\infty}}$,
$\|\cdot\|=\|\cdot\|_{L^{1}}+\mathrm{var}(\cdot)$ and
$|\cdot|=\|\cdot\|_{L^{1}}$, where $\mathrm{var}(\cdot)$ is the total
variation.

  * •

For random hyperbolic maps considered in [DFGV2019], the conditions are
verified with $\|\cdot\|_{\mathscr{E}}=\|\cdot\|_{\mathscr{C}^{r}}$ with
$r>2$, $\|\cdot\|=\|\cdot\|_{1,1}$ and $|\cdot|=\|\cdot\|_{0,2}$, where
$\|\cdot\|_{p,q}$ is the Gouëzel-Liverani’s scale of norms given in (2.2) of
[GL2006].

Furthermore, in Section 2 we show the following.

######  Theorem 1.5.

Let $T$ be the random U(1) extension of an expanding map on the circle with a
noise level $\epsilon>0$ described in Subsection 1.1. Assume that
$(\Omega,\mathcal{F},\mathbb{P})$ is a Lebesgue-Rokhlin probability space and
$T_{0}$ satisfies the partial captivity condition. Then for any sufficiently
small $\epsilon>0$ and any sufficiently large integer $m$, the usual Sobolev
space $H^{m}(\mathbb{T}^{2})$ is admissible for $T\equiv T(\epsilon)$, and
$H^{r}(\mathbb{T}^{2})$ is an associated space of observables for all $r\geq
m$. Moreover, $T$ satisfies the uniform spectral gap condition $\mathrm{(UG)}$
with respect to $H^{m}(\mathbb{T}^{2})$ and $H^{r}(\mathbb{T}^{2})$ and the
Lasota-Yorke inequality condition $\mathrm{(LY)}$ with respect to
$H^{m}(\mathbb{T}^{2})\subset H^{m-1}(\mathbb{T}^{2})$.

The definition of the Sobolev space will be given in Subsection 2.1. The above
regularity index $m$ only depends on the unperturbed map $T_{0}$, see Remark
2.2.

######  Remark.

For the above examples, the most difficult condition to verify is (1.9) in
(UG): note that the coefficient $C_{2}$ in (1.9) needs to be independent of
$\omega$. The techniques to show (1.9) in [DFGV2018], [DFGV2019] and this
paper are quite different. In particular, we will prove (1.9) for random U(1)
extensions of expanding maps without using any abstract perturbation lemma,
see Subsection 2.3 and compare it with [DFGV2019, (10)].

We also remark that for the analysis of random U(1) extensions of expanding
maps, we employ the setting (ii) in (LY). This is a difference from [DFGV2018]
and the piecewise hyperbolic part of [DFGV2019], that used the setting (i) and
needed in practice the restriction that $\\{T_{\omega}\mid\omega\in\Omega\\}$
is countable.

###  1.3. Limit theorems

Let $\\{\mu_{\omega}\\}_{\omega\in\Omega}$ be the measurable family of
probability measures on $M$ provided by (UG). Define a probability measure
$\mu$ on $\Omega\times M$ by $\mu(\Gamma\times
A)=\int_{\Gamma}\mu_{\omega}(A)\mathbb{P}(d\omega)$ for each measurable set
$\Gamma\subset\Omega$ and $A\subset M$. Fix also $g\in
L^{\infty}(\Omega,\mathscr{E})$ (i.e. $g$ is a measurable map from $\Omega$ to
$\mathscr{E}$ with
$\operatorname*{ess\,sup}_{\omega}\|g_{\omega}\|_{\mathscr{E}}<\infty$). For
simplicity, we will assume that

(1.12) $\mathbb{E}_{\mu_{\omega}}\left[g_{\omega}\right]=0\quad\text{for
$\mathbb{P}$-almost every $\omega\in\Omega$.}$

(Note that for any $g\in L^{\infty}(\Omega,\mathscr{E})$, if we let
$\widetilde{g}_{\omega}=g_{\omega}-\mathbb{E}_{\mu_{\omega}}[g_{\omega}]$,
called the centering, then
$\mathbb{E}_{\mu_{\omega}}[\widetilde{g}_{\omega}]=0$.) Our first main result
concerns exponential decay of correlation functions.

######  Theorem 1.6 (Exponential decay of correlations).

Let $(\mathscr{E},\|\cdot\|_{\mathscr{E}})$ be a seminormed vector space
associated with a $T$-admissible Banach space $(\mathscr{B},\|\cdot\|)$. Let
$g\in L^{\infty}(\Omega,\mathscr{E})$ satisfy (1.12). Assume that
$(\mathrm{UG})$ holds. Then, one can find a constant $C_{g}>0$ such that for
all $u\in\mathscr{E}$, $n\geq 1$ and $\mathbb{P}$-almost every
$\omega\in\Omega$,

(1.13) $\left|\mathbb{E}_{\mu_{\omega}}\big{[}g_{\sigma^{n}\omega}\circ
T_{\omega}^{(n)}u\big{]}\right|\leq C_{g}\rho^{n}\|u\|_{\mathscr{E}}.$

For random U(1) extensions of expanding maps on the circle, exponential decay
of correlation functions was investigated in [NW2015] via a spectral approach.
However, in [NW2015, Theorem 1.6], the coefficient $C_{g}$ in Theorem 1.6
(with $T$ described in §1.1) depended on $\omega$ and we were unable to remove
this dependency. (Note that the dependency prevents one from obtaining (1.14)
below.) By a more careful analysis we are now able to overcome this adversity
mainly by establishing (1.9) in §2.3.

###### Proof of Theorem 1.6.

By (UG) we have $\int u\,d\mu_{\omega}=\langle h_{\omega},u\rangle$ for
$u\in\mathscr{E}$, so

$\mathbb{E}_{\mu_{\omega}}\big{[}g_{\sigma^{n}\omega}\circ
T_{\omega}^{(n)}u\big{]}=\int ug_{\sigma^{n}\omega}\circ
T_{\omega}^{(n)}\,d\mu_{\omega}=\langle uh_{\omega},g_{\sigma^{n}\omega}\circ
T_{\omega}^{(n)}\rangle,$

where the right-hand side equals
$\langle\mathcal{L}^{(n)}_{\omega}(uh_{\omega}),g_{\sigma^{n}\omega}\rangle$
by (1.6). We would like to use (1.9) to estimate
$\mathcal{L}^{(n)}_{\omega}(uh_{\omega})$, but $\langle
uh_{\omega},1_{M}\rangle\neq 0$ in general so this requires modification.
Thus, we note that (1.11) and the centering assumption (1.12) gives

$\langle\mathcal{L}^{(n)}_{\omega}(h_{\omega}),g_{\sigma^{n}\omega}\rangle=\int
g_{\sigma^{n}\omega}\circ T^{(n)}_{\omega}\,d\mu_{\omega}=\int
g_{\sigma^{n}\omega}\,d\mu_{\sigma^{n}\omega}=\mathbb{E}_{\mu_{\sigma^{n}\omega}}[g_{\sigma^{n}\omega}]=0,$

which means that

$\displaystyle\mathbb{E}_{\mu_{\omega}}\big{[}g_{\sigma^{n}\omega}\circ
T_{\omega}^{(n)}u\big{]}$
$\displaystyle=\langle\mathcal{L}^{(n)}_{\omega}(uh_{\omega}),g_{\sigma^{n}\omega}\rangle-\langle\mathcal{L}^{(n)}_{\omega}(h_{\omega}),g_{\sigma^{n}\omega}\rangle\mathbb{E}_{\mu_{\omega}}[u]$
$\displaystyle=\langle\mathcal{L}^{(n)}_{\omega}((u-\mathbb{E}_{\mu_{\omega}}[u])h_{\omega}),g_{\sigma^{n}\omega}\rangle.$

Since $\langle(u-\mathbb{E}_{\mu_{\omega}}[u])h_{\omega},1_{M}\rangle=\langle
h_{\omega},u\rangle-\mathbb{E}_{\mu_{\omega}}[u]\langle
h_{\omega},1_{M}\rangle=0$, we can now use $\mathrm{(A2)}$ and (1.9) to obtain

$\displaystyle\left|\mathbb{E}_{\mu_{\omega}}\big{[}g_{\sigma^{n}\omega}\circ
T_{\omega}^{(n)}u\big{]}\right|$ $\displaystyle\leq
C_{1}\lVert\mathcal{L}^{(n)}_{\omega}((u-\mathbb{E}_{\mu_{\omega}}[u])h_{\omega})\rVert\lVert
g_{\sigma^{n}\omega}\rVert_{\mathscr{E}}$ $\displaystyle\leq
C_{1}C_{2}\rho^{n}G(1+H)H\lVert u\rVert_{\mathscr{E}}$

where $G=\operatorname*{ess\,sup}_{\omega}\|g_{\omega}\|_{\mathscr{E}}$ and
$H=\operatorname*{ess\,sup}_{\omega}\|h_{\omega}\|$. This completes the proof.
∎

Obviously, Theorem 1.6 implies exponential decay of annealed correlation
functions:

(1.14) $\left|\mathbb{E}_{\mu}\big{[}g\circ\mathsf{T}^{n}u\big{]}\right|\leq
C_{g}\rho^{n}\|u\|_{\mathscr{E}}$

for every $u\in\mathscr{E}$ and $n\geq 1$, where $\mathsf{T}:\Omega\times
M\to\Omega\times M$ is the skew-product map induced by $(T,\sigma)$, i.e.
$\mathsf{T}(\omega,x)=(\sigma\omega,T_{\omega}(x))$ for
$(\omega,x)\in\Omega\times M$. Furthermore, it easily follows from Theorem 1.6
that for $\mu$-almost every $(\omega,x)\in\Omega\times M$,

$\lim_{n\to\infty}\frac{(S_{n}g)_{\omega}(x)}{n}=0,$

if for each $A\in\mathcal{G}$, $\tilde{u}\in\mathscr{E}$ and
$\mathbb{P}$-almost every $\omega$ there is a sequence $(u_{n})_{n\geq
1}\subset\mathscr{E}$ such that
$\mathbb{E}_{\mu_{\omega}}\left[\tilde{u}(1_{A}-u_{n})\right]\to 0$ as
$n\to\infty$ (examples in/above Theorem 1.5 indeed satisfy this condition).
That is, the quenched strong law of large numbers (or ergodicity) holds under
a mild condition. We refer to [BDV04] for the relation between fast decay of
correlation functions and various limit theorems for deterministic systems.

To state our other results, we further assume that $g$ is $\mathbb{P}$-almost
surely a real-valued function. We define the asymptotic variance $V$ for $g$
given by Furthermore, I originally defined $V$ in a more “intuitive” manner
and planed to show that it coincides with this new $V$, which looks more
“technical” but naturally appears in the proof.

(1.15) $V=\mathbb{E}_{\mathbb{P}}[v],\quad
v_{\omega}=\mathbb{E}_{\mu_{\omega}}\left[g_{\omega}^{2}+2\sum_{n=1}^{\infty}g_{\omega}\cdot
g_{\sigma^{n}\omega}\circ T_{\omega}^{(n)}\right]$

(the existence and boundedness of the sum is ensured by (A2) and Theorem 1.6).
We say that $g$ is _non-degenerate_ if $V>0$, which was shown in [DFGV2018b]
to be equivalent to the non-coboundary condition of $g$ (i.e. non-existence of
$\phi\in L^{2}(\Omega\times M)$ such that $g=\phi-\phi\circ\mathsf{T}$) when
$T_{\omega}$ is a piecewise expanding map.

######  Theorem 1.7 (Central limit theorem).

Assume that the conditions in Theorem 1.6 together with $(\mathrm{LY})$ hold.
Assume also that $g$ is non-degenerate. Then, for $\mathbb{P}$-almost every
$\omega\in\Omega$, $\frac{(S_{n}g)_{\omega}}{\sqrt{n}}$ convergences in
distribution to a normal distribution with mean $0$ and variance $V$ as
$n\to\infty$, i.e. for any $a\in\mathbb{R}$, we have

$\lim_{n\to\infty}\mu_{\omega}\left((S_{n}g)_{\omega}\leq
a\sqrt{n}\right)=\frac{1}{\sqrt{2\pi
V}}\int_{-\infty}^{a}e^{-\frac{z^{2}}{2V}}dz.$

######  Theorem 1.8 (Large deviation principle).

Assume that the conditions in Theorem 1.7 hold. Then, one can find a
nonnegative, continuous, strictly convex function
$c:(-\delta_{0},\delta_{0})\to\mathbb{R}$ with some $\delta_{0}>0$ such that
$c$ vanishes at $0$ and that for $\mathbb{P}$-almost every $\omega\in\Omega$
and any $\delta\in(0,\delta_{0})$,

(1.16)
$\lim_{n\to\infty}\frac{1}{n}\log\mu_{\omega}((S_{n}g)_{\omega}>n\delta)=-c(\delta).$

######  Theorem 1.9 (Local central limit theorem).

Assume that the conditions in Theorem 1.7 hold. Assume also that the condition
$\mathrm{(L)}$ (see Definition 1.13) holds. Then, for $\mathbb{P}$-almost
every $\omega\in\Omega$ and every bounded interval $J\subset\mathbb{R}$, we
have

$\lim_{n\to\infty}\sup_{z\in\mathbb{R}}\left|\sqrt{nV}\mu_{\omega}\left(z+(S_{n}g)_{\omega}\in
J\right)-\frac{1}{\sqrt{2\pi}}e^{-\frac{z^{2}}{2nV}}|J|\right|=0,$

where $|J|$ denotes the length of $J$.

The proofs of Theorems 1.7, 1.8 and 1.9 will be given in Section 3. An
excellent survey paper for limit theorems (including central limit theorems,
large deviation principles and local central limit theorems) for deterministic
dynamical systems is [Gouezel2015]. For limit theorems of random dynamical
systems, see [ANV2015, DFGV2018, DFGV2019, DH2020] and reference therein.
These results were established by the Nagaev-Guivarc’h perturbative spectral
method explained in §1.4, which indicates that a variety of other limit
theorems (such as almost sure invariance principles, moderate deviations
principles and Berry-Esseen theorems) also may hold for random dynamical
systems in our abstract setting.

###  1.4. The Nagaev-Guivarc’h perturbative spectral method

Here we briefly recall the idea of the Nagaev-Guivarc’h perturbative spectral
method and the difficulties that have to be overcome in order for it to be
used for quenched schemes. This will heuristically explain why both (UG) and
(LY) are needed, and provide preparation for definition of condition (L).

The _twisted transfer operator_
$\mathcal{L}_{\theta,\omega}:\mathscr{D}^{\prime}\to\mathscr{D}^{\prime}$ (of
$T_{\omega}$ associated with $g_{\omega}$) with $\theta\in\mathbb{C}$ and
$\omega\in\Omega$ is given by

(1.17) $\mathcal{L}_{\theta,\omega}\varphi=\mathcal{L}_{\omega}(e^{\theta
g_{\omega}}\varphi),\quad\varphi\in\mathscr{D}^{\prime}.$

(Recall (1.8) for the multiplication of $e^{\theta g_{\omega}}$ and
$\varphi$.) By $\mathrm{(A1)}$ and $\mathrm{(A2)}$,
$\mathcal{L}_{\theta,\omega}$ is bounded on $\mathscr{B}$ for
$\mathbb{P}$-almost every $\omega\in\Omega$ and

(1.18)
$\operatorname*{ess\,sup}_{\omega\in\Omega}\|\mathcal{L}_{\theta,\omega}\varphi-\mathcal{L}_{\omega}\varphi\|\leq
C_{0}|\theta|Ge^{|\theta|C_{1}G}\|\varphi\|,\quad\varphi\in\mathscr{B},$

where $G=\operatorname*{ess\,sup}_{\omega}\|g_{\omega}\|_{\mathscr{E}}$. In a
manner similar to the one in the definition of $(\mathcal{L},\sigma)$, we
define a cocycle
$\mathbb{N}_{0}\times\Omega\times\mathscr{B}\ni(n,\omega,\varphi)\mapsto\mathcal{L}_{\theta,\omega}^{(n)}\varphi$
over $\sigma$, and denote it by $(\mathcal{L}_{\theta},\sigma)$. By repeating
the arguments in the proof of Lemma 3.3 of [DFGV2019], one can show that for
any $\theta\in\mathbb{C}$, $\mathbb{P}$-almost every $\omega\in\Omega$ and any
$\varphi\in\mathscr{B}$,

(1.19)
$\langle\varphi,e^{\theta(S_{n}g)_{\omega}}\rangle=\langle\mathcal{L}^{(n)}_{\theta,\omega}\varphi,1_{M}\rangle.$

For a random process $\\{u_{n}\\}_{n\geq 1}$ on a probability space
$(M,\mathcal{G},\nu)$, the moment generating function $\mathbb{R}\ni
t\mapsto\mathbb{E}_{\nu}[e^{tu_{n}}]$ and the characteristic function
$\mathbb{R}\ni t\mapsto\mathbb{E}_{\nu}[e^{itu_{n}}]$ of $u_{n}$’s (around
$t=0$) play a fundamental role in the study of the asymptotic behavior of
$\\{u_{n}\\}_{n\geq 1}$ (see e.g. [Durrett2019]). Hence, to understand the
asymptotic behavior of $(S_{n}g)_{\omega}$ with respect to $\mu_{\omega}$, it
is of great importance to investigate

$\mathbb{E}_{\mu_{\omega}}[e^{\theta(S_{n}g)_{\omega}}]=\langle
h_{\omega},e^{\theta(S_{n}g)_{\omega}}\rangle=\langle\mathcal{L}^{(n)}_{\theta,\omega}h_{\omega},1_{M}\rangle$

around $\theta=0$. (The first identity follows from (UG) and the second from
(1.19).) Therefore, it is natural to expect that several limit theorems should
follow from $(\mathcal{L}_{\theta},\sigma)$ having nice spectral properties
around $\theta=0$. This idea is called _Nagaev-Guivarc’h perturbative spectral
method_ , and has been broadly applied to show limit theorems for abundant
deterministic dynamical systems (see [HH2000, Gouezel2015, ANV2015, DFGV2018]
and the references therein; this method was originally applied to Markov
chains [HH2000]).

The difficulty in applying a (Nagaev-Guivarc’h type) spectral method for
quenched schemes is that one should consider a spectral analysis for operator
_cocycles_ , and the usual notion of spectrum for a single operator is not
useful. This was overcome in [DFGV2018] by using the theory of Lyapunov
spectrum. The notion of Lyapunov spectrum was also used in [NW2015] to study
decay of correlation functions for quenched schemes.

Let $\mathscr{H}$ be a Banach space and let $\mathcal{A}$ be a measurable
mapping from $\Omega$ to the set of bounded operators on $\mathscr{H}$
(endowed with the operator norm) such that
$\omega\mapsto\log^{+}\|\mathcal{A}_{\omega}\|$ is $\mathbb{P}$-integrable,
and consider the cocycle
$(n,\omega,\varphi)\mapsto\mathcal{A}_{\omega}^{(n)}\varphi=\mathcal{A}_{\sigma^{n-1}\omega}\circ\cdots\circ\mathcal{A}_{\omega}$
over $\sigma$ (denoted by $(\mathcal{A},\sigma)$). Then, it follows from
Kingman’s sub-additive ergodic theorem that the following limits (called the
_maximal Lyapunov exponent_ and the _index of compactness_ of $\mathcal{A}$)
exist and are $\mathbb{P}$-almost surely independent of $\omega\in\Omega$:

(1.20)
$\Lambda(\mathcal{A})=\lim_{n\to\infty}\frac{1}{n}\log\|\mathcal{A}^{(n)}_{\omega}\|,\quad\kappa(\mathcal{A})=\lim_{n\to\infty}\frac{1}{n}\log\mathrm{ic}(\mathcal{A}^{(n)}_{\omega}),$

where $\mathrm{ic}(\mathcal{A}_{0})$ is the index of compactness of a bounded
operator $\mathcal{A}_{0}:\mathscr{H}\to\mathscr{H}$ (i.e.
$\mathrm{ic}(\mathcal{A}_{0})$ is the infimum of $r>0$ such that the image of
the unit ball of $\mathscr{H}$ by $\mathcal{A}_{0}$ can be covered with
finitely many balls of radius $r$). The cocycle $(\mathcal{A},\sigma)$ is
called _quasi-compact_ if $\Lambda(\mathcal{A})>\kappa(\mathcal{A})$. The
notion of quasi-compactness for cocycles of linear operators was introduced by
Thieullen [Thieullen1987], and was key for establishing the multiplicative
ergodic theorem on Banach spaces (see [FLQ2013, GQ2014] and reference
therein). We shall use the fact that quasi-compact cocycles admit an Oseledets
decomposition in the following sense.

######  Definition 1.10.

We say that there is an Oseledets decomposition of $(\mathcal{A},\sigma)$ on
$\mathscr{H}$ if there is a real number
$\widetilde{\Lambda}(\mathcal{A})<\Lambda(\mathcal{A})$, a measurable
splitting of $\mathscr{H}$ into closed subspaces
$\mathscr{H}=\mathscr{F}_{\omega}\oplus\mathscr{R}_{\omega}$ with
$\dim{\mathscr{F}_{\omega}}<\infty$, and a $\sigma$-invariant subset
$\tilde{\Omega}\subset\Omega$ of full measure such that for each
$\omega\in\tilde{\Omega}$, the following holds:

  1. $\mathrm{(1)}$

The splitting is semi-invariant:

$\mathcal{A}_{\omega}\mathscr{F}_{\omega}=\mathscr{F}_{\sigma\omega}\quad\text{and}\quad\mathcal{A}_{\omega}\mathscr{R}_{\omega}\subset\mathscr{R}_{\sigma\omega}.$

  2. $\mathrm{(2)}$

If $\varphi\in\mathscr{F}_{\omega}\backslash\\{0\\}$, then

$\lim_{n\rightarrow\infty}\frac{1}{n}\log\lVert\mathcal{A}_{\omega}^{(n)}\varphi\rVert=\Lambda(\mathcal{A}).$

  3. $\mathrm{(3)}$

If $\varphi\in\mathscr{R}_{\omega}$, then

$\limsup_{n\rightarrow\infty}\frac{1}{n}\log\lVert\mathcal{A}_{\omega}^{(n)}\varphi\rVert\leq\widetilde{\Lambda}(\mathcal{A}).$

It can be easily shown that (UG) implies that $(\mathcal{L},\sigma)$ admits an
Oseledets decomposition
$\mathscr{B}=\mathscr{F}_{\omega}\oplus\mathscr{R}_{\omega}$ with
$\mathscr{F}_{\omega}=\\{\pi_{\omega}(f)\mid f\in\mathscr{B}\\}$,
$\mathscr{R}_{\omega}=\\{f-\pi_{\omega}(f)\mid f\in\mathscr{B}\\}$,
$\Lambda(\mathcal{L})=1$ and $\widetilde{\Lambda}(\mathcal{L})=\rho$, where
$\pi_{\omega}(f)=\langle h_{\omega},f\rangle h_{\omega}$ and $\rho$ is the
decay rate in (1.9). On the other hand,
$\mathbb{E}_{\mathbb{P}}[\log^{+}\lVert\mathcal{L}\rVert]<\infty$ by (A1),
where
$\log^{+}\|\mathcal{L}\|(\omega)=\max\left\\{\log\|\mathcal{L}_{\omega}\|,0\right\\}$
for $\omega\in\Omega$. Hence, using the multiplicative ergodic theorems in
[FLQ2013, GQ2014], it can also be shown that (LY) implies that
$(\mathcal{L},\sigma)$ admits an Oseledets decomposition on $\mathscr{B}$
([FLQ2013] for the case (i) of (LY) and [GQ2014] for the case (ii)). We
consider the following condition (recall that $\mathcal{L}_{\theta}$ depends
on $g$).

######  Definition 1.11 (Oseledets decomposition condition).

We say that $(T,g)$ satisfies the Oseledets decomposition condition (OD) if
$\mathcal{L}_{\theta}$ admits an Oseledets decomposition on $\mathscr{B}$ for
any $\theta\in\mathbb{C}$ with sufficiently small absolute value.

We note that _the proofs of Theorems 1.7, 1.8, 1.9 work with_ (OD) _instead
of_ (LY). However, it is unclear to us whether the Oseledets decomposition of
$(\mathcal{L},\sigma)$ is stable under the twisting perturbation, i.e. whether
the Oseledets decomposition of $(\mathcal{L},\sigma)$ implies (OD). On the
other hand, the stability of Lasota-Yorke inequality is easily obtained as in
the following proposition, which is the reason why (not only (UG) but also)
(LY) is assumed. In particular, if $T$ satisfies (LY) then $(T,g)$ satisfies
(OD) for _arbitrary $g\in L^{\infty}(\Omega,\mathscr{E})$_.

######  Proposition 1.12.

Assume that $T$ satisfies $(\mathrm{LY})$. Then the Lasota-Yorke inequality
(1.10) holds with $\mathcal{L}_{\theta}$ instead of $\mathcal{L}$ for any
$\theta\in\mathbb{C}$ with sufficiently small absolute value.

###### Proof.

Note that for any $n\geq 1$ and $\omega\in\Omega$,

$\mathcal{L}_{\theta,\omega}^{(n)}-\mathcal{L}_{\omega}^{(n)}=\sum_{j=0}^{n-1}\mathcal{L}_{\theta,\sigma^{n-j}\omega}^{(j)}(\mathcal{L}_{\theta,\sigma^{n-j-1}\omega}-\mathcal{L}_{\sigma^{n-j-1}\omega})\mathcal{L}_{\omega}^{(n-j-1)}.$

Hence, it follows from $\mathrm{(A1)}$ and (1.18) that there is a constant
$C\equiv C(g,n)>0$ such that

$\|\mathcal{L}_{\theta,\omega}^{(n)}\varphi-\mathcal{L}_{\omega}^{(n)}\varphi\|\leq
C|\theta|e^{C|\theta|}\|\varphi\|,\quad\varphi\in\mathscr{B}.$

This immediately leads to the implication that $\mathcal{L}_{\theta}$
satisfies (1.10) (with $\alpha$ replaced by $\alpha+C|\theta|e^{C|\theta|}$)
for any $\theta$ satisfying $E_{\mathbb{P}}[\alpha]+C|\theta|e^{C|\theta|}<1$.

We next show that in the case when $(\Omega,\mathcal{F},\mathbb{P})$ is a
Polish space, the $(\mathcal{F},\mathcal{B}_{L(\mathscr{B})})$-measurability
of $\mathcal{L}$ implies that of $\mathcal{L}_{\theta}$. Recall that, since
$(\Omega,\mathcal{F},\mathbb{P})$ is a Polish space, for any topological space
$\mathscr{V}$ and any map $\mathcal{A}:\Omega\to\mathscr{V}$, the
$(\mathcal{F},\mathcal{B}_{\mathscr{V}})$-measurability of $\mathcal{A}$ is
equivalent to the $\mathbb{P}$-continuity of $\mathcal{A}$, see [FLQ2013,
Remark 13]. Assume that $\mathcal{L}:\Omega\to L(\mathscr{B})$ is
$\mathbb{P}$-continuous. Then, since $g:\Omega\to\mathscr{E}$ is also
measurable, one can find a countable collection of Borel sets
$\\{A_{n}\\}_{n\in\mathbb{N}}$ such that $\bigcup_{n\in\mathbb{N}}A_{n}$ has
$\mathbb{P}$-full measure and both the restriction of $\mathcal{L}$ and $g$ on
$A_{n}$ are continuous for each $n\in\mathbb{N}$. Notice that for each
$n\in\mathbb{N}$, the restriction of $\mathcal{L}_{\theta}$ on $A_{n}$ is the
composition of a map $A_{n}\times\mathscr{E}\to
L(\mathscr{B}):(\omega,g_{0})\mapsto\mathcal{L}_{\omega}(e^{\theta
g_{0}}\cdot)$ and a continuous map $A_{n}\to
A_{n}\times\mathscr{E}:\omega\mapsto(\omega,g_{\omega})$. On the other hand,
it is easy to see that for each $\omega\in A_{n}$ the map
$g_{0}\mapsto\mathcal{L}_{\omega}(e^{\theta g_{0}}\cdot)$ is a continuous map
from $\mathscr{E}$ to $L(\mathscr{B})$ because

$\|\mathcal{L}_{\omega}(e^{\theta
g_{0}}\varphi)-\mathcal{L}_{\omega}(e^{\theta g_{0}^{\prime}}\varphi)\|\leq
C_{0}\|e^{\theta
g_{0}}\|_{\mathscr{E}}|\theta|\|g_{0}-g_{0}^{\prime}\|_{\mathscr{E}}\times
e^{|\theta|C_{1}\|g_{0}-g_{0}^{\prime}\|_{\mathscr{E}}}$

for each $\varphi\in\mathscr{B}$ with $\|\varphi\|=1$ and
$g_{0},g_{0}^{\prime}\in\mathscr{E}$ (recall the calculation to obtain
(1.18)). Moreover, for each $g_{0}\in\mathscr{E}$ the map
$\omega\mapsto\mathcal{L}_{\omega}(e^{\theta g_{0}}\cdot)$ is a continuous map
from $A_{n}$ to $L(\mathscr{B})$ due to the continuity of the restriction of
$\mathcal{L}$ on $A_{n}$ and (A2). In conclusion, $\mathcal{L}_{\theta}$ is
continuous on each $A_{n}$, which implies the
$(\mathcal{F},\mathcal{B}_{L(\mathscr{B})})$-measurability of
$\mathcal{L}_{\theta}$.

Finally, we show that in the case when $(\Omega,\mathcal{F},\mathbb{P})$ is a
Lebesgue-Rokhlin probability space and $\mathscr{B}$ is separable, the
$(\mathcal{F},\mathcal{S}_{L(\mathscr{B})})$-measurability of $\mathcal{L}$
implies that of $\mathcal{L}_{\theta}$. Recall that the
$(\mathcal{F},\mathcal{S}_{L(\mathscr{B})})$-measurability of a map
$\mathcal{A}:\Omega\to L(\mathscr{B})$ is equivalent to the strong
measurability of $\mathcal{A}$ (i.e. for any $\varphi\in\mathscr{B}$, the map
$\omega\mapsto\mathcal{A}(\omega)\varphi$ is
$(\mathcal{F},\mathcal{B}_{\mathscr{B}})$-measurable), see [GQ2014, Appendix
A]. Assume the strong measurability of $\mathcal{L}$, and fix
$\varphi\in\mathscr{B}$. As in the previous paragraph,
$\Omega\to\mathscr{B}:\omega\mapsto\mathcal{L}_{\theta,\omega}\varphi$ is the
composition of
$\Omega\times\mathscr{E}\to\mathscr{B}:(\omega,g_{0})\mapsto\mathcal{L}_{\omega}(e^{\theta
g_{0}}\varphi)$ and a measurable map
$\Omega\to\Omega\times\mathscr{E}:\omega\mapsto(\omega,g_{\omega})$.
Furthermore, for each $\omega\in\Omega$ the map $\mathscr{E}\ni
g_{0}\mapsto\mathcal{L}_{\omega}(e^{\theta g_{0}}\varphi)$ is continuous, and
for each $g_{0}\in\mathscr{E}$ the map
$\Omega\ni\omega\mapsto\mathcal{L}_{\omega}(e^{\theta g_{0}}\varphi)$ is
measurable due to the strong measurability of $\mathcal{L}$. Hence, by [CV77,
Lemma 3.14], $(\omega,g_{0})\mapsto\mathcal{L}_{\omega}(e^{\theta
g_{0}}\varphi)$ is measurable, and thus $\mathcal{L}_{\theta}$ is
$(\mathcal{F},\mathcal{S}_{L(\mathscr{B})})$-measurable. ∎

###  1.5. Local central limit theorem

After the preparation in Subsection 1.4, we can precisely give sufficient
conditions under which the local central limit theorem (LCLT) in Theorem 1.9
holds.

For recent progress on LCLT for dynamical systems (with a neutral direction),
see [DN2020, AT2020] and reference therein. As usual in the study of LCLT, we
start this subsection from considering a dichotomy between periodic (lattice,
arithmetic) and aperiodic (non-lattice, non-arithmetic) cases. We define
$\mathcal{U}_{g}$ by

$\mathcal{U}_{g}=\\{t\in\mathbb{R}\mid\Lambda(\mathcal{L}_{it})=0\\}$

(recall that $\mathcal{L}_{it}$ depends on $g$). In the case when $T_{\omega}$
is $\mathbb{P}$-almost surely a mixing piecewise expanding map and
$\mathscr{B}$ is the space of BV functions (i.e. real-valued functions whose
total variation is bounded), Dragičević et al. showed in [DFGV2018, Subsection
4.4] that $\mathcal{U}_{g}$ is a subgroup of $(\mathbb{R},+)$. Furthermore, it
follows from the remark in the proof of Lemma 4.13 of [DFGV2018] that, by
assuming that $g$ is non-degenerate (i.e. $V>0$), $\mathcal{U}_{g}$ is one of
the following two cases:

  * •

$\mathcal{U}_{g}=a\mathbb{Z}$ for some $a>0$ (periodic);

  * •

$\mathcal{U}_{g}=\\{0\\}$ (aperiodic).

This dichotomy is known to hold for wide classes of Markov processes or
deterministic dynamical systems [Durrett2019, HH2000, Iwata2008]. The
conclusion in Theorem 1.9 is a standard form of the LCLT in the aperiodic
case, while the LCLT in the periodic case always needs some modification in
the term $|J|$ (see [Durrett2019, HK, DFGV2018] for a precise description of
the LCLT in the periodic case).

Furthermore, Dragičević et al. also showed that (for their random dynamics and
functional space) the aperiodicity of $g$ is equivalent to the following
useful spectral condition, see [DFGV2018, Subsection 4.3.2]. To express our
respect to [DFGV2019], we keep using their terminology “condition (L)” for the
condition.

######  Definition 1.13.

We say that $T$ satisfies _condition_ (L) for $g$ if for every bounded
interval $J\subset\mathbb{R}\setminus\\{0\\}$, there exist a real number
$\kappa\in(0,1)$ and a random variable $C:\Omega\to(0,\infty)$ such that for
$\mathbb{P}$-almost every $\omega\in\Omega$,

$\|\mathcal{L}_{it,\omega}^{(n)}\|\leq C_{\omega}\kappa^{n}\quad\text{for
$t\in J$ and $n\geq 0$}.$

The deterministic version of condition (L) has been also considered and shown
to be equivalent to the deterministic version of the aperiodicity condition,
see e.g. [HH2000].

Although condition (L) is useful for a functional analytic proof of the LCLT
(and indeed we prove Theorem 1.9 by assuming condition (L)), as pointed out in
[DFGV2019] it may not be easy to verify this condition directly. Therefore, as
in [DFGV2019], we employ a more tractable sufficient condition for condition
(L) by Hafouta and Kifer [HK].

######  Definition 1.14.

We say that $T$ satisfies the _Hafouta-Kifer condition_ (HK) for $g$ if the
following holds:

  * (HK1)

$\mathcal{F}$ is the Borel $\sigma$-algebra on a metric space $\Omega$,
$\mathbb{P}(U)>0$ for any open set $U$, and $\sigma$ is a homeomorphism with a
periodic point $\omega_{0}$ with period $\ell_{0}$ (i.e.
$\sigma^{\ell_{0}}\omega_{0}=\omega_{0}$). Moreover for each
$j\in\\{0,1,\ldots,\ell_{0}-1\\}$, there exists a neighborhood of
$\sigma^{j}\omega_{0}$ on which the map $\omega\mapsto T_{\omega}$ is
constant.

  * (HK2)

For any compact interval $J\subset\mathbb{R}$, the family of maps
$\\{\omega\mapsto\mathcal{L}_{it,\omega}\\}_{t\in J}$ is equicontinuous on the
orbit of $\omega_{0}$. Furthermore, there exists a constant $B\geq 1$ such
that for $\mathbb{P}$-almost every $\omega\in\Omega$,

$\|\mathcal{L}_{it,\omega}^{(n)}\|\leq B$

for any $n\in\mathbb{N}$ and $t\in J$.

  * (HK3)

For any compact interval $J\subset R\setminus\\{0\\}$, there exist constants
$c>0$ and $b\in(0,1)$ such that

$\left\|\left(\mathcal{L}_{it,\omega_{0}}^{(\ell_{0})}\right)^{n}\right\|\leq
cb^{n}$

for any $n\in\mathbb{N}$ and $t\in J$.

In [HK, Lemma 2.10.4], Hafouta and Kifer showed (in a more abstract form) that
condition (HK) implies condition (L). Note that (HK1) is rather a restriction
on noise, than a hypothesis of dynamics to be verified. See [DFGV2019, Section
9] where it is explained that the restriction of (HK1) is mild.

Let $m$ be the integer given in Theorem 1.5 and fix $r\geq m$, so that $g$
belongs to $L^{\infty}(\Omega,H^{r}(\mathbb{T}^{2}))$. Assume that $g$
satisfies the centering condition (1.12).

Our final result is the following.

######  Theorem 1.15.

Let $T$ be the random U(1) extension of an expanding map on the circle given
in Theorem 1.5. Furthermore, we assume the following:

  * $\mathrm{(i)}$

$\mathrm{(HK1)}$ holds;

  * $\mathrm{(ii)}$

$g:\Omega\to H^{r}(\mathbb{T}^{2})$ is continuous on
$\\{\sigma^{j}\omega_{0}\mid 1\leq j\leq\ell_{0}\\}$;

  * $\mathrm{(iii)}$

$(S_{\ell_{0}}g)_{\omega_{0}}$ is not cohomologous to a constant with respect
to $T_{\omega_{0}}^{(\ell_{0})}$, that is, for any $c\in\mathbb{R}$ and
integrable function $\psi:\mathbb{T}^{2}\to\mathbb{R}$, it does not hold that

$(S_{\ell_{0}}g)_{\omega_{0}}=\psi\circ T_{\omega_{0}}^{(\ell_{0})}-\psi+c.$

  * $\mathrm{(iv)}$

The second part of $\mathrm{(HK2)}$ holds. Furthermore,
$\mathcal{L}_{\omega_{0}}^{(\ell_{0})}:H^{m}(\mathbb{T}^{2})\to
H^{m}(\mathbb{T}^{2})$ is a quasi-compact operator whose spectral radius is
bounded by $1$.

Then, $T$ satisfies the Hafouta-Kifer condition $\mathrm{(HK)}$ for $g$. In
particular, $T$ satisfies the condition $\mathrm{(L)}$ for $g$.

Notice that, as (HK1), condition (ii) in Theorem 1.15 is rather a restriction
on observables than a condition to be verified. See [DFGV2019, Remark 9.2] for
the argument that condition (ii) is preserved under the centering by virtue of
(HK1), so the centering condition on $g$ is not essential. A class of examples
satisfying condition (iii) is that $\ell_{0}=1$ and
$g_{\omega_{0}}(x,s)=\widetilde{g}(x)-\mathbb{E}_{\mu_{\omega_{0}}}[\widetilde{g}]$
with some $\widetilde{g}\in\mathscr{V}_{\varrho}$, $1<\varrho<k$, where
$\mathscr{V}_{\varrho}$ is the open and dense subset of
$\mathscr{C}^{r}(\mathbb{S}^{1})$ given in Theorem A.1. Indeed, supposing that
$(S_{m_{0}}g)_{\omega_{0}}=\psi\circ T_{\omega_{0}}-\psi+c$ with some
$c\in\mathbb{R}$ and $\psi\in L^{1}(\mathbb{T}^{2})$ implies that
$\widetilde{g}=\widetilde{\psi}\circ
E_{\omega_{0}}-\widetilde{\psi}+c+\mathbb{E}_{\mu_{\omega_{0}}}[\widetilde{g}]$
with $\widetilde{\psi}=\int\psi(\cdot,s)ds$ due to the translation invariance
of $\mathrm{Leb}_{\mathbb{S}^{1}}$. This contradicts to the fact that each
$\widetilde{g}\in\mathscr{V}_{\varrho}$ is not cohomologous to a constant (see
Appendix A). Therefore, this observable satisfies condition (iii) of Theorem
1.15.

######  Remark 1.16.

We will see in Subsection 2.5 that, under conditions (i) and (iv) of Theorem
1.15, (ii) implies (HK2) and (iii) implies (HK3).

Comparing with [DFGV2019, Corollary 9.5], one may hope to remove condition
(iv). However, it is unclear to us that (iv) holds for general $g$ because
twisted transfer operators do not preserve the Fourier mode (see Subsection
2.5). On the other hand, we will see in Subsection 2.5 that if
$g_{\omega}(x,s)$ does not depend on $s$, then (iv) holds.

##  2\. Hypothesis verification

In this section, we will see that the random U(1) extension of an expanding
map in Section 1 satisfies the conditions appearing in the statements of
Theorems 1.6–1.9, together with the proof of Theorem 1.15. Let therefore $E$,
$\tau$, $T$ be the ones given in (1.4). In particular, $\tau$ is a random
perturbation of a function $\tau_{0}$, where $\tau_{0}$ is guaranteed not to
be cohomologous to a constant function by the assumption of partial captivity.
Throughout this section we assume that $(\Omega,\mathcal{F},\mathbb{P})$ is a
Lebesgue-Rokhlin probability space.

### Notation

In Subsection 1.4, we used the double-indexed operator
$\mathcal{L}_{\theta,\omega}$ with $\theta\in\mathbb{C}$, $\omega\in\Omega$.
However, this notation will not appear in this section, so another double-
indexed operator $\mathcal{L}_{\nu,\omega}$ with $\nu\in\mathbb{Z}$,
$\omega\in\Omega$ given in (2.3) should cause no confusion. In particular,
$\mathcal{L}_{0,\omega}$ in this section always means
$\mathcal{L}_{\nu,\omega}$ at $\nu=0$. Furthermore, we will use $C$ and
$C_{n}$ ($n\geq 1$) as positive constants which do not depend on
$\omega\in\Omega$, $\nu\in\mathbb{Z}$ nor $\epsilon\geq 0$, and may change on
occasion.

###  2.1. Preliminaries

We first note that $T_{\omega}$ preserves $L^{2}(\mathbb{T}^{2})\equiv
L^{2}(\mathbb{T}^{2},\mathrm{Leb}_{\mathbb{T}^{2}})$ $\mathbb{P}$-almost
surely, so that we can define the transfer operator $\mathcal{L}_{\omega}$ of
$T_{\omega}$ on $L^{2}(\mathbb{T}^{2})\cong(L^{2}(\mathbb{T}^{2}))^{\prime}$
by (1.6). As in [NW2015] (originally in [Faure]), we employ the following
decomposition in Fourier modes,

(2.1) $L^{2}(\mathbb{T}^{2})=\bigoplus_{\nu\in
2\pi\mathbb{Z}}\mathcal{H}_{\nu},\quad\mathcal{H}_{\nu}=\\{(x,s)\mapsto\psi(x)e^{i\nu
s}:\psi\in L^{2}(\mathbb{S}^{1})\\}.$

The spaces
$(\mathcal{H}_{\nu},\lVert\phantom{i}\rVert_{L^{2}(\mathbb{T}^{2})})$ and
$L^{2}(\mathbb{S}^{1})$ are isometrically isomorphic, and the decomposition
(2.1) is preserved by the pullback operator $u\mapsto u\circ T_{\omega}$.

For given $\nu\in 2\pi\mathbb{Z}$ and fixed $\omega\in\Omega$, let
$\mathcal{M}_{\nu,\omega}^{(n)}$ denote the operator $u\mapsto u\circ
T^{(n)}_{\omega}$ restricted to $\mathcal{H}_{\nu}$. It is straightforward to
check that by identifying $\mathcal{H}_{\nu}$ with $L^{2}(\mathbb{S}^{1})$ we
can view $\mathcal{M}_{\nu,\omega}^{(n)}$ for $\mathbb{P}$-almost every
$\omega$ as an operator on $L^{2}(\mathbb{S}^{1})$ given by

(2.2)
$\mathcal{M}_{\nu,\omega}^{(n)}\psi(x)=\psi\big{(}E^{(n)}_{\omega}(x)\big{)}e^{i\nu\tau^{(n)}_{\omega}(x)},\quad\psi\in
L^{2}(\mathbb{S}^{1}),$

Here $E_{\omega}^{(n)}=E_{\sigma^{n-1}\omega}\circ
E_{\sigma^{n-2}\omega}\circ\cdots\circ E_{\omega}$ and $\tau^{(n)}_{\omega}$
is the quenched Birkhoff sum of $\tau$ given by

$\tau^{(n)}_{\omega}=\sum_{j=0}^{n-1}\tau_{\sigma^{j}\omega}\circ
E_{\omega}^{(j)}\quad(n\geq 1,\ \omega\in\Omega).$

Let $\mathcal{L}_{\nu,\omega}^{(n)}:L^{2}(\mathbb{S}^{1})\to
L^{2}(\mathbb{S}^{1})$ be the adjoint operator of
$\mathcal{M}_{\nu,\omega}^{(n)}$. It is straightforward to check that

(2.3)
$\mathcal{L}_{\nu,\omega}^{(n)}\psi(x)=\sum_{E^{(n)}_{\omega}(y)=x}\frac{e^{-i\nu\tau^{(n)}_{\omega}(y)}}{\frac{d}{dy}E^{(n)}_{\omega}(y)}\psi(y),\quad\psi\in
L^{2}(\mathbb{S}^{1}).$

Note also that for $\varphi\in L^{2}(\mathbb{T}^{2})$ we have

$\mathcal{L}_{\omega}\varphi(x,s)=\sum_{\nu\in
2\pi\mathbb{Z}}e^{is\nu}\mathcal{L}_{\nu,\omega}\varphi_{\nu}(x),\quad\varphi(x,s)=\sum_{\nu\in
2\pi\mathbb{Z}}\varphi_{\nu}(x)e^{i\nu s},$

where $\varphi_{\nu}(x)=\int_{\mathbb{S}^{1}}e^{-is\nu}\varphi(x,s)\,ds$ and
thus

(2.4)
$(\mathcal{L}_{\omega}^{(n)}\varphi)_{\nu}(x)=\int_{\mathbb{S}^{1}}e^{-is\nu}\mathcal{L}_{\omega}^{(n)}\varphi(x,s)ds=\mathcal{L}_{\nu,\omega}^{(n)}\varphi_{\nu}(x),\quad
n\geq 1.$

For $m\geq 0$ we let $H^{m}(\mathbb{S}^{1})\subset L^{2}(\mathbb{S}^{1})$
denote the Sobolev spaces of regularity index $m$, defined as the set of
measurable functions $\psi$ satisfying

$\lVert\psi\rVert_{H^{m}(\mathbb{S}^{1})}^{2}=\sum_{\xi\in
2\pi\mathbb{Z}}(1+\lvert\xi\rvert^{2})^{m}\lvert\hat{\psi}(\xi)\rvert^{2}<\infty.$

The Fourier coefficients of $\psi\in L^{2}(\mathbb{S}^{1})$ are defined by

$\hat{\psi}(\xi)=\langle\psi,\phi_{-\xi}\rangle,\quad\xi\in 2\pi\mathbb{Z},$

where the functions $\phi_{\xi}\in{\mathscr{C}}^{\infty}(\mathbb{S}^{1})$ are
given by $\phi_{\xi}(x)=e^{ix\xi}$ for $\xi\in 2\pi\mathbb{Z}$ and
$x\in\mathbb{S}^{1}$.

We also introduce an alternative $\nu$-dependent norm
$\lVert\cdot\rVert_{H_{\nu}^{m}}$ on $H^{m}(\mathbb{S}^{1})$ defined by

(2.5) $\lVert\psi\rVert_{H_{\nu}^{m}}^{2}=\sum_{\xi\in
2\pi\mathbb{Z}}(1+(\xi/\nu)^{2})^{m}\lvert\hat{\psi}(\xi)\rvert^{2},\quad\psi\in
L^{2}(\mathbb{S}^{1}),\quad 0\neq\nu\in\mathbb{Z}.$

Let $H_{\nu}^{m}(\mathbb{S}^{1})$ denote the space $H^{m}(\mathbb{S}^{1})$
equipped with the norm $\lVert\phantom{i}\rVert_{H_{\nu}^{m}}$.

Finally, we let $H^{r}(\mathbb{T}^{2})$ denote the Sobolev space on
$\mathbb{T}^{2}$ with regularity $r$, defined similarly as the set of
measurable functions $\varphi$ satisfying

$\lVert\varphi\rVert_{H^{r}(\mathbb{T}^{2})}^{2}=\sum_{\xi,\nu\in
2\pi\mathbb{Z}}(1+\lvert\xi\rvert^{2}+\lvert\nu\rvert^{2})^{r}\lvert\hat{\varphi}(\xi,\nu)\rvert^{2}<\infty,$

where

$\hat{\varphi}(\xi,\nu)=\int_{\mathbb{T}^{2}}e^{-i(x\xi+s\nu)}\varphi(x,s)dxds,\quad\xi,\nu\in
2\pi\mathbb{Z}.$

We shall work with the space $\mathscr{H}^{m}(\mathbb{T}^{2})\subset
L^{2}(\mathbb{T}^{2})$ defined as the set of measurable functions $\varphi$
satisfying

(2.6)
$\lVert\varphi\rVert_{\mathscr{H}^{m}}^{2}=\lVert\varphi_{0}\rVert_{H^{m}(\mathbb{S}^{1})}^{2}+\sum_{\nu\in
2\pi\mathbb{Z},\;\nu\neq
0}(1+\nu^{2})^{m}\lVert\varphi_{\nu}\rVert_{H^{m}_{\nu}(\mathbb{S}^{1})}^{2}<\infty$

where $\varphi_{\nu}(x)=\int_{\mathbb{S}^{1}}e^{-is\nu}\varphi(x,s)\,ds$ as
before. (This functional space did not appear in [NW2015].) Observe that
computing $\lVert\varphi_{\nu}\rVert_{H^{m}_{\nu}(\mathbb{S}^{1})}$ involves
the Fourier coefficients $\hat{\varphi}_{\nu}(\xi)$ of
$x\mapsto\varphi_{\nu}(x)$. Naturally, these are precisely the Fourier
coefficients $\hat{\varphi}(\xi,\nu)$ of $(x,s)\mapsto\varphi(x,s)$, since
$\hat{\varphi}_{\nu}(\xi)=\int_{\mathbb{S}^{1}}e^{-ix\xi}\varphi_{\nu}(x)\,dx=\int_{\mathbb{S}^{1}}e^{-ix\xi}\int_{\mathbb{S}^{1}}e^{-is\nu}\varphi(x,s)\,dxds$.
In fact, the somewhat odd-looking norm in (2.6) is equivalent to the usual
Sobolev norm:

######  Lemma 2.1.

$\lVert\cdot\rVert_{\mathscr{H}^{m}}$ is equivalent to
$\lVert\cdot\rVert_{H^{m}(\mathbb{T}^{2})}$.

###### Proof.

Note that for any $\xi,\nu\in 2\pi\mathbb{Z}$ with $\nu\neq 0$,

$\frac{(1+\xi^{2}+\nu^{2})}{(1+\nu^{2})(1+(\xi/\nu)^{2})}=\frac{1+\xi^{2}+\nu^{2}}{\xi^{2}+\nu^{2}}\frac{\nu^{2}}{1+\nu^{2}}=\frac{1+\frac{1}{\xi^{2}+\nu^{2}}}{1+\frac{1}{\nu^{2}}}$

is in an interval $(\frac{1}{2},1]$. Indeed, the right-hand side is is clearly
bounded by 1, and by inserting $\nu=1$ and letting $\lvert\xi\rvert\to\infty$
we see it is bounded from below by $\frac{1}{2}$. (Note that since $\nu\in
2\pi\mathbb{Z}$, the choice $\nu=1$ is actually illegal but the estimate is
clearly valid.) Hence the lemma immediately follows in view of the definitions
of the two norms. ∎

###  2.2. Conditions $\mathrm{(A1)}$ and $\mathrm{(A2)}$

We begin by showing that conditions $\mathrm{(A1)}$ and $\mathrm{(A2)}$ are
satisfied when $\mathscr{D}=L^{2}(\mathbb{T}^{2})$,
$\mathscr{B}=\mathscr{H}^{m}(\mathbb{T}^{2})$ and
$\mathscr{E}=H^{r}(\mathbb{T}^{2})$ for some sufficiently large $m$ depending
only on the unperturbed system $T_{0}$ given by (1.3), see Remark 2.2. The
dual pairing in (1.6) can thus be expressed via the standard inner product
$(\cdot,\cdot)_{L^{2}}$ on $L^{2}(\mathbb{T}^{2})$ as

$\langle\varphi,u\rangle=(\varphi,\bar{u})_{L^{2}}.$

######  Remark 2.2.

Assume that the map $E_{0}$ in (1.3) is expanding with an expansion rate
$\lambda_{0}=\min_{x}E_{0}^{\prime}(x)$. For sufficiently small noise levels
$\epsilon$ we have in view of (1.5) that $E_{\omega}$ is an expanding map
$\mathbb{P}$-almost surely with an expansion rate strictly larger than
$\lambda=\frac{1}{2}(\lambda_{0}+1)>1$. Let $\lambda^{-\frac{1}{2}}<\rho<1$.
By combining [NW2015, Theorem 1.4] and [NW2015, Theorem 1.5] we can find a
positive integer $m_{0}>1$ together with numbers
$\epsilon_{0}=\epsilon_{0}(m)$ and $\nu_{0}=\nu_{0}(m)$ depending also on an
integer $m\geq m_{0}$ for which those results are in force. For the rest of
this section we assume this to be the case and only consider noise levels
$\epsilon$ such that $0\leq\epsilon\leq\epsilon_{0}$. We make sure to pick $m$
large enough so that

(2.7) $\log(\lambda^{-m-\frac{1}{2}}k^{\frac{1}{2}})<0$

where $k\geq 2$ is the angle multiplication factor of $E_{0}$. Then $m$ only
depends on the unperturbed system $T_{0}$. Fixing $r\geq m>1$, the same is
true for the Sobolev regularity index $r$ appearing in
$\mathscr{E}=H^{r}(\mathbb{T}^{2})$.

######  Proposition 2.3.

$\mathscr{H}^{m}(\mathbb{T}^{2})$ is $T$-admissible.

###### Proof.

In view of Definition 1.1 we need to prove $\mathrm{(A1)}$, i.e., that
$\|\mathcal{L}_{\omega}\varphi\|_{\mathscr{H}^{m}}\leq
C\|\varphi\|_{\mathscr{H}^{m}}$ for all $\varphi\in\mathscr{H}^{m}$ and some
$C>0$. By (2.4) and the definition of the norm in $\mathscr{H}^{m}$ we have

(2.8)
$\|\mathcal{L}_{\omega}\varphi\|_{\mathscr{H}^{m}}^{2}=\lVert\mathcal{L}_{0,\omega}\varphi_{0}\rVert_{H^{m}(\mathbb{S}^{1})}^{2}+\sum_{\nu\in
2\pi\mathbb{Z},\;\nu\neq
0}(1+\nu^{2})^{m}\lVert\mathcal{L}_{\nu,\omega}\varphi_{\nu}\rVert_{H^{m}_{\nu}(\mathbb{S}^{1})}^{2}.$

It follows from [NW2015, Theorem 1.5] that there is a $\rho_{0}\in(0,1)$ as in
Remark 2.2 together with a $\nu_{0}\in 2\pi\mathbb{N}$ and a constant $c_{0}$
such that

(2.9) $\|\mathcal{L}_{\nu,\omega}^{(n)}\varphi_{\nu}\|_{H^{m}_{\nu}}\leq
c_{0}\rho_{0}^{n}\|\varphi_{\nu}\|_{H^{m}_{\nu}},\quad n\geq 1,$

for any $\nu$ with $|\nu|>\nu_{0}$ and $\mathbb{P}$-almost every
$\omega\in\Omega$. On the other hand, for any $\nu\neq 0$ with
$|\nu|\leq\nu_{0}$, we have

$\nu_{0}^{-2}(1+\xi^{2})\leq\nu^{-2}(\nu^{2}+\xi^{2})=1+(\xi/\nu)^{2}\leq
1+\xi^{2},$

so that

(2.10)
$\nu_{0}^{-m}\|\varphi_{\nu}\|_{H^{m}}\leq\|\varphi_{\nu}\|_{H^{m}_{\nu}}\leq\|\varphi_{\nu}\|_{H^{m}}.$

Hence, combining (2.9) for $n=1$ with [NW2015, Theorem 1.4] it follows that
there is a constant $C$ such that

$\|\mathcal{L}_{\nu,\omega}\varphi_{\nu}\|_{H^{m}_{\nu}}\leq
C\|\varphi_{\nu}\|_{H^{m}_{\nu}}$

for all $\nu\neq 0$, and that

$\|\mathcal{L}_{0,\omega}\varphi_{0}\|_{H^{m}}\leq C\|\varphi_{0}\|_{H^{m}}.$

Combining these estimates with (2.8) we conclude that $\mathrm{(A1)}$ holds. ∎

We shall need the following well-known structure result for Sobolev spaces. We
include a short direct proof for convenience.

######  Lemma 2.4.

Let $r>1$. Then there is a constant $C_{r}$ such that for all $u,\varphi\in
H^{r}(\mathbb{T}^{2})$ we have $u\varphi\in H^{r}(\mathbb{T}^{2})$ and

$\lVert u\varphi\rVert_{H^{r}(\mathbb{T}^{2})}\leq C_{r}\lVert
u\rVert_{H^{r}(\mathbb{T}^{2})}\lVert\varphi\rVert_{H^{r}(\mathbb{T}^{2})}.$

###### Proof.

To shorten notation we shall denote $(\xi,\nu)$ by $\xi=(\xi_{1},\xi_{2})$,
and write $\langle x\rangle=(1+\lvert x\rvert^{2})^{\frac{1}{2}}$ for the
Japanese bracket. Following Sjöstrand [sjostrand2009eigenvalue, Proposition
2.1], we pass to the Fourier side and see that the result follows if we show
that

(2.11)
$\sum_{\xi\in(2\pi\mathbb{Z})^{2}}\langle\xi\rangle^{r}w(\xi)(\langle\cdot\rangle^{-r}\widetilde{\varphi}\ast\langle\cdot\rangle^{-r}\widetilde{u})(\xi)\leq
C_{r}\lVert w\rVert\lVert\widetilde{\varphi}\rVert\lVert\widetilde{u}\rVert$

for all non-negative $\widetilde{u},\widetilde{\varphi},w\in\ell^{2}$, where
$\ast$ denotes convolution and the norms are the ones in $\ell^{2}$. Indeed,

$\lVert
u\varphi\rVert_{H^{r}(\mathbb{T}^{2})}=\lVert\langle\cdot\rangle^{r}\widehat{u\varphi}\rVert=\sup_{0\neq
w\in\ell^{2}}\frac{\langle\langle\cdot\rangle^{r}\widehat{u\varphi},w\rangle_{\ell^{2}}}{\lVert
w\rVert},$

so writing $\widetilde{\varphi}=\langle\cdot\rangle^{r}\widehat{\varphi}$ and
$\widetilde{u}=\langle\cdot\rangle^{r}\widehat{u}$ the claim easily follows.

To establish (2.11) we see that the left-hand side is equal to

$\sum_{\xi\in(2\pi\mathbb{Z})^{2}}\sum_{\eta+\zeta=\xi}\langle\xi\rangle^{r}w(\xi)\langle\eta\rangle^{-r}\widetilde{\varphi}(\eta)\langle\zeta\rangle^{-r}\widetilde{u}(\zeta)\leq\mathrm{I}+\mathrm{II},$

where $\mathrm{I},\mathrm{II}$ are the sums over
$\\{\lvert\eta\rvert\geq\lvert\xi\rvert/2\\}$ and
$\\{\lvert\zeta\rvert\geq\lvert\xi\rvert/2\\}$, respectively. In $\mathrm{I}$
we have $\langle\xi\rangle^{r}\leq 2^{r}\langle\eta\rangle^{r}$, so

$\mathrm{I}\leq
2^{r}\sum_{\zeta}\bigg{(}\sum_{\xi}w(\xi)\widetilde{\varphi}(\xi-\zeta)\bigg{)}\langle\zeta\rangle^{-r}\widetilde{u}(\zeta)\leq
2^{r}\lVert
w\rVert\lVert\widetilde{\varphi}\rVert\lVert\langle\cdot\rangle^{-r}\widetilde{u}\rVert_{\ell^{1}},$

where $\lVert\langle\cdot\rangle^{-r}\widetilde{u}\rVert_{\ell^{1}}\leq
C_{r}\lVert\widetilde{u}\rVert$ by the Cauchy-Schwartz inequality since $r>1$.
Hence $\mathrm{I}$ is bounded by a constant times $\lVert
w\rVert\lVert\widetilde{\varphi}\rVert\lVert\widetilde{u}\rVert$. In
$\mathrm{II}$ we have $\langle\xi\rangle^{r}\leq 2^{r}\langle\zeta\rangle^{r}$
so by symmetry the same estimate holds for $\mathrm{II}$, and (2.11) follows.
∎

######  Proposition 2.5.

$H^{r}(\mathbb{T}^{2})$ is associated with $\mathscr{H}^{m}(\mathbb{T}^{2})$
for all $r\geq m>1$.

###### Proof.

By Lemma 2.4 the multiplication map $H^{r}(\mathbb{T}^{2})\times
H^{r}(\mathbb{T}^{2})\to H^{r}(\mathbb{T}^{2})$ is continuous. Hence, it
suffices to prove that the first inequality in $\mathrm{(A2)}$ is satisfied
when $\lVert\cdot\rVert_{\mathscr{B}}=\lVert\cdot\rVert_{\mathscr{H}^{m}}$ and
$\lVert\cdot\rVert_{\mathscr{E}}=\lVert\cdot\rVert_{H^{r}}$. It is even enough
to show that

(2.12)
${\left|\kern-1.07639pt\left|u\varphi\right|\kern-1.07639pt\right|_{\mathscr{H}^{m}}}\leq
C_{m,r}{\left|\kern-1.07639pt\left|\varphi\right|\kern-1.07639pt\right|_{\mathscr{H}^{m}}}\lVert
u\rVert_{H^{r}}.$

Indeed, if $\psi\in H^{m}_{\nu}(\mathbb{S}^{1})$ and $\nu\neq 0$ then
$\lVert\psi\rVert_{L^{2}(\mathbb{S}^{1})}\leq\langle\nu\rangle^{m}\lVert\psi\rVert_{H_{\nu}^{m}(\mathbb{S}^{1})}$,
which implies that
$\lVert\varphi\rVert_{L^{2}(\mathbb{T}^{2})}\leq\lVert\varphi\rVert_{\mathscr{H}^{m}(\mathbb{T}^{2})}$.
Hence

$\langle\varphi,u\rangle=\langle u\varphi,1_{\mathbb{T}^{2}}\rangle\leq\lVert
u\varphi\rVert_{L^{2}(\mathbb{T}^{2})}\leq\lVert
u\varphi\rVert_{\mathscr{H}^{m}(\mathbb{T}^{2})}$

so we always have
$\max\\{\langle\varphi,u\rangle,{\left|\kern-1.07639pt\left|u\varphi\right|\kern-1.07639pt\right|_{\mathscr{H}^{m}}}\\}={\left|\kern-1.07639pt\left|u\varphi\right|\kern-1.07639pt\right|_{\mathscr{H}^{m}}}$
in this case, which proves the claim.

But by Lemma 2.1, $\mathscr{H}^{m}(\mathbb{T}^{2})$ coincides with
$H^{m}(\mathbb{T}^{2})$, so if $r\geq m>1$ then

${\left|\kern-1.07639pt\left|u\varphi\right|\kern-1.07639pt\right|_{\mathscr{H}^{m}}}\leq
C_{m}{\left|\kern-1.07639pt\left|\varphi\right|\kern-1.07639pt\right|_{\mathscr{H}^{m}}}{\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|_{\mathscr{H}^{m}}}\leq
C_{m}{\left|\kern-1.07639pt\left|\varphi\right|\kern-1.07639pt\right|_{\mathscr{H}^{m}}}\lVert
u\rVert_{H^{r}}$

by Lemma 2.4 which establishes (2.12) and the proof is complete. ∎

###  2.3. Condition (UG)

Having established that $H^{r}(\mathbb{T}^{2})$ is associated with the
$T$-admissible Banach space $\mathscr{H}^{m}(\mathbb{T}^{2})$ we proceed to
prove that $T$ satisfies the uniform spectral gap condition with respect to
$\mathscr{H}^{m}(\mathbb{T}^{2})$ and observables in $H^{r}(\mathbb{T}^{2})$.
We begin by verifying the existence of invariant measures.

######  Proposition 2.6.

For any sufficiently small noise level $\epsilon>0$ there exists a unique
$h\in L^{\infty}(\Omega,\mathscr{H}^{m}(\mathbb{T}^{2}))$ and a family of
probability measures $\\{\mu_{\omega}\\}_{\omega\in\Omega}$ such that

$\mathcal{L}_{\omega}h_{\omega}=h_{\sigma\omega},\quad\langle
h_{\omega},u\rangle=\int
u\,d\mu_{\omega},\quad\operatorname*{ess\,inf}_{\omega\in\Omega}\|h_{\omega}\|_{\mathscr{H}^{m}}\geq
1$

for $\mathbb{P}$-almost every $\omega\in\Omega$ and any observable $u\in
H^{r}(\mathbb{T}^{2})$.

###### Proof.

For sufficiently small $\epsilon>0$ there is by [NW2015, Theorem 1.3] a
uniquely defined function $h=h(\epsilon):\omega\mapsto h_{\omega}$ such that
$\mathcal{L}_{\omega}h_{\omega}=h_{\sigma\omega}$ and $\langle
h_{\omega},1_{\mathbb{T}^{2}}\rangle=1$, together with a family of probability
measures $\\{\mu_{\omega}\\}_{\omega\in\Omega}$ such that
$d\mu_{\omega}=h_{\omega}d\mathrm{Leb}_{\mathbb{T}^{2}}$. Hence, if $u\in
H^{r}(\mathbb{T}^{2})\subset L^{2}(\mathbb{T}^{2})$ we have $\langle
h_{\omega},u\rangle=\int u\,d\mu_{\omega}$. Note that by [NW2015, Theorem
1.3], $h_{\omega}(x,s)$ for $(x,s)\in\mathbb{T}^{2}$ has the form of a tensor
product $h_{\omega}(x,s)=h_{\omega}(x)\otimes 1_{\mathbb{S}^{1}}(s)$ with
$x\mapsto h_{\omega}(x)\in\mathscr{C}^{\infty}(\mathbb{S}^{1})$ satisfying
$\int_{\mathbb{S}^{1}}h_{\omega}(x)\,dx=1$ and
$\operatorname*{ess\,sup}_{\omega}\lVert
h_{\omega}\rVert_{H^{m}(\mathbb{S}^{1})}\leq C_{m}$ for any $m\in\mathbb{N}$.
In particular, decomposing $h_{\omega}(x,s)$ into Fourier modes we find that
the mode corresponding to $\nu=0$ satisfies $(h_{\omega})_{0}(x)=\int
h_{\omega}(x,s)\,ds=h_{\omega}(x)$, while $(h_{\omega})_{\nu}(x)=0$ for all
$\nu\in 2\pi\mathbb{Z}$ such that $\nu\neq 0$. Inspecting the definition of
the norm $\lVert\cdot\rVert_{\mathscr{H}^{m}(\mathbb{T}^{2})}$ this implies
that

$\lVert
h_{\omega}\rVert_{\mathscr{H}^{m}(\mathbb{T}^{2})}^{2}=\lVert(h_{\omega})_{0}\rVert_{H^{m}(\mathbb{S}^{1})}^{2}\leq
C_{m}^{2}$

$\mathbb{P}$-almost surely, which shows that $h\in
L^{\infty}(\Omega,\mathscr{H}^{m}(\mathbb{T}^{2}))$. Moreover, by the Cauchy-
Schwartz inequality we have

$1=\int_{\mathbb{S}^{1}}h_{\omega}(x)\,dx\leq\lVert
h_{\omega}\rVert_{L^{2}(\mathbb{S}^{1})}=\lVert(h_{\omega})_{0}\rVert_{L^{2}(\mathbb{S}^{1})}$

and since $m\geq 0$ we thus find that

$\lVert
h_{\omega}\rVert_{\mathscr{H}^{m}(\mathbb{T}^{2})}^{2}=\lVert(h_{\omega})_{0}\rVert_{H^{m}(\mathbb{S}^{1})}^{2}\geq\lVert(h_{\omega})_{0}\rVert_{L^{2}(\mathbb{S}^{1})}^{2}\geq
1,$

so
$\operatorname*{ess\,inf}_{\omega\in\Omega}\|h_{\omega}\|_{\mathscr{H}^{m}}\geq
1$. ∎

We next record a perturbation lemma.

######  Lemma 2.7.

There is a $\rho\in(0,1)$, an arbitrarily large integer $n_{0}$ and a number
$\epsilon(n_{0})$ such that for all $0\leq\epsilon<\epsilon(n_{0})$ we have

$\|\mathcal{L}_{\omega}^{(n_{0})}\varphi\|_{\mathscr{H}^{m}}\leq\rho^{n_{0}}\lVert\varphi\rVert_{\mathscr{H}^{m}}+C_{n_{0}}\lVert\varphi_{0}\rVert_{H^{m-1}(\mathbb{S}^{1})}$

where $\varphi_{0}(x)=\int_{\mathbb{S}^{1}}\varphi(x,s)\,ds$. If
$\langle\varphi,1_{\mathbb{T}^{2}}\rangle=0$ we may take $C_{n_{0}}=0$.

###### Proof.

By (2.4) together with (2.8)–(2.9) we have

(2.13)
$\|\mathcal{L}_{\omega}^{(n)}\varphi\|_{\mathscr{H}^{m}}^{2}\leq\lVert\mathcal{L}_{0,\omega}^{(n)}\varphi_{0}\rVert_{H^{m}}^{2}+\sum_{0<\lvert\nu\rvert\leq\nu_{0}}\langle\nu\rangle^{2m}\lVert\mathcal{L}_{\nu,\omega}^{(n)}\varphi_{\nu}\rVert_{H^{m}_{\nu}}^{2}+\sum_{\lvert\nu\rvert>\nu_{0}}c_{0}\rho_{0}^{2n}\langle\nu\rangle^{2m}\lVert\varphi_{\nu}\rVert_{H^{m}_{\nu}}^{2}$

where $\rho_{0}\in(0,1)$. Let $\mathcal{L}_{T_{0}}$ denote the transfer
operator induced by the unperturbed map $T_{0}$ and
$(\mathcal{L}_{T_{0}})_{\nu}$ its restriction corresponding to
$\mathcal{L}_{\nu,\omega}$. By applying [NW2015, Proposition 4.5] to each
$\mathcal{L}_{\nu,\omega}$ with $\lvert\nu\rvert\leq\nu_{0}$ we can find a
$\rho_{1}\in(0,1)$ together with an integer $n_{0}$ (which can be chosen
arbitrarily large) and a number $\epsilon(n_{0})$ such that for all
$\lvert\nu\rvert\leq\nu_{0}$ and $0\leq\epsilon<\epsilon(n_{0})$ we have

(2.14)
$\operatorname*{ess\,sup}_{\omega}\lVert(\mathcal{L}_{\nu,\omega}^{(n_{0})}-(\mathcal{L}_{T_{0}}^{n_{0}})_{\nu})\varphi_{\nu}\rVert_{H^{m}}\leq\rho_{1}^{n_{0}}\lVert\varphi_{\nu}\rVert_{H^{m}}.$

By [NW2015, Theorem 4.4] the spectral radius of each
$(\mathcal{L}_{T_{0}})_{\nu}:H^{m}\to H^{m}$ is strictly less than 1 when
$\nu\neq 0$. Hence, by increasing $n_{0}$ if necessary, we have for some
$\rho_{2}\in(0,1)$ and all $0<\lvert\nu\rvert\leq\nu_{0}$ that

(2.15)
$\lVert(\mathcal{L}_{T_{0}}^{n_{0}})_{\nu}\varphi_{\nu}\rVert_{H^{m}}\leq\rho_{2}^{n_{0}}\lVert\varphi_{\nu}\rVert_{H^{m}}.$

When $\nu=0$, the transfer operator $(\mathcal{L}_{T_{0}})_{\nu=0}$ is the
standard transfer operator induced by the uniformly expanding map $E_{0}$ on
the circle. Thus, by [BaillifBaladi, Lemma 15] we can (by taking $n_{0}$
sufficiently large and increasing $\rho_{2}$ if necessary) find a constant
$C_{n_{0}}$ such that

(2.16)
$\lVert(\mathcal{L}_{T_{0}}^{n_{0}})_{0}\varphi_{0}\rVert_{H^{m}}\leq\rho_{2}^{n_{0}}\lVert\varphi_{0}\rVert_{H^{m}}+C_{n_{0}}\lVert\varphi_{0}\rVert_{H^{m-1}}$

for all $\varphi_{0}\in H^{m}$. On the other hand, if
$\langle\varphi,1_{\mathbb{T}^{2}}\rangle=0$ then the Fourier mode
$\varphi_{0}(x)$ corresponding to $\nu=0$ satisfies

(2.17)
$\int_{\mathbb{S}^{1}}\varphi_{0}(x)\,dx=\int_{\mathbb{S}^{1}}\bigg{(}\int_{\mathbb{S}^{1}}\varphi(x,s)\,ds\bigg{)}\,dx=\langle\varphi,1_{\mathbb{T}^{2}}\rangle=0.$

This implies that (2.16) holds with $C_{n_{0}}=0$ for such $\varphi$. Indeed,
it is well-known that the only eigenvalue of $(\mathcal{L}_{T_{0}})_{\nu=0}$
on the unit circle is the simple eigenvalue 1, and the Lyapunov subspace
associated to the eigenvalues of modulus 1 is one-dimensional and spanned by
the function $h(\epsilon=0)$ given in the first paragraph of the proof, see
[Mane, Chapter 3] or [NW2015, Theorem 4.4]. Let us write $h_{0}$ for this
function, indicating that $h_{0}$ does not depend on $\omega$ in contrast to
$h_{\omega}=h_{\omega}(\epsilon)$ for $\epsilon>0$. If
$\pi_{0}:H^{m}(\mathbb{S}^{1})\to H^{m}(\mathbb{S}^{1})$ denotes the
projection $\pi_{0}(\psi)=h_{0}(x)\int_{\mathbb{S}^{1}}\psi(x)\,dx$ then
$\varphi_{0}$ is in the kernel of $\pi_{0}$ by (2.17). In the notation of
Definition 1.10 we thus have a splitting
$H^{m}=\mathscr{F}_{0}\oplus\mathscr{R}_{0}$ with
$\mathscr{F}_{0}=\mathbb{C}h_{0}$ and if $\psi\in\mathscr{R}_{0}$ then

$\lim_{n\rightarrow\infty}\frac{1}{n}\log\lVert(\mathcal{L}_{T_{0}}^{n})_{0}\psi\rVert_{H^{m}}\leq\widetilde{\Lambda}((\mathcal{L}_{T_{0}})_{0})$

for some number $\widetilde{\Lambda}((\mathcal{L}_{T_{0}})_{0})<0$. After
increasing $\rho_{2}$ if necessary we may assume that
$\widetilde{\Lambda}((\mathcal{L}_{T_{0}})_{0})<\log\rho_{2}$ which proves the
claim since $\varphi_{0}\in\mathscr{R}_{0}$.

By combining (2.14)–(2.16) we see that

$\lVert\mathcal{L}_{0,\omega}^{(n_{0})}\varphi_{0}\rVert_{H^{m}}^{2}\leq
K_{1}(n_{0})\lVert\varphi_{0}\rVert_{H^{m}}^{2}+C_{n_{0}}^{2}\lVert\varphi_{0}\rVert_{H^{m-1}}^{2}$

where
$K_{1}(n_{0})=(\rho_{1}^{n_{0}}+\rho_{2}^{n_{0}})(\rho_{1}^{n_{0}}+\rho_{2}^{n_{0}}+2C_{n_{0}})$
and by using (2.10) twice we also get

$\sum_{0<\lvert\nu\rvert\leq\nu_{0}}\langle\nu\rangle^{2m}\lVert\mathcal{L}_{\nu,\omega}^{(n_{0})}\varphi_{\nu}\rVert_{H^{m}_{\nu}}^{2}\leq
K_{2}(n_{0})\sum_{0<\lvert\nu\rvert\leq\nu_{0}}\langle\nu\rangle^{2m}\lVert\varphi_{\nu}\rVert_{H^{m}_{\nu}}^{2}$

where $K_{2}(n_{0})=\nu_{0}^{2m}(\rho_{1}^{n_{0}}+\rho_{2}^{n_{0}})^{2}$.
Since $0<\rho_{1},\rho_{2}<1$ we can choose $n_{0}$ so large that
$K_{1}(n_{0}),K_{2}(n_{0})<1$, which means that there is a $\rho_{3}\in(0,1)$
such that both $K_{1}(n_{0})$ and $K_{1}(n_{0})$ are bounded by
$\rho_{3}^{2n_{0}}$. Inserting this into (2.13) we find that
$\lVert\mathcal{L}_{\omega}^{(n_{0})}\varphi\rVert_{\mathscr{H}^{m}}^{2}$ is
bounded by

$\displaystyle\rho_{3}^{2n_{0}}\bigg{(}\lVert\varphi_{0}\rVert_{H^{m}}^{2}+\sum_{0<\lvert\nu\rvert\leq\nu_{0}}\langle\nu\rangle^{2m}\lVert\varphi_{\nu}\rVert_{H^{m}_{\nu}}^{2}\bigg{)}+\sum_{\lvert\nu\rvert>\nu_{0}}c_{0}\rho_{0}^{2n_{0}}\langle\nu\rangle^{2m}\lVert\varphi_{\nu}\rVert_{H^{m}_{\nu}}^{2}+C_{n_{0}}^{2}\lVert\varphi_{0}\rVert_{H^{m-1}}^{2}.$

Since $\rho_{0}<1$ we may assume that
$c_{0}\rho_{0}^{2n_{0}}=\rho_{4}^{2n_{0}}$ for some $\rho_{4}\in(0,1)$ by
increasing $n_{0}$ if necessary. The result now follows by picking
$\rho=\max(\rho_{3},\rho_{4})$. ∎

######  Proposition 2.8.

There is an $\epsilon_{0}$ such that for all $0\leq\epsilon<\epsilon_{0}$, $T$
satisfies the uniform spectral gap condition with respect to $\mathscr{H}^{m}$
and $H^{r}(\mathbb{T}^{2})$.

###### Proof.

In view of Proposition 2.6 we only need to verify (1.9), namely, that there
are constants $C>0$ and $\rho\in(0,1)$ such that if
$\varphi\in\mathscr{H}^{m}(\mathbb{T}^{2})$ satisfies
$\langle\varphi,1_{\mathbb{T}^{2}}\rangle=0$, then

(2.18)
$\operatorname*{ess\,sup}_{\omega\in\Omega}\|\mathcal{L}_{\omega}^{(n)}\varphi\|_{\mathscr{H}^{m}}\leq
C\rho^{n}\|\varphi\|_{\mathscr{H}^{m}}\quad\text{for all $n\geq 1$.}$

By Lemma 2.7 there is a $\rho\in(0,1)$ together with an integer $n_{0}$ and a
number $\epsilon(n_{0})$ such that for all $0\leq\epsilon<\epsilon(n_{0})$ we
have

(2.19)
$\|\mathcal{L}_{\omega}^{(n_{0})}\varphi\|_{\mathscr{H}^{m}}\leq\rho^{n_{0}}\lVert\varphi\rVert_{\mathscr{H}^{m}}$

for all $\varphi\in\mathscr{H}^{m}$ such that
$\langle\varphi,1_{\mathbb{T}^{2}}\rangle=0$.

Now let $n\geq 1$ be arbitrary and write $n=kn_{0}+r$ where $0\leq r<n_{0}$.
Using $\mathrm{(A1)}$ and (2.19) this immediately gives

$\|\mathcal{L}_{\omega}^{(n)}\varphi\|_{\mathscr{H}^{m}}\leq
C_{0}^{r}\rho^{kn_{0}}\lVert\varphi\rVert_{\mathscr{H}^{m}}=\bigg{(}\frac{C_{0}}{\rho}\bigg{)}^{r}\rho^{kn_{0}+r}\lVert\varphi\rVert_{\mathscr{H}^{m}}\leq
C\rho^{n}\lVert\varphi\rVert_{\mathscr{H}^{m}}$

where $C$ is independent of $0\leq r<n_{0}$. This establishes (2.18) and the
proof is complete. ∎

###  2.4. Condition (LY)

Here we show that $T$ satisfies the Lasota-Yorke inequality condition. Recall
that $(\Omega,\mathcal{F},\mathbb{P})$ is assumed to be a Lebesgue-Rokhlin
probability space. Since $\mathscr{H}^{m}(\mathbb{T}^{2})$ is equivalent to
$H^{m}(\mathbb{T}^{2})$ which is separable, condition (ii) in Definition 1.4
holds as long as $\mathcal{L}$ is
$(\mathcal{F},\mathcal{S}_{L(H^{m}(\mathbb{T}^{2}))})$-measurable, i.e., if
$\mathcal{L}:\Omega\to L(H^{m}(\mathbb{T}^{2}))$ is strongly measurable. The
fact that this holds can easily be shown by adapting the proof on page 981 in
[NW2015] of strong measurability of the reduced transfer operator
$\mathcal{L}_{\nu,\omega}$. Thus it remains to verify that there is a Banach
space $(\mathscr{B}_{+},|\cdot|)$ such that $\mathscr{H}^{m}(\mathbb{T}^{2})$
is compactly embedded in $\mathscr{B}_{+}$ and $\mathscr{B}_{+}$ is
continuously embedded in $L^{2}(\mathbb{T}^{2})$, that $\mathcal{L}$ is
$\mathbb{P}$-almost surely bounded on $\mathscr{B}_{+}$ and that (1.10) holds,
namely,

(2.20)
$\lVert\mathcal{L}_{\omega}^{(n_{0})}\varphi\rVert_{\mathscr{H}^{m}}\leq\alpha_{\omega}\lVert\varphi\rVert_{\mathscr{H}^{m}}+\beta_{\omega}\lvert\varphi\rvert,\quad\varphi\in\mathscr{H}^{m},$

for some positive integer $n_{0}$, where $\alpha$ and $\beta$ are random
variables with values in $\mathbb{R}_{+}$ with
$\mathbb{E}_{\mathbb{P}}\left[\alpha\right]<1$ and
$\mathbb{E}_{\mathbb{P}}\left[\beta\right]<\infty.$ By the following result we
can take $\mathscr{B}_{+}=H^{m-1}(\mathbb{T}^{2})$.

######  Proposition 2.9.

There is an $\epsilon_{0}$ such that for all $0\leq\epsilon<\epsilon_{0}$, $T$
satisfies the Lasota-Yorke inequality condition with respect to
$\mathscr{H}^{m}\subset H^{m-1}(\mathbb{T}^{2})$.

###### Proof.

By Lemma 2.7 we have

(2.21)
$\|\mathcal{L}_{\omega}^{(n_{0})}\varphi\|_{\mathscr{H}^{m}}\leq\rho^{n_{0}}\lVert\varphi\rVert_{\mathscr{H}^{m}}+C_{n_{0}}\lVert\varphi_{0}\rVert_{H^{m-1}(\mathbb{S}^{1})}$

where $\rho\in(0,1)$. Let $H^{j,k}(\mathbb{T}^{2})$ be the anisotropic Sobolev
space with norm

$\|\varphi\|_{H^{j,k}}^{2}=\sum_{\xi,\nu\in
2\pi\mathbb{Z}}((1+\xi^{2})^{j}+(1+\nu^{2})^{k})\lvert\hat{\varphi}(\xi,\nu)\rvert^{2}.$

If $\varphi_{0}$ is the Fourier mode
$\varphi_{0}(x)=\int_{\mathbb{S}^{1}}e^{-i\nu s}\varphi(x,s)\,ds$ then clearly

$\lVert\varphi_{0}\rVert_{H^{m-1}(\mathbb{S}^{1})}\leq\|\varphi\|_{H^{m-1,0}}\leq\|\varphi\|_{H^{m-1,m-1}}$

where the right-hand side is equivalent to the usual norm in
$H^{m-1}(\mathbb{T}^{2})$. Hence (2.21) gives

$\|\mathcal{L}_{\omega}^{(n_{0})}\varphi\|_{\mathscr{H}^{m}}\leq\rho^{n_{0}}\lVert\varphi\rVert_{\mathscr{H}^{m}}+C_{n_{0}}\lVert\varphi\rVert_{H^{m-1}(\mathbb{T}^{2})}.$

It is well-known that $H^{m}(\mathbb{T}^{2})$ is compactly embedded in
$H^{m-1}(\mathbb{T}^{2})$, and since $\lVert\cdot\rVert_{\mathscr{H}^{m}}$ is
equivalent to $\lVert\cdot\rVert_{H^{m}(\mathbb{T}^{2})}$ we can thus take
$\mathscr{B}_{+}=H^{m-1}(\mathbb{T}^{2})$. (We also need to check that
$\mathcal{L}_{\omega}$ is $\mathbb{P}$-almost surely bounded on
$H^{m-1}(\mathbb{T}^{2})$, but this is immediate from $\mathrm{(A1)}$ by the
same reasoning.) In particular, (2.20) holds with
$\alpha_{\omega}\equiv\rho^{n_{0}}$ and $\beta_{\omega}\equiv C_{n_{0}}$. ∎

###### Proof of Theorem 1.5.

The result follows by virtue of Propositions 2.3, 2.5, 2.8 and 2.9. ∎

###  2.5. Proof of Theorem 1.15

In this subsection, we will show the first part of (HK2) and (HK3) under the
assumption of Theorem 1.15, together with the claim in Remark 1.16.

The first part of (HK2). We first prove the first part of (HK2) under the
assumption of Theorem 1.15. For this purpose, it suffices to show that there
is a constant $C>0$ such that for any bounded interval $J$,

(2.22) $\|\mathcal{L}_{it,\omega}-\mathcal{L}_{it,\omega_{0}}\|_{H^{m}}\leq
C\|g_{\omega}-g_{\omega_{0}}\|_{H^{r}}$

for any $t\in J$ and $\omega$ in a neighborhood of $\omega_{0}$.

We first note that due to condition (i) of Theorem 1.15,
$\mathcal{L}_{\omega}=\mathcal{L}_{\omega_{0}}$ for all $\omega$ in a
neighborhood of $\omega_{0}$. Thus, for each $\varphi\in
H^{m}(\mathbb{T}^{2})$, it follows from (A1) and (A2) (with
$\mathscr{B}=H^{m}(\mathbb{T}^{2})$ and $\mathscr{E}=H^{r}(\mathbb{T}^{2})$)
that

(2.23)
$\|(\mathcal{L}_{it,\omega}-\mathcal{L}_{it,\omega_{0}})\varphi\|_{H^{m}}\leq
C_{0}\|e^{itg_{\omega}}-e^{itg_{\omega_{0}}}\|_{H^{r}}\|\varphi\|_{H^{m}}.$

Since it holds

$e^{itg_{\omega}}-e^{it\omega_{0}}=\sum_{k=0}^{\infty}\frac{(it)^{k}(g_{\omega}^{k}-g_{\omega_{0}}^{k})}{k!}=(g_{\omega}-g_{\omega_{0}})\sum_{k=1}^{\infty}\frac{(it)^{k}\sum_{\ell=0}^{k-1}g_{\omega}^{\ell}g_{\omega_{0}}^{k-\ell}}{k!},$

by using (A2) with $\mathscr{E}=H^{r}(\mathbb{T}^{2})$ again, we get

$\|e^{itg_{\omega}}-e^{it\omega_{0}}\|_{H^{r}}\leq\|g_{\omega}-g_{\omega_{0}}\|_{H^{r}}|t|e^{|t|G}$

with $G=\operatorname*{ess\,sup}_{\omega}\|g_{\omega}\|_{H^{r}}$. The desired
estimate (2.22) immediately follows from this inequality and (2.23).

(HK3). We next show (HK3) under the condition of Theorem 1.15. Fix a bounded
interval $J\subset\mathbb{R}\setminus\\{0\\}$. Denote the essential spectral
radius of $\mathcal{L}_{\omega_{0},it}^{(\ell_{0})}$ by $b_{t}$ and let
$b=\sup_{t\in J}b_{t}$. Then due to condition (iv) of Theorem 1.15, since $J$
is compact, $b<1$. Therefore, it suffices to show that for each
$t\in\mathbb{R}\setminus\\{0\\}$, $\mathcal{L}_{\omega_{0},it}^{(\ell_{0})}$
has no eigenvalue with radius $1$.

Arguing by contradiction, assume that there exist $c\in\mathbb{R}$ and
$\gamma\in H^{m}(\mathbb{T}^{2})$ such that

$\sum_{z=T_{\omega_{0}}^{(\ell_{0})}(\zeta)}\frac{e^{it(S_{\ell_{0}}g)_{\omega_{0}}(\zeta)}\gamma(\zeta)}{|\det
DT_{\omega_{0}}^{(\ell_{0})}|}=e^{ic}\gamma(z).$

In fact, according to the remark on p. 1479 in [Faure] we even have
$\gamma\in\mathscr{C}^{\infty}(\mathbb{T}^{2})$. Then, repeating the argument
in [NW2015, Theorem 4.4], we obtain

$t(S_{\ell_{0}}g)_{\omega_{0}}(\zeta)=\psi(\zeta)-\psi(T_{\omega_{0}}^{(\ell_{0})}(\zeta))-c+2\pi
n(\zeta)$

with $\psi(\zeta)=\operatorname{Arg}(\gamma(\zeta))$ where
$\operatorname{Arg}$ is the principal value, for some integer valued function
$\zeta\mapsto n(\zeta)$. However, $n(\zeta)$ must be a constant function of
$\zeta$ since it can be written as a linear combination of smooth functions.
So, we conclude that $(S_{\ell_{0}}g)_{\omega_{0}}$ is cohomologous to a
constant, which is a contradiction to condition (iii). This completes the
proof.

Remark 1.16. We finally show the claims in Remark 1.16. Assume that there
exists $\widetilde{g}\in L^{\infty}(\Omega,\mathscr{C}^{m}(\mathbb{S}^{2}))$
such that $g_{\omega}(x,s)=\widetilde{g}_{\omega}(x)$ for all
$(\omega,x,s)\in\Omega\times\mathbb{T}^{2}$. As in Subsection 2.1, given
$\theta\in\mathbb{C}$, $\nu\in\mathbb{Z}$ and $n\in\mathbb{N}$, we introduce
the Fourier-decomposed twisted transfer operator cocycle
$\mathcal{L}_{\theta,\nu,\omega}:L^{2}(\mathbb{S}^{1})\to
L^{2}(\mathbb{S}^{1})$ by

$\mathcal{L}_{\theta,\nu,\omega}^{(n)}\psi(x)=\sum_{E_{\omega}^{(n)}(y)=x}\frac{e^{\theta(S_{n}\widetilde{g})_{\omega}-i\nu\tau_{\omega}^{(n)}}\>\psi}{\frac{d}{dy}E_{\omega}^{(n)}}(y).$

Then by virtue of the independence of $g_{\omega}$ on $s$, we have that for
each $\varphi$ of the form $\varphi(x,s)=\sum_{\nu\in\mathbb{Z}}e^{i\nu
s}\psi_{\nu}(x)$,

$\mathcal{L}_{\theta,\omega}^{(n)}\varphi(x,s)=\sum_{\nu\in\mathbb{Z}}e^{i\nu
s}\mathcal{L}_{\theta,\omega}^{(n)}\psi_{\nu}(x).$

Fix $t\in\mathbb{R}$. Since $(\mathcal{L}_{it,\nu},\sigma)$ can be considered
as the transfer operators cocycle of the random expanding map $E$ with weight
$e^{i(t\widetilde{g}-\nu\tau)}$ whose real part is bounded by $1$, due to
[Baladibook2] together with the argument in the proof of [NW2015, Proposition
4.1], one can conclude that the maximal Lyapunov exponent of
$(\mathcal{L}_{it,\nu},\sigma)$ is bounded by $1$ and the index of compactness
of $(\mathcal{L}_{it,\nu},\sigma)$ is strictly less than $1$. Moreover, the
uniform exponential decay in the semiclassical limit $\nu\to\infty$ in
[NW2015, Theorem 1.5] that holds for $\mathcal{L}_{0,\nu}$ also holds for
$\mathcal{L}_{it,\nu}$: for any $\rho>\lambda^{-1/2}$ and sufficiently large
$m$, there are $\nu_{0}=\nu_{0}(t)$ and $c_{0}=c_{0}(t)$ such that (if the
noise level $\epsilon$ is sufficiently small)

$\|\mathcal{L}_{it,\nu,\omega}^{(n)}\|\leq c_{0}\rho^{n},\quad n\geq
1,\;|\nu|\geq\nu_{0}.$

This fact is obtained by repeating the proof of [NW2015, Theorem 1.5]; the
only difference is that the function $(dE_{\omega}^{(n)}/dx)^{-1}$ is replaced
by $e^{it(S_{n}\widetilde{g})_{\omega}}(dE_{\omega}^{(n)}/dx)^{-1}$ which does
not lead to any complications in the proof.

Therefore, repeating the argument in the previous subsection, we get the
second part of (HK2). A similar argument leads to the quasi-compactness of
$\mathcal{L}_{\omega_{0}}^{(\ell_{0})}$.

##  3\. The proof of Theorems 1.7, 1.8 and 1.9

###  3.1. Regularity of the top Oseledets space

As in [DFGV2018], we start the proof of Theorems 1.7, 1.8 and 1.9 by
establishing the regularity of the top Oseledets space of the twisted transfer
operator cocycles.

Due to the observation in Subsection 1.4, it follows from (LY) that $(T,g)$
satisfies the Oseledets decomposition condition (OD): there is a real number
$K$ such that for any $\theta$ with sufficiently small absolute value, one can
find a real number $\Lambda_{\theta}>K$ and a measurable splitting of
$\mathscr{B}$ into closed subspaces
$\mathscr{B}=\mathscr{F}_{\theta,\omega}\oplus\mathscr{R}_{\theta,\omega}$
with $\dim{\mathscr{F}_{\theta,\omega}}<\infty$ such that $\mathbb{P}$-almost
surely the following holds:

  1. $\mathrm{(1)}$

$\mathcal{L}_{\theta,\omega}\mathscr{F}_{\theta,\omega}=\mathscr{F}_{\theta,\sigma\omega}$
and
$\mathcal{L}_{\theta,\omega}\mathscr{R}_{\theta,\omega}\subset\mathscr{R}_{\theta,\sigma\omega},$

  2. $\mathrm{(2)}$

If $\varphi\in\mathscr{F}_{\theta,\omega}\backslash\\{0\\}$, then
$\lim_{n\rightarrow\infty}\frac{1}{n}\log\lVert\mathcal{L}_{\theta,\omega}^{(n)}\varphi\rVert=\Lambda_{\theta},$

  3. $\mathrm{(3)}$

If $\varphi\in\mathscr{R}_{\theta,\omega}$, then
$\limsup_{n\rightarrow\infty}\frac{1}{n}\log\lVert\mathcal{L}_{\theta,\omega}^{(n)}\varphi\rVert\leq
K.$

For $\theta\in\mathbb{C}$ and $\omega\in\Omega$, let
$\mathcal{L}_{\theta,\omega}^{*}:\mathscr{B}^{\prime}\to\mathscr{B}^{\prime}$
be the adjoint operator of $\mathcal{L}_{\theta,\omega}$ given by

$(\mathcal{L}_{\theta,\omega}^{*}u)(\varphi)=u(\mathcal{L}_{\theta,\omega}\varphi),\quad
u\in\mathscr{B}^{\prime},\quad\varphi\in\mathscr{B}.$

Recall that given a map $\mathbf{A}$ from an open subset $U$ of a normed
vector space $X$ to a normed vector space $Y$, the Fréchet derivative
$D\mathbf{A}(x)$ of $\mathbf{A}$ at $x\in U$ is a bounded operator from $X$ to
$Y$ such that

$\lim_{\|y\|_{X}\to
0}\frac{\|\mathbf{A}(x+y)-\mathbf{A}(x)-D\mathbf{A}(x)(y)\|_{Y}}{\|y\|_{X}}=0.$

The Fréchet derivative of $D\mathbf{A}:U\to L(X,Y)$ at $x\in U$ is denoted by
$D^{2}\mathbf{A}(x)$, where $L(X,Y)$ is the space of bounded linear operators
from $X$ to $Y$. $\mathbf{A}$ is said to be of class $\mathscr{C}^{1}$ (resp.
$\mathscr{C}^{2}$) if the map $D\mathbf{A}$ (resp. $D^{2}\mathbf{A}$) is
continuous. We also use the notation $\mathbf{A}^{\prime}$,
$\mathbf{A}^{\prime\prime}$ for $D\mathbf{A}$, $D^{2}\mathbf{A}$.

######  Remark 3.1.

In this section we use $\mathbb{B}$ as a sufficiently small ball in
$\mathbb{C}$ centered at $0$ on which the maps given below are well-defined,
even if it may change between occurrences. Furthermore, for simple
description, we identify each $\varphi\in X$ with a map $t\mapsto t\varphi$
from $\mathbb{C}$ to $X$ for $X=L^{\infty}(\Omega,\mathscr{B})$ or
$L^{\infty}(\Omega,\mathbb{C})$, if it makes no confusion. Moreover, for a
$\mathscr{C}^{1}$ map $\varphi:\mathbb{B}\to X$, we permit us to write
$\varphi^{\prime}(\theta)(\omega)=\psi(\omega)$ with some $\psi\in X$ to mean
$\varphi^{\prime}(\theta)(t)(\omega)=t\psi(\omega)$ for every
$t\in\mathbb{B}$. Finally, in this section we hope to use the notation $h$ as
the $\mathscr{C}^{1}$ map from $\mathbb{B}$ to
$L^{\infty}(\Omega,\mathscr{B})$ (in Theorem 3.2), which conflicts to another
map $h:\Omega\to\mathscr{B}$ given in (UG). To avoid notational confusion, we
instead use notation $h_{0}$ for the map $h:\Omega\to\mathscr{B}$ of (UG).

The following theorem is crucial in this section.

######  Theorem 3.2.

There are a ball $\mathbb{B}$ in $\mathbb{C}$ centered at $0$,
$\mathscr{C}^{1}$ maps $h:\mathbb{B}\to L^{\infty}(\Omega,\mathscr{B})$,
$w:\mathbb{B}\to L^{\infty}(\Omega,\mathscr{B}^{\prime})$, and a
$\mathscr{C}^{2}$ map $\lambda:\mathbb{B}\to L^{\infty}(\Omega,\mathbb{C})$
such that the following hold:

  1. $\mathrm{(1)}$

For every $\theta\in\mathbb{B}$ and $\mathbb{P}$-almost every
$\omega\in\Omega$,

$\mathcal{L}_{\theta,\omega}h_{\theta,\omega}=\lambda_{\theta,\omega}h_{\theta,\sigma\omega},\quad\text{
and}\quad\langle h_{\theta,\omega},1_{M}\rangle=1.$

Furthermore, $h(0)=h_{0}$ and $\lambda(0)=1$ $\mathbb{P}$-almost surely.

  2. $\mathrm{(2)}$

If we define $\widehat{\Lambda}:\mathbb{B}\to L^{\infty}(\Omega,\mathbb{R})$
by $\widehat{\Lambda}(\theta)(\omega)=\log|\lambda(\theta)(\omega)|$ for
$\theta\in\mathbb{B}$ and $\omega\in\Omega$, then

$\widehat{\Lambda}(0)=\widehat{\Lambda}^{\prime}(0)=0\quad\text{$\mathbb{P}$-almost
surely}$

and

$\mathbb{E}_{\mathbb{P}}\left[\widehat{\Lambda}^{\prime\prime}(0)\right]=V$

(recall (1.15) for the definition of $V$).

  3. $\mathrm{(3)}$

$\mathscr{F}_{\theta,\omega}$ is spanned by $h_{\theta,\omega}$ for every
$\theta\in\mathbb{B}$ and $\mathbb{P}$-almost every $\omega\in\Omega$, and

$\mathbb{E}_{\mathbb{P}}\left[\widehat{\Lambda}(\theta)\right]=\Lambda_{\theta}.$

  4. $\mathrm{(4)}$

For every $\theta\in\mathbb{B}$ and $\mathbb{P}$-almost every
$\omega\in\Omega$,

$\mathcal{L}_{\theta,\omega}^{*}w_{\theta,\sigma\omega}=\lambda_{\theta,\omega}w_{\theta,\omega},\quad\text{
and}\quad w_{\theta,\omega}(h_{\theta,\omega})=1.$

We used the notation
$h_{\theta,\omega},w_{\theta,\omega},\lambda_{\theta,\omega}$ for
$h(\theta)(\omega),w(\theta)(\omega),\lambda(\theta)(\omega)$, respectively.

The proof of the item (1) and the former statement of the item (2) in Theorem
3.2 will be given in Subsection 3.2. Subsections 3.3, 3.4 and 3.5 are,
respectively, dedicated for the proof of the later statement of the items (2),
(3) and (4).

######  Remark 3.3.

Note that $\widehat{\Lambda}(\theta)(\omega)$ in Theorem 3.2 corresponds to
$Z(\theta,\omega)$ in [DFGV2018] (see the proof of Lemma 3.9 of [DFGV2018]),
and $\mathbb{E}_{\mathbb{P}}[\widehat{\Lambda}(\theta)]$ in our context is
written as $\widehat{\Lambda}(\theta)$ in that of [DFGV2018].

Before starting the proof of Theorem 3.2, we give necessary definitions as
well as a brief explanation for the strategy of the proof of Theorem 3.2.
Define $\mathbf{I}:L^{\infty}(\Omega,\mathscr{B})\to
L^{\infty}(\Omega,\mathbb{C})$ and
$\mathbf{L}_{\theta}:L^{\infty}(\Omega,\mathscr{B})\to
L^{\infty}(\Omega,\mathscr{B})$ by

$\mathbf{I}(\varphi)(\omega)=\langle\varphi_{\omega},1_{M}\rangle,\quad\mathbf{L}_{\theta}(\varphi)(\omega)=\mathcal{L}_{\theta,\sigma^{-1}\omega}\varphi_{\sigma^{-1}\omega}$

for $\varphi\in L^{\infty}(\Omega,\mathscr{B})$ and $\omega\in\Omega$.
$\mathbf{L}_{\theta}$ is well-defined for any $\theta\in\mathbb{C}$ by virtue
of (1.18). We simply write $\mathbf{L}$ for $\mathbf{L}_{0}$. Note that
$(\mathbf{I}\circ\mathbf{L}(h_{0}))_{\omega}=1$ for $\mathbb{P}$-almost every
$\omega\in\Omega$. We will see that
$\theta\mapsto\mathbf{I}\circ\mathbf{L}_{\theta}(h_{0})$ is a continuous map
from $\mathbb{C}$ to $L^{\infty}(\Omega,\mathbb{C})$ in the proof of Lemma
3.5. Hence, one can find a small neighborhood $\mathbb{B}$ of $0$ in
$\mathbb{C}$ such that $\mathbf{I}\circ\mathbf{L}_{\theta}(\varphi)\neq 0$
$\mathbb{P}$-almost surely and
$\mathbf{F}(\theta,\varphi):\Omega\to\mathscr{B}$ given by

(3.1)
$\mathbf{F}(\theta,\varphi)=\frac{\mathbf{L}_{\theta}(\varphi)}{\mathbf{I}\circ\mathbf{L}_{\theta}(\varphi)}-\varphi$

is well-defined for any
$(\theta,\varphi)\in\mathbb{B}\times(h_{0}+\mathrm{Ker}(\mathbf{I}))$. Notice
that $\mathbf{F}(\theta,\varphi)\in\mathrm{Ker}(\mathbf{I})$ for each
$(\theta,\varphi)\in\mathbb{B}\times(h_{0}+\mathrm{Ker}(\mathbf{I}))$. It also
holds that

(3.2)
$\mathbf{F}(\theta,h_{\theta})=0\quad\text{implies}\quad\mathcal{L}_{\theta,\omega}h_{\theta,\omega}=\lambda_{\theta,\omega}h_{\theta,\sigma\omega}\;\text{$\mathbb{P}$-almost
surely},$

where
$\lambda_{\theta,\omega}=\langle\mathcal{L}_{\theta,\omega}h_{\theta,\omega},1_{M}\rangle$.
Therefore, in Subsections 3.2 and 3.3, we will apply the implicit function
theorem to a map
$\widetilde{\mathbf{F}}:\mathbb{B}\times\mathrm{Ker}(\mathbf{I})\to\mathrm{Ker}(\mathbf{I})$
given by

(3.3)
$\widetilde{\mathbf{F}}(\theta,\varphi)=\mathbf{F}(\theta,h_{0}+\varphi),\quad(\theta,\varphi)\in\mathbb{B}\times\mathrm{Ker}(\mathbf{I}),$

in order to prove the regularities of $h:\mathbb{B}\to
L^{\infty}(\Omega,\mathscr{B})$ and $\lambda:\mathbb{B}\to
L^{\infty}(\Omega,\mathbb{C})$ in Theorem 3.2. Furthermore, in Subsection 3.4,
we will apply the implicit function theorem to another map $\mathbf{G}$
induced by the adjoint twisted operators $\mathcal{L}_{\theta,\omega}^{*}$ to
obtain the regularity of $w:\mathbb{B}\to
L^{\infty}(\Omega,\mathscr{B}^{\prime})$ in Theorem 3.2.

###  3.2. First order regularity

In this subsection, we show the existence of $h:\mathbb{B}\to
L^{\infty}(\Omega,\mathscr{B})$ satisfying the item (1) of Theorem 3.2. We
first consider Fréchet derivatives of $\mathbf{M}:\mathbb{C}\times
L^{\infty}(\Omega,\mathscr{B})\to
L^{\infty}(\Omega,\mathscr{B}):\mathbf{(}\theta,\varphi)\mapsto\mathbf{L}_{\theta}(\varphi)$
(both with respect to $\theta$ and $\varphi$) and $\mathbf{I}$. For any
$\theta\in\mathbb{C}$ and $\varphi\in L^{\infty}(\Omega,\mathscr{B})$, since
$\mathbf{M}(\theta,\cdot)=\mathbf{L}_{\theta}:L^{\infty}(\Omega,\mathscr{B})\to
L^{\infty}(\Omega,\mathscr{B})$ is a linear operator,

(3.4) the Fréchet derivative of $\mathbf{M}(\theta,\cdot)$ at $\varphi$,
denoted by $\partial_{2}\mathbf{M}(\theta,\varphi)$, is $\mathbf{L}_{\theta}$.

Similarly,

(3.5) the Fréchet derivative $D\mathbf{I}(\varphi)$ of $\mathbf{I}$ at
$\varphi$ is $\mathbf{I}$

for any $\varphi\in L^{\infty}(\Omega,\mathscr{B})$ because $\mathbf{I}$ is
linear.

Given $(\theta,\varphi)\in\mathbb{C}\times L^{\infty}(\Omega,\mathscr{B})$, we
denote by $\partial_{1}\mathbf{M}(\theta,\varphi)$ the Fréchet derivative of
$\mathbf{M}(\cdot,\varphi)$ at $\theta\in\mathbb{C}$. Define
$\mathbf{L}_{j,\theta}:L^{\infty}(\Omega,\mathscr{B})\to
L^{\infty}(\Omega,\mathscr{B})$ with $j=1,2$ by
$\left(\mathbf{L}_{j,\theta}\varphi\right)_{\omega}=\mathcal{L}_{j,\theta,\sigma^{-1}\omega}\left(\varphi_{\sigma^{-1}\omega}\right)$
for $\varphi\in L^{\infty}(\Omega,\mathscr{B})$ and $\omega\in\Omega$, where
$\mathcal{L}_{j,\theta,\omega}:\mathscr{B}\to\mathscr{B}$ is given by the
duality

(3.6)
$\left\langle\mathcal{L}_{j,\theta,\omega}\varphi,u\right\rangle=\left\langle\varphi,(g_{\omega})^{j}e^{\theta
g_{\omega}}\cdot u\circ T_{\omega}\right\rangle.$

This is well-defined due to (A2). Note that
$\mathcal{L}_{j,\theta,\omega}\varphi=\mathcal{L}_{\omega}((g_{\omega})^{j}e^{\theta
g_{\omega}}\varphi)$.

######  Lemma 3.4.

For each $\theta\in\mathbb{C}$ and $\varphi\in
L^{\infty}(\Omega,\mathscr{B})$,

(3.7) $\partial_{1}\mathbf{M}(\theta,\varphi)=\mathbf{L}_{1,\theta}(\varphi)$

under the identification in Remark 3.1.

###### Proof.

Let $\widetilde{g}_{t,\omega}=(e^{tg_{\omega}}-1)/t-g_{\omega}$ with
$t\in\mathbb{C}\setminus\\{0\\}$ and $\widetilde{g}_{0,\omega}=0$ for each
$\omega\in\Omega$. We first show that there is a positive constant $c_{t}$
converging to $0$ as $t\to 0$ such that

(3.8)
$\displaystyle\operatorname*{ess\,sup}_{\omega}\left\|\widetilde{g}_{t,\omega}\varphi\right\|\leq
c_{t}\|\varphi\|\quad\text{for $\varphi\in\mathscr{B}$}.$

By Taylor expansion of $e^{tg_{\omega}}$,

$\widetilde{g}_{t,\omega}=\frac{1}{t}\left(\sum_{k=0}^{\infty}\frac{(tg_{\omega})^{k}}{k!}-1\right)-g_{t,\omega}=t\sum_{k=0}^{\infty}\frac{t^{k}g_{\omega}^{k+2}}{(k+2)!}.$

Therefore, it follows from (A2) that

$\|\widetilde{g}_{t,\omega}\varphi\|\leq
C_{1}|t|G^{2}\|\varphi\|\sum_{k=0}^{\infty}\frac{|t|^{k}G^{k}}{k!}=C_{1}|t|G^{2}e^{tG}\|\varphi\|$

(recall that
$G=\operatorname*{ess\,sup}_{\omega}\|g_{\omega}\|_{\mathscr{E}}$), which
concludes (3.8).

For each $\theta\in\mathbb{C}$ and $\varphi\in
L^{\infty}(\Omega,\mathscr{B})$, it follows from (A1) that

$\frac{\left\|\left(\mathbf{M}(\theta+t,\varphi)-\mathbf{M}(\theta,\varphi)-t\mathbf{L}_{1,\theta}(\varphi)\right)_{\sigma\omega}\right\|}{|t|}\leq
C_{0}\left\|\widetilde{g}_{t,\omega}e^{\theta
g_{\omega}}\varphi_{\omega}\right\|.$

Hence, by (A2) and (3.8) we get that

(3.9)
$\frac{\|\mathbf{M}(\theta+t,\varphi)-\mathbf{M}(\theta,\varphi)-t\mathbf{L}_{1,\theta}(\varphi)\|_{L^{\infty}(\Omega,\mathscr{B})}}{|t|}\\\
\leq
C_{0}C_{1}e^{|\theta|G}c_{t}\|\varphi\|_{L^{\infty}(\Omega,\mathscr{B})},$

which converges to $0$ as $t\to 0$. This completes the proof. ∎

We let
$\widetilde{\mathbf{L}}_{n,\theta}(\varphi)=\mathbf{L}_{n,\theta}(h_{0}+\varphi)$
and $\widetilde{\mathbf{M}}(\theta,\varphi)=\mathbf{M}(\theta,h_{0}+\varphi)$,
and define Fréchet derivatives
$\partial_{j}\widetilde{\mathbf{F}}(\theta,\varphi)$ and
$\partial_{j}\widetilde{\mathbf{M}}(\theta,\varphi)$ ($j=1,2$) in a similar
manner to the definition of $\partial_{j}\mathbf{M}(\theta,\varphi)$. By
(3.7), (3.4) and the chain rule for Fréchet derivatives, it is obvious that

(3.10)
$\partial_{1}\widetilde{\mathbf{M}}(\theta,\varphi)=\widetilde{\mathbf{L}}_{1,\theta}(\varphi)\quad\text{and}\quad\partial_{2}\widetilde{\mathbf{M}}(\theta,\varphi)=\mathbf{L}_{\theta}.$

######  Lemma 3.5.

For each $\theta\in\mathbb{B}$ and $\varphi\in\mathrm{Ker}(\mathbf{I})$,

$\partial_{1}\widetilde{\mathbf{F}}(\theta,\varphi)=\frac{\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}(\varphi)\cdot\widetilde{\mathbf{L}}_{1,\theta}(\varphi)-\widetilde{\mathbf{L}}_{\theta}(\varphi)\cdot\mathbf{I}\circ\widetilde{\mathbf{L}}_{1,\theta}(\varphi)}{(\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}(\varphi))^{2}}$

and

$\partial_{2}\widetilde{\mathbf{F}}(\theta,\varphi)=\frac{\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}(\varphi)\cdot\mathbf{L}_{\theta}-\widetilde{\mathbf{L}}_{\theta}(\varphi)\cdot\mathbf{I}\circ\mathbf{L}_{\theta}}{(\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}(\varphi))^{2}}-\mathbf{id},$

under the identification in Remark 3.1. In particular, for each
$t\in\mathbb{B}$ and $\psi\in\mathrm{Ker}(\mathbf{I})$,

$\partial_{1}\widetilde{\mathbf{F}}(0,0)(t)=t\mathbf{L}_{1,0}(h_{0})\quad\text{$\mathbb{P}$-almost
surely}$

and

$\partial_{2}\widetilde{\mathbf{F}}(0,0)(\psi)=\mathbf{L}(\psi)-\psi.$

###### Proof.

By the chain rule for Fréchet derivatives, the Fréchet derivatives
$D\left[\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}\right](\varphi)$ of
$\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}=\mathbf{I}\circ\widetilde{\mathbf{M}}(\theta,\cdot)$
at $\varphi$ is calculated as

(3.11)
$D\left[\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}\right](\varphi)=D\mathbf{I}(\widetilde{\mathbf{L}}_{\theta}(\varphi))\circ\partial_{2}\widetilde{\mathbf{M}}(\theta,\varphi)=\mathbf{I}\circ\mathbf{L}_{\theta}.$

In the last equality, we used (3.4), (3.5) and (3.10). Similarly, for each
$\varphi\in\mathrm{Ker}(\mathbf{I})$, the Fréchet derivatives
$D\left[\mathbf{I}\circ\widetilde{\mathbf{L}}_{(\cdot)}(\varphi)\right](\theta)$
of
$\mathbf{I}\circ\widetilde{\mathbf{L}}_{(\cdot)}(\varphi)=\mathbf{I}\circ\widetilde{\mathbf{M}}(\cdot,\varphi)$
at $\theta\in\mathbb{B}$ is

(3.12)
$D\left[\mathbf{I}\circ\widetilde{\mathbf{L}}_{(\cdot)}(\varphi)\right](\theta)=D\mathbf{I}(\widetilde{\mathbf{L}}_{\theta}(\varphi))\circ\partial_{1}\widetilde{\mathbf{M}}(\theta,\varphi)=\mathbf{I}\circ\widetilde{\mathbf{L}}_{1,\theta}(\varphi).$

Here the right-hand side means a map $\mathbb{B}\ni
t\mapsto\mathbf{I}(t\widetilde{\mathbf{L}}_{1,\theta}(\varphi))$. We used
(3.5), (3.7) and (3.10).

By quotient rule for Fréchet derivatives,

$\partial_{2}\widetilde{\mathbf{F}}(\theta,\varphi)=\frac{\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}(\varphi)\cdot
D\widetilde{\mathbf{L}}_{\theta}(\varphi)-\widetilde{\mathbf{L}}_{\theta}(\varphi)\cdot
D\left[\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}\right](\varphi)}{\left(\mathbf{I}\circ\widetilde{\mathbf{L}}_{\theta}(\varphi)\right)^{2}}-D\mathbf{id}(\varphi).$

This implies the second claim of this lemma by (3.4) and (3.11). The first
claim also can be proven in a similar manner, together with (3.7) and (3.12).

On the other hand, $\widetilde{\mathbf{L}}_{0}(0)=\mathbf{L}(h_{0})=h_{0}$,
and thus $\mathbf{I}\circ\widetilde{\mathbf{L}}_{0}(0)=\mathbf{I}(h_{0})=1$.
Similarly, $\mathbf{I}\circ\mathbf{L}(\psi)=\mathbf{I}(\psi)=0$ for each
$\psi\in\mathrm{Ker}(\mathbf{I})$. Moreover, by the assumption on $g$, we have
that

(3.13)
$\mathbf{I}\circ\widetilde{\mathbf{L}}_{1,0}(0)(\sigma\omega)=\langle\mathcal{L}_{\omega}(g_{\omega}h_{0,\omega}),1_{M}\rangle=\mathbb{E}_{\mu_{\omega}}[g_{\omega}]=0\quad\text{$\mathbb{P}$-almost
surely}.$

The last claim of Lemma 3.5 immediately follows from these observations. ∎

As a final preparation to apply the implicit function theorem to
$\widetilde{\mathbf{F}}$ around $(\theta,\varphi)=(0,0)$, we check that
$\partial_{2}\widetilde{\mathbf{F}}(0,0):\mathrm{Ker}(\mathbf{I})\to\mathrm{Ker}(\mathbf{I})$
is bijective. We first show injectivity. Arguing by contradiction, we assume
that

$\partial_{2}\widetilde{\mathbf{F}}(0,0)(\psi_{1})=\partial_{2}\widetilde{\mathbf{F}}(0,0)(\psi_{2})$

for some $\psi_{1},\psi_{2}\in\mathrm{Ker}(\mathbf{I})$ with
$\psi_{1}\neq\psi_{2}$. Let $\psi=\psi_{1}-\psi_{2}$. Then, $\psi$ is a
nonzero mapping in $\mathrm{Ker}(\mathbf{I})$, and it follows from Lemma 3.5
that $\mathbf{L}(\psi)=\psi$. That is,
$\psi_{\omega}\in\mathscr{F}_{0,\omega}$ $\mathbb{P}$-almost surely. On the
other hand, $\psi_{\omega}\not\in\mathbb{C}h_{0,\omega}$ because
$\langle\psi_{\omega},1_{M}\rangle=0$ while $\langle
h_{0,\omega},1_{M}\rangle=1$. This contradicts to (UG), and we get that
$\partial_{2}\widetilde{\mathbf{F}}(0,0)(\psi)$ is injective.

Next we show surjectivity. We note that a linear opeartor
$\mathbf{N}:\mathrm{Ker}(\mathbf{I})\to\mathrm{Ker}(\mathbf{I})$ given by

$\mathbf{N}\varphi=-\sum_{n=0}^{\infty}\mathbf{L}^{n}\varphi\quad\text{for
$\varphi\in\mathrm{Ker}$}$

is well-defined due to the assumption (UG) and the fact that
$\mathbf{I}\circ\mathbf{L}=\mathbf{I}$ on $L^{\infty}(\Omega,\mathscr{B})$.
Therefore, since it is easy to see that

(3.14)
$\left(\partial_{2}\widetilde{\mathbf{F}}(0,0)\right)^{-1}=\mathbf{N}\quad\text{
on $\mathrm{Ker}(\mathbf{I})$}$

by Lemma 3.5, we obtain the surjectivity.

Note that $\widetilde{\mathbf{F}}(0,0)=0$. It follows from the implicit
function theorem for Banach spaces together with the estimates in this
subsection that there is a small ball $\mathbb{B}$ in $\mathbb{C}$ centered at
$0$ and a $\mathscr{C}^{1}$ map $\eta:\mathbb{B}\to\mathrm{Ker}(\mathbf{I})$
such that $\widetilde{\mathbf{F}}(\theta,\eta(\theta))=0$. Hence, by the
observation in (3.2), if we let

(3.15)
$h(\theta)=h_{0}+\eta(\theta)\quad\text{and}\quad\lambda(\theta)(\omega)=\langle\mathbf{M}(\theta,h(\theta))(\sigma\omega),1_{M}\rangle$

for $\theta\in\mathbb{B}$ and $\omega\in\Omega$, then $h:\mathbb{B}\to
L^{\infty}(\Omega,\mathscr{B})$ and $\lambda:\mathbb{B}\to
L^{\infty}(\Omega,\mathbb{C})$ are $\mathscr{C}^{1}$ maps and satisfy the
first equality in the item (1) of Theorem 3.2, and it $\mathbb{P}$-almost
surely holds that

(3.16)
$h(0)=h_{0}\quad\text{and}\quad\lambda(0)(\omega)=\langle\mathcal{L}_{0,\omega}h_{0,\omega},1_{M}\rangle=\langle
h_{0,\omega},1_{M}\rangle=1$

by (UG). This completes the proof of the item (1) of Theorem 3.2.

Next we show the first assertion of the item (2) of Theorem 3.2. By applying
the implicit function theorem for Banach spaces to $\widetilde{\mathbf{F}}$,
we have

(3.17) $\displaystyle h^{\prime}(0)(t)(\omega)=\eta^{\prime}(0)(t)(\omega)$
$\displaystyle=-\left(\partial_{2}\widetilde{\mathbf{F}}(0,0)\right)^{-1}\left(\partial_{1}\widetilde{\mathbf{F}}(0,0)(t)\right)(\omega)$
$\displaystyle=-t\mathbf{N}\circ\mathbf{L}_{1,0}(h_{0})(\omega)$
$\displaystyle=t\sum_{n=1}^{\infty}\mathcal{L}^{(n)}_{\sigma^{-n}\omega}(h_{0,\sigma^{-n}\omega}g_{\sigma^{-n}\omega})$

for each $t\in\mathbb{B}$ and $\mathbb{P}$-almost every $\omega\in\Omega$ by
Lemma 3.5 and (3.14). Since $\mathbf{N}$ is a bounded operator on
$\mathrm{Ker}(\mathbf{I})$ and
$\mathbf{L}_{1,0}(h_{0})\in\mathrm{Ker}(\mathbf{I})$ by the assumption on $g$,
we have

(3.18) $h^{\prime}(0)(t)\in\mathrm{Ker}(\mathbf{I})\quad\text{for all
$t\in\mathbb{B}$}.$

Using these estimates, we can show the following.

######  Lemma 3.6.

For each $\theta,t\in\mathbb{B}$ and $\mathbb{P}$-almost every
$\omega\in\Omega$,

(3.19) $\displaystyle\lambda^{\prime}(\theta)(t)(\omega)=\langle
t\mathbf{L}_{1,\theta}(h(\theta))(\sigma\omega),1_{M}\rangle+\langle\mathbf{L}_{\theta}(h^{\prime}(\theta)(t))(\sigma\omega),1_{M}\rangle.$

In particular, we have

(3.20) $\lambda^{\prime}(0)(t)(\omega)=0.$

###### Proof.

For each $\theta\in\mathbb{B}$ and $\varphi\in\mathrm{Ker}(\mathbf{I})$, we
define $\lambda(\theta,\varphi)\in L^{\infty}(\Omega,\mathbb{C})$ by

$\lambda(\theta,\varphi)(\omega)=\langle\mathbf{M}(\theta,\varphi)(\sigma\omega),1_{M}\rangle\quad(\omega\in\Omega).$

Then, for any $\theta,t\in\mathbb{B}$, $\varphi\in\mathrm{Ker}(\mathbf{I})$
and $\mathbb{P}$-almost every $\omega\in\Omega$, it follows from the
assumption (A2) and $1_{M}\in\mathscr{E}$ (given in Subsection 1.2) that

$|\lambda(\theta+t,\varphi)(\omega)-\lambda(\theta,\varphi)(\omega)-\langle
t\mathbf{L}_{1,\theta}(\varphi)(\sigma\omega),1_{M}\rangle|\\\ \leq
C_{M}\|\mathbf{M}(\theta+t,\varphi)(\sigma\omega)-\mathbf{M}(\theta,\varphi)(\sigma\omega)-t\mathbf{L}_{1,\theta}(\varphi)(\sigma\omega)\|$

with $C_{M}=\|1_{M}\|_{\mathscr{E}}>0$. Therefore, due to (3.7), we get

(3.21) $\partial_{1}\lambda(\theta,\varphi)(t)(\omega)=\langle
t\mathbf{L}_{1,\theta}(\varphi)(\sigma\omega),1_{M}\rangle.$

Similarly, by virtue of (3.4) and the assumption (A2) and
$1_{M}\in\mathscr{E}$, we can calculate $\partial_{2}\lambda(\theta,\varphi)$
as

(3.22)
$\partial_{2}\lambda(\theta,\varphi)(\psi)(\omega)=\langle\mathbf{L}_{\theta}(\psi)(\sigma\omega),1_{M}\rangle.$

The first assertion of Lemma 3.6 immediately follows from (3.21) and (3.22)
together with the chain rule for Fréchet derivatives. The second assertion
also immediately follows from the assumption on $g$, (3.6) and (3.18). ∎

Let $\widehat{\Lambda}(\theta)(\omega)=\log|\lambda(\theta)(\omega)|$ for each
$\theta\in\mathbb{C}$ and $\omega\in\Omega$. Then, since
$\widehat{\Lambda}(\theta)=\frac{1}{2}\log(\lambda(\theta)\overline{\lambda(\theta)})$
one can check that (by the chain rule for Fréchet derivatives)

(3.23)
$\displaystyle\widehat{\Lambda}^{\prime}(\theta)=\frac{\lambda(\theta)\overline{\lambda^{\prime}(\theta)}+\lambda^{\prime}(\theta)\overline{\lambda(\theta)}}{2|\lambda(\theta)|^{2}}=\Re\left(\frac{\lambda^{\prime}(\theta)\overline{\lambda(\theta)}}{|\lambda(\theta)|^{2}}\right)$

under the identification in Remark 3.1, where $\Re(z)=(z+\overline{z})/2$ is
the real part of a complex number $z$. Therefore, by virtue of (3.16) and
(3.20) we have that

$\widehat{\Lambda}(0)=0\quad\text{ and}\quad\widehat{\Lambda}^{\prime}(0)=0,$

which completes the proof of the first assertion in the item (2) of Theorem
3.2.

###  3.3. Second order regularity

For $(\theta,\varphi)\in\mathbb{C}\times L^{\infty}(\Omega,\mathscr{B})$, we
denote by $\partial_{1}^{2}\widetilde{\mathbf{M}}(\theta,\varphi)$ the Fréchet
derivative of $\partial_{1}\widetilde{\mathbf{M}}(\cdot,\varphi):\mathbb{C}\to
L(\mathbb{C},\mathrm{Ker}(\mathbf{I}))$ at $\theta\in\mathbb{C}$. Note that
$\partial_{1}^{2}\widetilde{\mathbf{M}}(\cdot,\varphi)$ is a map from
$\mathbb{C}$ to $L(\mathbb{C},L(\mathbb{C},\mathrm{Ker}(\mathbf{I})))$. In a
similar manner, we define
$\partial_{2}\partial_{1}\widetilde{\mathbf{M}}(\theta,\varphi)$,
$\partial_{1}\partial_{2}\widetilde{\mathbf{M}}(\theta,\varphi)$ and
$\partial_{2}^{2}\widetilde{\mathbf{M}}(\theta,\varphi)$.

######  Lemma 3.7.

For each $\theta\in\mathbb{C}$ and $\varphi\in
L^{\infty}(\Omega,\mathscr{B})$,

(3.24)
$\partial_{1}^{2}\widetilde{\mathbf{M}}(\theta,\varphi)(s)(t)=ts\mathbf{L}_{2,\theta}(\varphi)\quad(s,t\in\mathbb{C}),$
(3.25)
$\partial_{2}\partial_{1}\widetilde{\mathbf{M}}(\theta,\varphi)(\psi)(t)=t\mathbf{L}_{1,\theta}(\psi)\quad(\psi\in\mathrm{Ker}(\mathbf{I}),\;t\in\mathbb{C}).$

Furthermore, for each $\theta\in\mathbb{C}$ and $\varphi\in
L^{\infty}(\Omega,\mathscr{B})$,

$\partial_{1}\partial_{2}\widetilde{\mathbf{M}}(\theta,\varphi)(t)(\psi)=t\mathbf{L}_{1,\theta}(\psi)\quad(t\in\mathbb{C},\;\psi\in\mathrm{Ker}(\mathbf{I})),$
$\partial_{2}^{2}\widetilde{\mathbf{M}}(\theta,\varphi)(\phi)(\psi)=0\quad(\phi,\psi\in\mathrm{Ker}(\mathbf{I})).$

###### Proof.

One can prove (3.24) (resp. (3.25)) from the formula (3.7) as the proof of
(3.7) (resp. (3.4)), see also [DFGV2018, Lemma B.11]. The proof of the later
equalities also follows from the formula (3.4) by a similar argument (cf.
[DFGV2018, Lemma B.10]). ∎

It follows from (3.5) that $\mathbf{I}$ is of class $\mathscr{C}^{2}$, and so
is
$\widetilde{\mathbf{F}}:\mathbb{B}\times\mathrm{Ker}(\mathbf{I})\to\mathrm{Ker}(\mathbf{I})$
due to the form of $\widetilde{\mathbf{F}}$ given in (3.1), (3.3). Hence, by
the implicit function theorem, $h:\mathbb{B}$ given in (3.15) is of class
$\mathscr{C}^{2}$.

Furthermore, in a manner similar to one in the proof of Lemma 3.6, it follows
from (3.19) and Lemma 3.7 together with assumption (A2) and
$1_{M}\in\mathscr{E}$, we have

$\lambda^{\prime\prime}(\theta)(t)(s)(\omega)=\langle
ts\mathbf{L}_{2,\theta}(h(\theta))(\sigma\omega),1_{M}\rangle+\langle
t\mathbf{L}_{1,\theta}(h^{\prime}(\theta)(s))(\sigma\omega),1_{M}\rangle\\\
+\langle
s\mathbf{L}_{1,\theta}(h^{\prime}(\theta)(t))(\sigma\omega),1_{M}\rangle+\langle\mathbf{L}_{\theta}(h^{\prime\prime}(\theta)(t)(s))(\sigma\omega),1_{M}\rangle$

all $\theta,t,s\in\mathbb{C}$ and $\mathbb{P}$-almost every $\omega\in\Omega$.
On the other hand, since $h$ is a $\mathscr{C}^{2}$ map from $\mathbb{B}$ to
$\mathrm{Ker}(\mathbf{I})$,
$h^{\prime\prime}(0)(t)(s)\in\mathrm{Ker}(\mathbf{I})$ for all
$t,s\in\mathbb{B}$ and

$\langle\mathbf{L}_{\theta}(h^{\prime\prime}(\theta)(t)(s))(\sigma\omega),1_{M}\rangle=\langle
h^{\prime\prime}(0)(t)(s)(\omega),1_{M}\rangle=0.$

So, by (3.6) and (3.17),

(3.26) $\displaystyle\lambda^{\prime\prime}(0)(t)(s)(\omega)$
$\displaystyle=ts\langle h_{0,\omega},(g_{\omega})^{2}\cdot 1_{M}\circ
T_{\omega}\rangle+t\langle h^{\prime}(\theta)(s)(\omega),g_{\omega}\cdot
1_{M}\circ T_{\omega}\rangle$ $\displaystyle\quad\quad\quad\quad+s\langle
h^{\prime}(\theta)(t)(\omega),g_{\omega}\cdot 1_{M}\circ T_{\omega}\rangle$
$\displaystyle=ts\mathbb{E}_{\mu_{\omega}}\left[g_{\omega}^{2}\right]+2ts\left\langle\sum_{n=1}^{\infty}\mathcal{L}^{(n)}_{\sigma^{-n}\omega}(h_{0,\sigma^{-n}\omega}g_{\sigma^{-n}\omega}),g_{\omega}\right\rangle$
$\displaystyle=ts\mathbb{E}_{\mu_{\omega}}\left[g_{\omega}^{2}\right]+2ts\sum_{n=1}^{\infty}\mathbb{E}_{\mu_{\sigma^{-n}\omega}}\left[g_{\sigma^{-n}\omega}\cdot
g_{\omega}\circ T_{\sigma^{-n}\omega}^{(n)}\right].$

Thus it follows from the $\sigma$-invariance of $\mathbb{P}$ that
$\mathbb{E}_{\mathbb{P}}\left[\lambda^{\prime\prime}(0)\right]=V$ under the
identification of Remark 3.1 (recall (1.15)).

On the other hand, it follows from (3.23) that (by chain and quotient rules
for Fréchet derivatives)

$\widehat{\Lambda}^{\prime\prime}(\theta)=\Re\left(\frac{\lambda^{\prime\prime}(\theta)}{\lambda(\theta)}-\left(\frac{\lambda^{\prime}(\theta)}{\lambda(\theta)}\right)^{2}\right).$

Since $g_{\omega}$ is a real-valued function, (3.16), (3.20) and (3.26) leads
to that

$\widehat{\Lambda}^{\prime\prime}(0)=\Re\left(\lambda^{\prime\prime}(0)\right)=\lambda^{\prime\prime}(0)\quad\text{$\mathbb{P}$-almost
surely}.$

These observations immediately imply the later assertion of the item (3) of
Theorem 3.2.

###  3.4. Relation between $\Lambda_{\theta}$ and $\widehat{\Lambda}(\theta)$

We first show

(3.27)
$\mathbb{E}_{\mathbb{P}}\left[\widehat{\Lambda}(\theta)\right]\leq\Lambda_{\theta}$

for every $\theta$ with sufficiently small absolute value. It follows from the
item (1) of Theorem 3.2 and (UG) that for every $\theta$ with sufficiently
small absolute value and $\mathbb{P}$-almost every $\omega\in\Omega$,

$\Lambda_{\theta}\geq\lim_{n\to\infty}\frac{1}{n}\log\|\mathcal{L}_{\theta,\omega}^{(n)}h_{\theta,\omega}\|\geq\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\log|\lambda_{\theta,\sigma^{j}\omega}|$

and the claim immediately follows from the definition of $\widehat{\Lambda}$
and the ergodicity of $(\theta,\mathbb{P})$.

Let $\mathscr{B}=\mathscr{F}_{\theta,\omega}\oplus\mathscr{R}_{\theta,\omega}$
be the closed splitting for $\mathcal{L}_{\theta}$ given in Subsection 3.1.
Note that $\mathrm{dim}(\mathscr{F}_{0,\omega})=1$ due to (UG). On the other
hand, it immediately follows from (1.18) that for each $\omega\in\Omega$,
$\theta\mapsto\mathcal{L}_{\theta,\omega}:\mathbb{C}\to L(\mathscr{B})$ is
continuous. Hence, by the semi-continuity argument of Lyapunov exponents of
operator cocycles restricted on finite dimensional spaces (cf. [DFGV2018,
Theorem 3.12]), we get that

(3.28) $\mathrm{dim}(\mathscr{F}_{\theta,\omega})=1$

for every $\theta$ with sufficiently small absolute value and
$\mathbb{P}$-almost every $\omega\in\Omega$.

Finally, the item (3) of Theorem 3.2 immediately follows from (3.27), (3.28)
and the item (1) of Theorem 3.2.

###  3.5. Regularity of adjoint cocycles

Note that the map from $\mathbb{N}\times\Omega\times\mathscr{B}^{\prime}$ to
$\mathscr{B}^{\prime}$ given by

$(n,\omega,u)\mapsto\mathcal{L}_{\theta,\sigma^{-n}\omega}^{*}\circ\mathcal{L}_{\theta,\sigma^{-n+1}\omega}^{*}\circ\cdots\circ\mathcal{L}_{\theta,\sigma^{-1}\omega}^{*}u$

satisfies the cocycle property over the driving system
$\sigma^{-1}:\Omega\to\Omega$. Hence, as the cocycle
$(\mathcal{L}_{\theta},\sigma)$ in Subsection 3.1, we define
$\mathbf{J}:L^{\infty}(\Omega,\mathscr{B}^{\prime})\to
L^{\infty}(\Omega,\mathbb{C})$ and
$\mathbf{L}_{\theta}^{*}:L^{\infty}(\Omega,\mathscr{B}^{\prime})\to
L^{\infty}(\Omega,\mathscr{B}^{\prime})$ by

$\mathbf{J}(u)(\omega)=u_{\omega}(h_{0,\omega}),\quad\mathbf{L}_{\theta}^{*}(u)(\omega)=\mathcal{L}_{\theta,\omega}^{*}u_{\sigma\omega}$

for $u\in L^{\infty}(\Omega,\mathscr{B}^{\prime})$ and $\omega\in\Omega$. By
(UG), it holds that

$\mathbf{J}\circ\mathbf{L}^{*}_{0}(u)(\omega)=\mathbf{J}(u)(\sigma\omega)\quad\text{for
$\mathbb{P}$-almost every $\omega\in\Omega$},$

so $\mathbf{L}^{*}_{0}$ preserves $\mathrm{Ker}(\mathbf{J})$. Let $w_{0}\in
L^{\infty}(\Omega,\mathscr{B}^{\prime})$ such that

(3.29)
$\mathbf{L}^{*}_{0}(w_{0})=w_{0}\quad\text{and}\quad\mathbf{J}(w_{0})=1\quad\text{$\mathbb{P}$-almost
surely}$

(its existence will be proven in Lemma 3.8 below). In a similar manner to
Subsection 3.2 (by using the argument in the proof of Lemma 3.9 below), one
can find a small neighborhood $\mathbb{B}$ of $0$ in $\mathbb{C}$ such that
$\mathbf{G}(\theta,u):\Omega\to\mathscr{B}^{\prime}$ given by

$\mathbf{G}(\theta,u)=\frac{\mathbf{L}_{\theta}^{*}(u)}{\mathbf{J}\circ\mathbf{L}_{\theta}^{*}(u)}-u$

is well-defined and in $\mathrm{Ker}(\mathbf{J})$ for any
$(\theta,u)\in\mathbb{B}\times(w_{0}+\mathrm{Ker}(\mathbf{J}))$. Define
$\widetilde{\mathbf{G}}:\mathbb{B}\times\mathrm{Ker}(\mathbf{J})\to\mathrm{Ker}(\mathbf{J})$
by

$\widetilde{\mathbf{G}}(\theta,u)=\mathbf{G}(\theta,w_{0}+u)\quad\text{for
$(\theta,u)\in\mathbb{B}\times\mathrm{Ker}(\mathbf{J})$},$

and its derivatives
$\partial_{1}\widetilde{\mathbf{G}},\partial_{2}\widetilde{\mathbf{G}}$ as in
Subsection 3.2.

As in Subsection 3.2, we will apply the implicit function theorem to
$\widetilde{\mathbf{G}}$. Necessary estimates for it is the following.

######  Lemma 3.8.

There is a constant $C_{2}^{\prime}>0$ and $\rho\in(0,1)$ such that for any
$u\in\mathrm{Ker}(\mathbf{J})$ and $n\geq 0$,

(3.30)
$\left\|(\mathbf{L}^{*}_{0})^{n}(u)\right\|_{L^{\infty}(\Omega,\mathscr{B}^{\prime})}\leq
C_{2}^{\prime}\rho^{n}\|u\|_{L^{\infty}(\Omega,\mathscr{B}^{\prime})}.$

Furthermore, there exists $w_{0}\in L^{\infty}(\Omega,\mathscr{B}^{\prime})$
satisfying (3.29).

###### Proof.

Let $\Pi_{\omega}:\mathscr{B}\to\mathrm{Ker}(\mathbf{I})$ denote the
projection onto $\mathrm{Ker}(\mathbf{I})$ along the subspace spanned by
$h_{0,\omega}$. By (UG), we can take a positive number $K>0$ such that

$1\leq\|h_{0,\omega}\|\leq K\quad\text{$\mathbb{P}$-almost surely}.$

Let $n_{1}$ be an integer such that $K^{-1}-C_{2}\rho^{n_{1}}>0$, and $c$ a
positive number given by

$c=\frac{K^{-1}-C_{2}\rho^{n_{1}}}{C_{0}^{n_{1}}}.$

We will show that

(3.31) $\operatorname*{ess\,sup}_{\omega\in\Omega}\|\Pi_{\omega}\|\leq
2c^{-1}.$

Note that the invariant splitting
$\mathscr{B}=\mathscr{F}_{0,\omega}\oplus\mathscr{R}_{0,\omega}$ with
$\mathrm{dim}(\mathscr{F}_{0,\omega})=1$ with respect to the cocycle
$(\mathcal{L}_{0},\sigma)$ (given in Subsection 3.1) is written as
$\mathscr{F}_{0,\omega}=\mathbb{C}h_{0,\omega}$ and
$\mathscr{R}_{0,\omega}=\\{\varphi\in\mathscr{B}\mid\langle\varphi,1_{M}\rangle=0\\}$.
Let $\gamma_{\omega}$ be a real number given by

$\gamma_{\omega}=\inf\left\\{\|ah_{0,\omega}+\varphi\|\;:\;a\in\mathbb{C},\;\varphi\in\mathscr{R}_{0,\omega},\;\|ah_{0,\omega}\|=\|\varphi\|=1\right\\}.$

Then it is straightforward to see that $\|\Pi_{\omega}\|\leq
2\gamma_{\omega}^{-1}$ (cf. [DF2018, Lemma 1]). On the other hand, by (A1),
for any $a\in\mathbb{C}$ and $\varphi\in\mathscr{R}_{0,\omega}$,

$\|ah_{0,\omega}+\varphi\|\geq\frac{\|\mathcal{L}_{0,\omega}^{(n_{1})}(ah_{0,\omega}+\varphi)\|}{C_{0}^{n_{1}}}\geq\frac{\|\mathcal{L}_{0,\omega}^{(n_{1})}(ah_{0,\omega})\|-\|\mathcal{L}_{0,\omega}^{(n_{1})}\varphi\|}{C_{0}^{n_{1}}}.$

So, if $\|ah_{0,\omega}\|=\|\varphi\|=1$, then it follows from (UG) that

$\|ah_{0,\omega}+\varphi\|\geq\frac{1}{C_{0}^{n_{1}}}\left(\frac{\|h_{0,\sigma^{n_{1}}\omega}\|}{\|h_{0,\omega}\|}-C_{2}\rho^{n_{1}}\right)\geq
c,$

and we get (3.31).

It follows from (UG) and (3.31) that for any $u\in\mathrm{Ker}(\mathbf{J})$,

$\displaystyle\left\|(\mathbf{L}^{*}_{0})^{n}(u)(\omega)\right\|_{\mathscr{B}^{\prime}}$
$\displaystyle=\sup_{\varphi\in\mathscr{B},\;\|\varphi\|\leq
1}\left|u(\sigma^{n}\omega)(\mathcal{L}_{0,\omega}^{(n)}\varphi)\right|$
$\displaystyle=\sup_{\varphi\in\mathscr{B},\;\|\varphi\|\leq
1}\left|u(\sigma^{n}\omega)(\mathcal{L}_{0,\omega}^{(n)}\Pi_{\omega}\varphi)\right|$
$\displaystyle\leq
C_{2}\rho^{n}\|\Pi_{\omega}\|\|u\|_{L^{\infty}(\Omega,\mathscr{B}^{\prime})}$
$\displaystyle\leq
2c^{-1}C_{2}\rho^{n}\|u\|_{L^{\infty}(\Omega,\mathscr{B}^{\prime})},$

and the first assertion of Lemma 3.8 is proven with
$C_{2}^{\prime}=2c^{-1}C_{2}$.

Let $\mathscr{F}^{*}_{0,\omega}$ be the subspace of $\mathscr{B}^{\prime}$
given by

$\mathscr{F}^{*}_{0,\omega}=\left\\{u\in\mathscr{B}^{\prime}\mid
u(\varphi)=0\;\text{for all $\varphi\in\mathscr{R}_{0,\omega}$}\right\\}.$

Then, $\mathrm{dim}(\mathscr{F}^{*}_{0,\omega})=1$ due to [DFGV2018, Lemma
2.6]. Let $w_{0,\omega}\in\mathscr{F}^{*}_{0,\omega}$ such that
$\mathbf{J}(w_{0})=1$. If we denote by $\widetilde{\lambda}_{\omega}$ the
complex number such that
$\mathcal{L}_{0,\omega}^{*}w_{0,\sigma\omega}=\widetilde{\lambda}_{\omega}w_{0,\omega}$,
then

(3.32)
$\widetilde{\lambda}_{\omega}=\widetilde{\lambda}_{\omega}w_{0,\omega}(h_{0,\omega})=\mathcal{L}_{0,\omega}^{*}w_{0,\sigma\omega}(h_{0,\omega})=w_{0,\sigma\omega}(\mathcal{L}_{0,\omega}h_{0,\omega})=w_{0,\sigma\omega}(h_{0,\omega})=1.$

Furthermore, by (UG), for each $a\in\mathbb{C}$ and
$\varphi\in\mathscr{R}_{0,\omega}$,

(3.33) $w_{0,\omega}(ah_{0,\omega}+\varphi)=a=\langle
ah_{0,\omega}+\varphi,1_{M}\rangle.$

Hence, due to (A1) we $\mathbb{P}$-almost surely have

$\|w_{0,\omega}\|_{\mathscr{B}^{\prime}}\leq\|1_{M}\|_{\mathscr{E}}.$

By the assumption $1_{M}\in\mathscr{E}$, we obtain the second assertion of
Lemma 3.8. ∎

We need the following lemma to apply the implicit function theorem to
$\widetilde{G}$. For each $\theta\in\mathbb{C}$, define
$\widetilde{\mathbf{L}}_{1,\theta}^{*}:L^{\infty}(\Omega,\mathscr{B}^{\prime})\to
L^{\infty}(\Omega,\mathscr{B}^{\prime})$ by

$\widetilde{\mathbf{L}}_{1,\theta}^{*}(u)=\mathbf{L}_{1,\theta}^{*}(w_{0}+u),\quad\mathbf{L}_{1,\theta}^{*}(v)(\omega)(\varphi)=v(\sigma\omega)(\mathcal{L}_{1,\theta,\omega}\varphi)$

for $u,v\in L^{\infty}(\Omega,\mathscr{B}^{\prime})$, $\omega\in\Omega$ and
$\varphi\in\mathscr{B}$.

######  Lemma 3.9.

For each $\theta\in\mathbb{B}$ and $u\in\mathrm{Ker}(\mathbf{J})$,

$\partial_{1}\widetilde{\mathbf{G}}(\theta,u)=\frac{\mathbf{J}\circ\widetilde{\mathbf{L}}_{\theta}^{*}(u)\cdot\widetilde{\mathbf{L}}_{1,\theta}^{*}(u)-\widetilde{\mathbf{L}}_{\theta}^{*}(u)\cdot\mathbf{J}\circ\widetilde{\mathbf{L}}_{1,\theta}^{*}(u)}{(\mathbf{J}\circ\widetilde{\mathbf{L}}_{\theta}^{*}(u))^{2}}$

and

$\partial_{2}\widetilde{\mathbf{G}}(\theta,u)=\frac{\mathbf{J}\circ\widetilde{\mathbf{L}}_{\theta}^{*}(u)\cdot\mathbf{L}_{\theta}^{*}-\widetilde{\mathbf{L}}_{\theta}^{*}(u)\cdot\mathbf{J}\circ\mathbf{L}_{\theta}^{*}}{(\mathbf{J}\circ\widetilde{\mathbf{L}}_{\theta}^{*}(u))^{2}}-\mathbf{id}.$

In particular,

$\partial_{2}\widetilde{\mathbf{G}}(0,0)=\widetilde{\mathbf{L}}^{*}-\mathbf{id}.$

###### Proof.

Define $\mathbf{M}^{*}:\mathbb{C}\times
L^{\infty}(\Omega,\mathscr{B}^{\prime})\to
L^{\infty}(\Omega,\mathscr{B}^{\prime})$ by
$\mathbf{M}^{*}(\theta,u)=\mathbf{L}^{*}_{\theta}(u)$, and
$\widetilde{\mathbf{M}}^{*}:\mathbb{C}\times
L^{\infty}(\Omega,\mathscr{B}^{\prime})\to
L^{\infty}(\Omega,\mathscr{B}^{\prime})$ by
$\widetilde{\mathbf{M}}^{*}(\theta,u)=\mathbf{M}^{*}(\theta,w_{0}+u)$. Since
$\mathbf{J}$ and $\mathbf{L}^{*}_{\theta}$ are linear with
$\theta\in\mathbb{C}$,

(3.34)
$D\mathbf{J}(u)=\mathbf{J}\quad\text{and}\quad\partial_{2}\widetilde{\mathbf{M}}^{*}(\theta,u)=\widetilde{\mathbf{L}}^{*}_{\theta}$

for all $\theta\in\mathbb{C}$ and $u\in
L^{\infty}(\Omega,\mathscr{B}^{\prime})$.

We will show that

(3.35)
$\partial_{1}\widetilde{\mathbf{M}}^{*}(\theta,u)(t)=t\widetilde{\mathbf{L}}^{*}_{1,\theta}$

for all $\theta,t\in\mathbb{C}$ and $u\in
L^{\infty}(\Omega,\mathscr{B}^{\prime})$. For each $\omega\in\Omega$ and
$\varphi\in\mathscr{B}$ with $\|\varphi\|=1$,

$\left|\left(\mathbf{M}^{*}(\theta+t,u)-\mathbf{M}^{*}(\theta,u)-t\mathbf{L}_{1,\theta}^{*}(u)\right)(\omega)(\varphi)\right|\\\
=\left|u(\sigma\omega)\left(\mathcal{L}_{\theta+t,\omega}-\mathcal{L}_{\theta,\omega}-t\mathcal{L}_{1,\theta,\omega}\right)(\varphi)\right|\\\
\leq\|u\|_{L^{\infty}(\Omega,\mathscr{B}^{\prime})}\left\|\left(\mathcal{L}_{\theta+t,\omega}-\mathcal{L}_{\theta,\omega}-t\mathcal{L}_{1,\theta,\omega}\right)(\varphi)\right\|.$

Hence, it follows from (3.9) that

$\lim_{t\to
0}\frac{\left\|\mathbf{M}^{*}(\theta+t,u)-\mathbf{M}^{*}(\theta,u)-t\mathbf{L}_{1,\theta}^{*}(u)\right\|_{L^{\infty}(\Omega,\mathscr{B}^{\prime})}}{|t|}=0,$

and we get (3.35).

On the other hand, by (3.13) and (3.33), we have that

$\mathbf{J}\circ\widetilde{\mathbf{L}}^{*}_{1,0}(0)(\omega)=w_{0,\sigma\omega}\left(\mathcal{L}_{1,0,\omega}h_{0,\omega}\right)=\mathbf{I}\circ\widetilde{\mathbf{L}}_{1,0}(0)(\sigma\omega)=0$

for $\mathbb{P}$-almost every $\omega\in\Omega$. Therefore, Lemma 3.9 follows
from (3.34), (3.35) in a similar manner to the proof of Lemma 3.5. ∎

Now we can check that
$\partial_{2}\widetilde{\mathbf{G}}(0,0):\mathrm{Ker}(\mathbf{J})\to\mathrm{Ker}(\mathbf{J})$
is bijective by virtue of Lemma 3.9 in a manner similar to the proof of the
bijectivity of
$\partial_{2}\widetilde{\mathbf{F}}(0,0):\mathrm{Ker}(\mathbf{I})\to\mathrm{Ker}(\mathbf{I})$
(by using $\mathrm{dim}(\mathscr{F}^{*}_{0,\omega})=1$ shown in the proof of
Lemma 3.8 instead of $\mathrm{dim}(\mathscr{F}_{0,\omega})=1$ from (UG), and
(3.30) in Lemma 3.8 instead of (1.9) in (UG)).

Note that $\widetilde{\mathbf{G}}(0,0)=0$. It follows from the implicit
function theorem for Banach spaces together with the estimates in this
subsection that there is a small ball $\mathbb{B}$ in $\mathbb{C}$ centered at
$0$ and a $\mathscr{C}^{1}$ map $\xi:\mathbb{B}\to\mathrm{Ker}(\mathbf{J})$
such that $\widetilde{\mathbf{G}}(\theta,\xi(\theta))=0$. Hence, by the
definition of $\widetilde{\mathbf{G}}$ and the fact that $h:\mathbb{B}\to
L^{\infty}(\Omega,\mathscr{B})$ is a $\mathscr{C}^{1}$ map, one can check that
if we let

$\widetilde{w}_{\theta}=w_{0}+\xi(\theta),\quad
w_{\theta,\omega}=\frac{\widetilde{w}_{\theta,\omega}}{\widetilde{w}_{\theta,\omega}(h_{\theta,\omega})}\quad\text{and}\quad\widetilde{\lambda}_{\theta,\omega}=\frac{(\mathbf{J}\circ\mathbf{L}_{\theta}^{*}(\widetilde{w}_{\theta}))_{\omega}\widetilde{w}_{\theta,\omega}(h_{\theta,\omega})}{\widetilde{w}_{\theta,\sigma\omega}(h_{\theta,\sigma\omega})}$

for $\theta\in\mathbb{B}$ and $\omega\in\Omega$, then $w:\mathbb{B}\to
L^{\infty}(\Omega,\mathscr{B}^{\prime})$ and
$\widetilde{\lambda}:\mathbb{B}\to L^{\infty}(\Omega,\mathbb{C})$ are
$\mathscr{C}^{1}$ maps and satisfy

$\mathcal{L}_{\theta,\omega}^{*}\widetilde{w}_{\theta,\sigma\omega}=(\mathbf{J}\circ\mathbf{L}_{\theta}^{*}(\widetilde{w}_{\theta}))_{\omega}\widetilde{w}_{\theta,\omega}\quad\text{$\mathbb{P}$-almost
surely}$

and so,

$\mathcal{L}_{\theta,\omega}^{*}w_{\theta,\sigma\omega}=\frac{\mathcal{L}_{\theta,\omega}^{*}\widetilde{w}_{\theta,\sigma\omega}}{\widetilde{w}_{\theta,\sigma\omega}(h_{\theta,\sigma\omega})}=\frac{(\mathbf{J}\circ\mathbf{L}_{\theta}^{*}(\widetilde{w}_{\theta}))_{\omega}\widetilde{w}_{\theta,\omega}}{\widetilde{w}_{\theta,\sigma\omega}(h_{\theta,\sigma\omega})}=\widetilde{\lambda}_{\theta,\omega}w_{\theta,\omega}\quad\text{$\mathbb{P}$-almost
surely}.$

It also holds by construction that $w_{\theta,\omega}(h_{\theta,\omega})=1$
$\mathbb{P}$-almost surely. Therefore, repeating the argument in (3.32), we
can see that $\widetilde{\lambda}_{\theta,\omega}=\lambda_{\theta,\omega}$.
This completes the proof of the item (4) of Theorem 3.2.

###  3.6. Central limit theorem

Now we can prove Theorem 1.7 by using Theorem 3.2. By Lévy’s continuity
theorem, it suffices to show that

(3.36)