#

Wenmeng Zhang  School of Mathematical Sciences
Chongqing Normal University
Chongqing 401331, P. R. China<EMAIL_ADDRESS>,  Kening Lu  School of
Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China<EMAIL_ADDRESS>and  Weinian Zhang
School of Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China<EMAIL_ADDRESS>
# Smooth invariant foliations without a bunching condition and Belitskii’s
$C^{1}$ linearization for random dynamical systems

Wenmeng Zhang  School of Mathematical Sciences
Chongqing Normal University
Chongqing 401331, P. R. China<EMAIL_ADDRESS>,  Kening Lu  School of
Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China<EMAIL_ADDRESS>and  Weinian Zhang
School of Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China<EMAIL_ADDRESS>
###### Abstract.

Smooth linearization is one of the central themes in the study of dynamical
systems. The classical Belitskii’s $C^{1}$ linearization theorem has been
widely used in the investigation of dynamical behaviors such as bifurcations,
mixing, and chaotic behaviors due to its minimal requirement of partial second
order non-resonances and low regularity of systems. In this article, we
revisit Belitskii’s $C^{1}$ linearization theorem by taking an approach based
on smooth invariant foliations and study this problem for a larger class of
dynamical systems (random dynamical systems). We assumed that the linearized
system satisfies the condition of Multiplicative Ergodic Theorem and the
associated Lyapunov exponents satisfy Belitskii’s partial second order non-
resonant conditions. We first establish the existence of $C^{1,\beta}$ stable
and unstable foliations without assuming the bunching condition for Lyapunov
exponents, then prove a $C^{1,\beta}$ linearization theorem of Belitskii type
for random dynamical systems. As a result, we show that the classical
Belitskii’s $C^{1}$ linearization theorem for a $C^{2}$ diffeomorphism $F$
indeed holds without assuming all eigenspaces of the linear system $DF(0)$ are
invariant under the nonlinear system $F$, a requirement previously imposed by
Belitskii in his proof.

###### Key words and phrases:

Invariant foliation; invariant distribution; bunching condition; random normal
form; cohomological equation.
This work was partially supported by from NSFC (11922105,
12090010,12090013,11971330,12171336,11831012) and National Key R&D Program of
China 2022YFA1005900.

###### 2020 Mathematics Subject Classification:

Primary: 37C15, 37H15; Secondary: 37C86, 37D25, 37G05

###### Contents

  1. 1 Introduction
  2. 2 Main result
  3. 3 Random invariant distribution
  4. 4 Random invariant foliation
  5. 5 Decomposition of a cohomological equation
  6. 6 Random normal form
  7. 7 Solving the decomposed cohomological equation when $\alpha\in(0,1]$
  8. 8 Solving the decomposed cohomological equation when $\alpha=0$
  9. 9 Smoothness of the distribution
  10. 10 Proof of the main result

##  1\. Introduction

Reducing a nonlinear system to the simplest form through a smooth conjugacy is
one of the fundamental themes in the study of dynamical systems. When the
simplest form is a linear system, finding such a conjugacy is called the
smooth linearization problem, which has widely been investigated under
different frameworks such as local diffeormorphisms near fixed points ([20,
33, 51, 59]), circle diffeomorphisms ([2, 14, 25, 35, 47, 62]), and toral
diffeomorphisms ([16, 31, 37, 45, 53]).

The study of smooth linearization goes back to Poincaré ([51]). He proved that
analytic linearization can be realized for an analytic diffeomorphism
$F(x)=\Lambda x+O(|x|^{2})$ in $\mathbb{C}^{d}$ when the eigenvalues
$r_{1},...,r_{d}$ of $\Lambda$ satisfy the contraction (or expansion)
condition:

$\max_{i=1,...,d}|r_{i}|<1\;(\text{or}\;\min_{i=1,...,d}|r_{i}|>1)$

and the non-resonant conditions of all orders:

$\displaystyle r_{j}\neq r_{1}^{m_{1}}\cdots r_{d}^{m_{d}}$ (1.1)

for each $j=1,...,d$, where $m_{i}\geq 0$ are all possible integers such that
$\sum_{i=1}^{d}m_{i}\geq 2$. Siegel ([57]) replaced Poincaré’s conditions by a
small-divisor (Diophantine) condition:

$|r_{j}-r_{1}^{m_{1}}\cdots r_{d}^{m_{d}}|\geq\frac{C}{|m|^{\mu}},\quad 1\leq
j\leq d,\;m\in\mathbb{Z}^{d}_{+},\;|m|=m_{1}+\cdots+m_{d}\geq 2,$

and proved that the analytic diffeomorphism is analytically conjugated to its
linear part $\Lambda$. Siegel’s condition allows $\Lambda$ to be hyperbolic
having both contraction and expansion. The simpler proof of Siegel’s theorem
was later given by V. Arnold ([3]), Moser ([46]) and Zehnder ([63]) by using
the KAM method.

$C^{k}$ smooth linearization in $\mathbb{R}^{d}$ was firstly obtained by
Sternberg ([58, 59]) in 1950s. He proved that for each $k\in\mathbb{N}$ there
is a sufficiently large integer $N>0$ such that a $C^{N}$ local hyperbolic
diffeomorphism $F(x)=\Lambda x+O(|x|^{2})$ is $C^{k}$ conjugate to its linear
part $\Lambda$ if the eigenvalues $r_{1},\cdots,r_{d}$ of $\Lambda$ satisfy
the non-resonant condition (1.1) up to order $N$, i.e., (1.1) holds for all
$m_{i}\geq 0$ with $2\leq\sum_{i=1}^{d}m_{i}\leq N$. Following this work,
there is an extensive literature on this subject, see for example Nagumo-Isé
([48]), Chen ([12]), Nelson ([49]), Sell ([55]), and Banyaga, de la Llave, and
Wayne ([5]). A crucial condition in these works is the high order non-resonant
condition. In the meantime, it also requires high order regularity of systems.

A fundamental question is whether a smooth linearization can be realized with
the lowest order non-resonance, or even a part of the lowest order non-
resonance (we call it the partial lowest order non-resonance), together with
low order regularity of the systems. Without assuming any non-resonant
conditions, Hartman ([19]) and Grobman ([17]) independently showed in 1960’s
that a $C^{1}$ local hyperbolic diffeomorphism can be $C^{0}$ linearized.
However, the $C^{0}$ conjugacy usually does not preserve dynamical properties
such as the characteristic directions, the derivatives of the Poincaré mapping
of global orbits, the iteration rates of systems and the differentiable
structure of smooth manifolds. In order to maintain such properties, the
$C^{1}$ conjugacy is essential.

In 1960, Hartman ([18]) proved that a $C^{1,1}$ contractive diffeomorphism in
$\mathbb{R}^{d}$ admits local $C^{1}$ linearization without assuming any non-
resonances. In the same paper, he also proved that a $C^{1,1}$ typically
hyperbolic diffeomorphism admits local $C^{1}$ linearization under the
following condition:

$\displaystyle|r_{1}|/|r_{t}|<|r_{{t}+1}|^{-1},\qquad|r_{t+1}|/|r_{d}|<|r_{{t}}|,$
(1.2)

where the eigenvalues $r_{1},\cdots,r_{d}$ are ordered as

$\displaystyle|r_{1}|\geq\cdots\geq|r_{t}|>1>|r_{{t}+1}|\geq\cdots\geq|r_{d}|$
(1.3)

for ${t}\in\\{1,...,d-1\\}$. Condition (1.2) means that the contractive
spectrum and the expansive spectrum both lie in a “narrow-band” area in
$\mathbb{C}$, which is referred to as the bunching condition (see for example
[33, pp.603-604]. Since the bunching condition (1.2) holds automatically in
$\mathbb{R}^{2}$, as a result, a $C^{1,1}$ typically hyperbolic diffeomorphism
in $\mathbb{R}^{2}$ admits local $C^{1}$ linearization. Samovol ([54]) in 1972
proved that a $C^{N}$ typically hyperbolic diffeomorphism with sufficiently
large $N\in\mathbb{N}$ admits local $C^{1}$ linearization if the eigenvalues
satisfy a partial second order non-resonant condition:

$\displaystyle r_{j}\neq r_{i}r_{\kappa}$ (1.4)

for $i=1,...,{t}$, $\kappa={t}+1,...,d$ and $j=1,...,d$. Note that condition
(1.4) is not only of the lowest order but also a part of the second order non-
resonance. However, it retains the requirement of $C^{N}$ smoothness of the
system for large $N$. Samovol’s strategy is to transform the $C^{N}$ system to
its polynomial normal form by Sternberg’s normal form theorem, which requires
$N$ to be sufficiently large, and then to give an explicit form of the
conjugacy, which can linearize the polynomial normal form and can be proved to
be $C^{1}$ under (1.4).

In 1973, this high order regularity requirement was removed by Belitskii
([7]). He stated that a $C^{1,1}$ typical hyperbolic diffeomorphism $F$ in
$\mathbb{R}^{d}$ admits local $C^{1}$ linearization if a condition

$\displaystyle|r_{j}|\neq|r_{i}|\,|r_{\kappa}|$ (1.5)

holds for $i=1,...,{t}$, $\kappa={t}+1,...,d$ and $j=1,...,d$. Clearly, this
result gives the above Hartman’s $C^{1}$ linearization results since (1.2)
implies (1.5). We notice that condition (1.5) allows some second order
resonances to appear. Condition (1.5), called the strong non-resonance of
order 2, is a little stronger than (1.4), but counter examples given in [8,
pp.139-142] show that such a condition is sharp, i.e., it cannot be removed
for $C^{1}$ linearization of a $C^{2}$ system in general.

Belitskii’s theorem is a truly remarkable and optimal result for $C^{1}$
linearization, and has many applications to homoclinic bifurcations ([4, 28,
29]), heteroclinic tangencies ([36]), mixing of hyperbolic systems ([15]),
invariant measures of partially hyperbolic systems ([27]), and solutions of
higher order semilinear equations ([11, 13]). However, Belitskii’s proof was
commented “In our opinion, the proofs of Theorems given by Belitskii should be
recognized as insufficient and incomprehensible” by Bronstein and Kopanskii
([10, p.191]). Actually, in his proof, Belitskii needs to establish the
surjectivity of nonlinear operators deduced from the conjugacy equation, where
all eigenspaces of $\Lambda$ are in fact assumed to be invariant under
$F(x)=\Lambda x+O(|x|^{2})$. This is a very restrictive assumption, which will
be elaborated just after the following Corollary.

In this article, we revisit Belitskii’s $C^{1}$ linearization theorem, taking
a different approach based on the smooth invariant foliations, which was not
involved in Samovol and Belitskii’s proofs, and study this problem for random
dynamical systems, a larger class of dynamical systems. We first establish the
existence of $C^{1,\beta}$ stable and unstable foliations without assuming the
bunching condition (1.2) but only the partial second order non-resonant
condition (1.5). Then we prove a Belitskii type of $C^{1,\beta}$ linearization
theorem for random dynamical systems under a partial second order non-resonant
condition in terms of Lyapunov exponents.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and
$(\theta^{n})_{n\in\mathbb{Z}}$ be a measurable $\mathbb{P}$-measure
preserving dynamical system on $\Omega$. A random dynamical system (or a
cocycle) on the space $\mathbb{R}^{d}$ over the dynamical system
${\theta}^{n}$ is a measurable mapping

$F:\mathbb{Z}\times\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d},\quad(n,\omega,x)\mapsto
F(n,\omega,x),$

such that the mapping $F(n,\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$
forms a cocycle over $\theta^{n}$:

$F(0,\omega,\cdot)={\rm id}~{}~{}\mbox{(the identity mapping)},\quad\hbox{ for
all }\;\omega\in\Omega,$ $F(n+m,\omega,\cdot)=F(n,\theta^{m}\omega,\cdot)\circ
F(m,\omega,\cdot),\quad\hbox{ for all
}\;m,n\in\mathbb{Z},\quad\omega\in\Omega.$

Replacing $\mathbb{Z}$ by $\mathbb{R}$ gives a continuous time random
dynamical system. A typical example is the solution operator for a stochastic
differential equation ([1]).

We consider a $C^{2,\alpha}$ ($\alpha\in[0.1]$) random dynamical system
$F(n,\omega,x)$ (that is, $F$ is $C^{2,\alpha}$ in $x$ for each $n$ and
$\omega$ and its derivatives are measurable with respect to $\omega$), and
write the time-one mapping $F(1,\omega,x)$ as $F(\omega,x):=F(1,\omega,x)$.
Then $F(\omega,\cdot)$ is a random diffeomorphism, whose $i$-th order
derivative is denoted by $D^{i}F(\omega,\cdot)$ for every $i\in\mathbb{N}$.
This random diffeomorphism generates the random dynamical system

$F(n,\omega,\cdot)=\begin{cases}F(\theta^{n-1}\omega,\cdot)\circ\cdots\circ
F(\omega,\cdot),&n>0,\\\ I,&n=0,\\\
F^{-1}(\theta^{n}\omega,\cdot)\circ\cdots\circ
F^{-1}(\theta^{-1}\omega,\cdot),&n<0.\end{cases}$

We assume that $F(\omega,\cdot)$ has a fixed point $x=0$ for all
$\omega\in\Omega$, to which a large class of random dynamical systems can be
converted (see [1, p.310]), and is a locally tempered $C^{2,\alpha}$ random
diffeomorphism, that is, there is a tempered ball
$V(\omega)=B_{\rho(\omega)}(0)=\\{x\in\mathbb{R}^{d}:\|x\|<\rho(\omega)\\}$,
where $\rho(\omega)$ is a random variable tempered from below (i.e.,
$\lim_{n\to\pm\infty}\frac{1}{n}\min\\{0,\log\rho(\theta^{n}\omega)\\}=0,\;\mathbb{P}-a.s.$),
such that

$\displaystyle\|D^{i}F(\omega,x)\|\leq
M(\omega),\quad\|D^{2}F(\omega,x)-D^{2}F(\omega,y)\|\leq
L(\omega)\|x-y\|^{\alpha},\quad\forall x,y\in V(\omega),$

where $M(\omega),L(\omega)>0$ are random variable tempered from above (i.e.,
$\lim_{n\to\pm\infty}\frac{1}{n}\max\\{0,\log
M(\theta^{n}\omega)\\}=0,\;\mathbb{P}-a.s.$). The size of a tempered ball may
decrease as $\omega$ varies, but these changes along each orbit
$\theta^{n}\omega$ are at a subexponential rate. The upper bounds of
$M(\omega)$ and $L(\omega)$ may grow to infinity as $\omega$ varies, but along
each orbit $\theta^{n}\omega$, it may increase only at a subexponential rate.
This nonuniform behavior is one of the intrinsic features of random dynamical
systems.

Two local tempered random diffeomporhisms $F$ and $G$ with fixed point $x=0$
are said to be $C^{1,\beta}$ locally conjugate for $\beta\in[0,1]$ if there
exists a $C^{1,\beta}$ random diffeomorphism $\Phi(\omega,x)$ defined on a
tempered ball $V(\omega)$ with $\Phi(\omega,0)=0$ such that

$\Phi(\theta\omega,F(\omega,x))=G(\omega,\Phi(\omega,x))\quad\hbox{for}\quad
x\in V(\omega),\quad\mathbb{P}-a.s.$

This conjugacy relation implies that $h$ carries orbits of $\varphi$ to orbits
of $\psi$ when the orbits stay in the corresponding tempered ball.

We write $F(\omega,x)$ as

$F(\omega,x)=\Lambda(\omega)x+f(\omega,x)=\Lambda(\omega)x+O(\|x\|^{2}),$

where $\Lambda(\omega)=DF(\omega,0)\in Gl(d,\mathbb{R})$. Assume that
$\Lambda(\omega)$ satisfies the conditions of the Multiplicative Ergodic
Theorem, that is,

$\max\\{0,\log\|\Lambda(\cdot)\|\\}\in
L^{1}(\Omega,\mathcal{F},\mathbb{P})\quad\hbox{and}\quad\max\\{0,\log\|\Lambda^{-1}(\cdot)\|\\}\in
L^{1}(\Omega,\mathcal{F},\mathbb{P}),$

and that $\mathbb{P}$ is an ergodic invariant measure for $\theta$. By the
Multiplicative Ergodic Theorem (see Section 2 for details), there exists a
$\theta$-invariant set $\tilde{\Omega}\subset\Omega$ of full measure such that
for each $\omega\in\tilde{\Omega}$, $\Lambda(n,\omega)$ (the linear random
dynamical system generated by $\Lambda(\omega)$) has $p$ Lyapunov exponents
$\lambda_{j}$, $j=1,\cdots,p$. We assume that $\Lambda(n,\omega)$ is
hyperbolic, namely, there is no zero Lyapunov exponent. We order them as
follows:

$\lambda_{1}>\cdots\lambda_{\tau}>0>\lambda_{\tau+1}\cdots>\lambda_{p}.$

For the remainder of this article, we use $\Omega$ to denote $\tilde{\Omega}$
and assume that these statements are true for all $\omega\in\Omega$.

Our main result is the following theorem on $C^{1,\beta}$ foliations and
linearization for random dynamical systems in the typically hyperbolic case,
in which there are both positive and negative Lyapunov exponents.

Theorem. Let $F:\Omega\times V(\omega)\to\mathbb{R}^{d}$ be a tempered
$C^{2,\alpha}$ $(\alpha\in[0,1])$ random diffeomorphism with an
$\epsilon$-slowly continuous second order derivative when $\alpha=0$. Assume
its linearization $\Lambda(\omega)$ at hyperbolic fixed point $x=0$ satisfies
the condition of the Multiplicative Ergodic Theorem. If the Lyapunov exponents
satisfy a partial second order non-resonant condition:

$\displaystyle\lambda_{j}\neq\lambda_{i}+\lambda_{\kappa}$ (1.6)

for all $i=1,...,{\tau}$, $\kappa={\tau}+1,...,{{p}}$ and $j=1,...,{{p}}$,
then the following assertions are true:

  * (i)

Foliations: $F$ has $C^{1,\beta_{\alpha}}$ random stable and unstable
foliations with $C^{2,\alpha}$ leaves in a tempered ball $U(\omega)\subset
V(\omega)$, where $\beta_{\alpha}=0$ when $\alpha=0$ and
$\beta_{\alpha}\in(0,\alpha]$ when $\alpha\in(0,1]$.

  * (ii)

Conjugacy: $F(\omega,\cdot)$ is conjugate to its linear part $\Lambda(\omega)$
in a tempered ball $U(\omega)\subset V(\omega)$ by a $C^{1,\beta_{\alpha}}$
random diffeomorphism $\Phi:\Omega\times U(\omega)\to\mathbb{R}^{d}$
satisfying $\Phi(\omega,0)=0$ and $D\Phi(\omega,0)={\rm id}.$

Remark. (1) The assumption of $\epsilon$-slowly continuity means that
$e^{-\epsilon|n|}D^{2}F(\theta^{n}\omega,\cdot)$ is equicontinuous with
respect $n\in\mathbb{Z}$ for each fixed $\omega\in\Omega$. It is only
necessary for random dynamical systems in the case of $\alpha=0$ since, when
$\alpha\in(0,1]$ or $F$ is deterministic, the second order derivative of
$F(\omega,\cdot)$ is automatically $\epsilon$-slowly continuous. (2) The non-
resonant condition (1.6) obviously does not imply the bunching condition:

$\lambda_{1}-\lambda_{\tau}<-\lambda_{\tau+1}\quad\text{and}\quad\lambda_{\tau+1}-\lambda_{p}<\lambda_{\tau}.$
(1.7)

(3) When $F$ is deterministic, the relationship between Lyapunov exponent
$\lambda$ and its corresponding eigenvalue $r$ is $|r|=e^{\lambda}$. Thus
condition (1.6) becomes (1.5).

As a result, we have the following $C^{1,\beta_{\alpha}}$ linearization
theorem.

Corollary. Let $F:V\to\mathbb{R}^{d}$ be a $C^{2,\alpha}$ ($\alpha\in[0,1]$)
diffeomorphism over a neighborhood $V\subset\mathbb{R}^{d}$ of $x=0$. Suppose
the eigenvalues of its linearization $\Lambda:=DF(0)$ at the hyperbolic fixed
point $x=0$ satisfy (1.5). Then $F$ is conjugate to its linear part $\Lambda$
in a small neighborhood $U\subset V$ by a $C^{1,\beta_{\alpha}}$
diffeomorphism $\Phi:U\to\mathbb{R}^{d}$ satisfying $\Phi(0)=0$ and
$D\Phi(0)={\rm id},$ where $\beta_{\alpha}=0$ when $\alpha=0$ and
$\beta_{\alpha}\in(0,\alpha]$ when $\alpha\in(0,1]$.

When $\alpha=0$, this corollary is exactly the same as the above-mentioned
Belitskii’s theorem [7] except the $C^{2}$ smoothness, which is stronger than
his $C^{1,1}$ smoothness. In [7], Belitskii proved the above result under an
assumption that all eigenspaces of $\Lambda$ are invariant under the nonlinear
system $F$. To be more precise, in [7, Lemma 3.1], he assumed that the
subspaces $L_{I}:=\bigoplus_{i\in I\subset Q}L_{i}$ of $\mathbb{R}^{d}$ are
invariant under $F$ for all subsets $I$ of $Q=\\{1,\cdots,q\\}$ with
$q\in\mathbb{N}$, where each $L_{i}$ is the eigenspace of $\Lambda$
corresponding to the eigenvalue lying on the circle
$\\{z\in\mathbb{C}:|z|=\rho_{i}\\}$ and those $\rho_{i}$, being the moduli of
the eigenvalues of $\Lambda$, are ordered as

$\rho_{1}>\rho_{2}>\cdots>\rho_{k}>1>\rho_{k+1}>\cdots>\rho_{q}.$

In other words, Belitskii assumed that the invariant manifolds of $F$
associated with any spectral splitting according to their moduli are all flat,
i.e., linear spaces. This can be achieved only if one can construct a
$C^{1,1}$ invariant manifold of $F$ for any spectral splitting:

$\\{\rho_{1},\cdots,\rho_{q}\\}=\\{\rho_{i}:i\in I\\}\cup\\{\rho_{i}:i\in
Q\setminus I\\}.$

In this case, the invariant manifold is given by the graph of a $C^{1,1}$
mapping

$\phi:L_{I}=\bigoplus_{i\in I}L_{i}\mapsto L_{I^{\prime}}=\bigoplus_{i\in
Q\setminus I}L_{i}$

Using these manifolds to change the coordinate system, one obtain flat
invariant manifolds for the new system.

By the theory of invariant manifolds, $F$ has $C^{1,1}$ stable manifold and
unstable manifold, i.e., according to the spectral slitting:

$Q=\\{\rho_{i}:\rho_{i}<1\\}\cup\\{\rho_{i}:\rho_{i}>1\\}.$

However, in order to have $C^{1,1}$ pseudo-stable and pseudo-unstable
manifolds for $F$, a spectral gap condition is needed. de la Llave and Wayne
[39, Example 5.1] gave an example of a 2-dimensional $C^{\infty}$
diffeomorphism with a small spectral gap, whose pseudo-stable manifold is not
$C^{(\ln 3/\ln 2)+\varepsilon}$ for any $\varepsilon>0$, so the pseudo-stable
manifold is not $C^{1,1}$ either. To better understand the issue here, we
borrow their Example 5.1 and add one trivial stable component to get the
diffeomorphism

$F(x)=\begin{pmatrix}1/2&0&0\\\ 0&2&0\\\
0&0&3\end{pmatrix}\begin{pmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix}+\begin{pmatrix}0\\\ 0\\\ \varphi(x_{2})\end{pmatrix},$
(1.8)

where $\varphi$ is a $C^{\infty}$ function with a compact support. Note that
the eigenvalues satisfy the non-resonant condition (1.5), but the pseudo-
stable manifold associated with the splitting $\\{1/2,2\\}\cup\\{3\\}$ is not
$C^{1,1}$. Consequently, one can not transform equation (1.8) by a conjugacy
to a $C^{1,1}$ system with invariant manifolds all being linear spaces.
Furthermore, a spectral splitting such as

$Q=\\{\rho_{1},\rho_{q}\\}\cup\\{\rho_{i}:i\in Q,i\not=1,q\\}$

may not yield a $C^{1,1}$ invariant manifold (as a graph over $L_{1}\bigoplus
L_{q}$) either. The existence of such a $C^{1,1}$ invariant manifold may need
much stronger non-resonant conditions (see .e.g. [38]). In short, it is quite
restricted to assume that all the invariant manifolds of $F$ based on spectral
decompositions are linear spaces. We emphasize that the non-resonant condition
(1.5) implies neither the spectral gap condition needed for $C^{1,1}$
invariant manifolds as we see for example (1.8) nor the bunching condition
(1.2) required in Hartman’s theorem.

On the other hand, Bronstein and Kopanskii stated in [10, p.191] that “Theorem
9.4 coincides, in fact, with Theorem 5 (i.e., Belitskii’s theorem), but the
proof is new.” However, we find that there is no “Theorem 9.4” in their
monograph [10]. In fact, there are only Theorems 9.1 and 9.6 in [10, Chapter
II, Section 9] for the smooth linearization of mappings. The two theorems are
the same type as Sternberg’s theorem since $F$ was required to be sufficiently
smooth, and the derivatives of $F$ up to sufficiently high order were required
to be equal to $0$ at the origin, which is equivalent to a corresponding non-
resonant condition. These conditions on high order smoothness and vanishing
derivatives make it possible to get a contractive operator, whose fixed point
is used to construct the conjugacy, by the homotopy method (for Sternberg’s
theorem) or by the conjugacy relation (for Bronstein and Kopanskii’s results).

In this article, we do not make any assumption on the invariant manifolds of
$F$, except for the non-resonant condition (1.6) and the $C^{2,\alpha}$
regularity of $F$. We take a different approach from Belitskii’s. We construct
$C^{1,\beta_{\alpha}}$ random stable and unstable foliations with
$C^{2,\alpha}$ leaves and utilize them to decouple the system into a
contraction and an expansion. The $C^{2,\alpha}$ smoothness of the leaves can
be obtained by using the Hadamard graph transformation or the Lyapunov-Perron
method. However, without the bunching condition on the Lyapunov exponents, it
is more complicated to prove the $C^{1,\beta_{\alpha}}$ smoothness of the
foliations with respect to the base point only under the non-resonant
condition (1.6).

To obtain the $C^{1,\beta_{\alpha}}$ smoothness, it is sufficient to show that
the tangent spaces of leaves of the stable and unstable foliations are
$C^{1,\beta_{\alpha}}$ distributions thanks to Frobenius’ theorem. We
construct continuous bases for these stable and unstable distributions which
are uniformly $C^{1,\beta_{\alpha}}$ along each local leaf of the stable and
unstable foliations. This will be done by solving a random cohomological
equation along each leaf of all intermediate (pseudo-center) foliations, which
is the main step to overcome the difficulty of no bunching condition (see the
end of section 7). The non-resonant condition (1.6) plays a key role in
establishing the smoothness, which will be used to prove the result on random
normal form and to show the invertibility of cohomological operators (see
(7.36) and (8.10) below). To deal with the non-uniformity of hyperbolicity,
the Lyapunov norm is used in solving the random cohomology equation. Our proof
also involves the roughness of non-uniformly exponential dichotomy and
Journé’s lemma ([32]).

When the Lyaponov exponents satisfy the bunching condition (1.7) instead of
(1.6), the $C^{1,\beta_{\alpha}}$ smoothness of the random stable and unstable
foliations with respect to the base point can be obtained by using the
Lyapunov-Perron approach, see for example [56]. There is an extensive study of
regularity of invariant foliations. The optimal regularity of invariant
foliations for hyperbolic dynamical systems under bunching conditions was
obtained by Hasselblatt ([21, 22, 23]), and Hasselblatt and Wilkinson ([24]).
Moreover, Push, Shub and Wilkinson ([52]) established the regularity of
invariant foliations of partially hyperbolic diffeomorphisms under bunching
conditions.

Finally, we mention some existing results which are closely relevant to the
results derived in this article. The theorems of random version were
established for Poincaré’s analytic linearization, Siegel’s analytic
linearization, Sternberg’s smooth linearization and Hartman-Grobman’s $C^{0}$
linearization ([6, 40, 41, 42, 60]), but no result for random dynamical
systems on $C^{1}$ linearization until very recently. In [44], the authors of
the current paper proved a random version of Hartman’s $C^{1,\beta_{\alpha}}$
linearization for a $C^{1,\alpha}$ ($\alpha\in(\alpha_{0},1]$) random
contraction (or expansion) in $\mathbb{R}^{d}$, where $\alpha_{0}\in(0,1)$ is
a constant depending on the Lyapunov exponents.

##  2\. Main result

In this section, we first review some of the basic concepts and facts for
random dynamical systems taken from Arnold’s book [1]. We then state our main
results.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with a sample set
$\Omega$, a $\sigma$-algebra $\mathcal{F}\subset 2^{\Omega}$ and a probability
measure $\mathbb{P}$ on $\mathcal{F}$, and let
$(\Omega,\mathcal{F},\mathbb{P},(\theta^{n})_{n\in\mathbb{Z}})$ (denoted by
$\theta$ for short) be a metric dynamical system (see [1, p.536]). A mapping

${F}:\mathbb{Z}\times\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d},\quad(t,\omega,x)\mapsto{F}(t,\omega,x),$

where $\mathbb{Z}$ is endowed with its Borel $\sigma$-algebra
${\mathcal{B}}(\mathbb{Z})$, is called a $C^{N}$ ($N\geq 1$) random dynamical
system (abbreviated as RDS) on the measurable space ($\mathbb{R}^{d}$,
${\mathcal{B}}$) over $\theta$ if

  * (1)

$F(n,\omega,\cdot)$ is
$\mathcal{B}(\mathbb{Z})\otimes\mathcal{F}\otimes\mathcal{B}(\mathbb{R}^{d})$-measurable
and $F(n,\omega,\cdot)$ is $C^{N}$ with measurable derivatives, and

  * (2)

the mappings ${F}(n,\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$,
$n\in\mathbb{Z}$, form a cocycle over $\theta$, i.e.,

$\displaystyle{F}(0,\omega,\cdot)={\rm id}~{}\mbox{{\rm(}the identity
mapping{\rm)}},\quad\forall\omega\in\Omega,$
$\displaystyle{F}(n+m,\omega,\cdot)={F}(n,\theta^{m}\omega,\cdot)\circ{F}(m,\omega,\cdot),\quad\forall
n,m\in\mathbb{Z},\quad\forall\omega\in\Omega.$

Clearly, the time-one mapping
${F}(\omega,\cdot):={F}(1,\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a
$C^{N}$ random diffeomorphism. Conversely, a $C^{N}$ random diffeomorphism
${F}(\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ generates a $C^{N}$ RDS by

$\displaystyle{F}(n,\omega,\cdot):=\begin{cases}{F}(\theta^{n-1}\omega,\cdot)\circ\cdots\circ{F}(\omega,\cdot),&n>0,\\\
{\rm id},&n=0,\\\
{F}^{-1}(\theta^{n}\omega,\cdot)\circ\cdots\circ{F}^{-1}(\theta^{-1}\omega,\cdot),&n<0.\end{cases}$
(2.1)

When $\theta$ is the identity mapping, the RDS becomes a usual deterministic
system.

A random variable $K:\Omega\to(0,\infty)$ is said to be tempered if it is
tempered from above, i.e.,

$\lim_{n\to\pm\infty}\frac{1}{n}\max\\{0,\log
K(\theta^{n}\omega)\\}=0,\qquad\mathbb{P}-a.s.,$

and is tempered from below, i.e.,

$\lim_{n\to\pm\infty}\frac{1}{n}\min\\{0,\log
K(\theta^{n}\omega)\\}=0,\qquad\mathbb{P}-a.s.$

Note that $K(\omega)$ is tempered from above (or below) if and only if
$1/K(\omega)$ is tempered from below (or above). When $K(\omega)$ is tempered
above, it follows from [1] that for any constant $\epsilon>0$ there is a
tempered random variable $K_{\epsilon}(\omega)$ such that
$K_{\epsilon}(\omega)\geq K(\omega)$ and

$\displaystyle e^{-\epsilon|n|}K_{\epsilon}(\omega)\leq
K_{\epsilon}(\theta^{n}\omega)\leq
e^{\epsilon|n|}K_{\epsilon}(\omega),\quad\forall n\in\mathbb{Z}.$ (2.2)

Such $K_{\epsilon}(\omega)$ is called an $\epsilon$-slowly varying random
variable. We call $V(\omega):=\\{x\in\mathbb{R}^{d}:\|x\|\leq\rho(\omega)\\}$
a tempered ball if $\rho(\omega)>0$ is a random variable tempered from below.
For an integer $N\in\mathbb{N}$ and a real number $\alpha\in[0,1]$, we call
${F}(\omega,x)$ a tempered $C^{N,\alpha}$ random diffeomorphism (see [42]) if
${F}(\omega,\cdot)$ is a $C^{N}$ diffeomorphism and the $i$-th derivative
$D^{i}{F}(\omega,x)$ are measurable with respect to $\omega$ such that

$\displaystyle\begin{split}&\|D^{i}{F}(\omega,x)\|\leq M(\omega),\quad\forall
i=0,1,...,N,\quad\forall\omega\in\Omega,\\\
&\|D^{N}{F}(\omega,x)-D^{N}{F}(\omega,y)\|\leq
L(\omega)\|x-y\|^{\alpha},\quad\forall x,y\in V(\omega),\end{split}$ (2.3)

where $M(\omega),L(\omega)>0$ are random variables tempered from above. In
particular, if $\alpha=0$ then it means that $F(\omega,x)$ is $C^{N}$.
Moreover, in the case of $\alpha=0$, we can further assume that
$D^{N}F(\omega,\cdot)$ is $\epsilon$-slowly continuous with respect to $x$ for
each fixed $\omega\in\Omega$, where $\epsilon>0$ is a small constant, i.e.,
$e^{-\epsilon|n|}D^{N}F(\theta^{n}\omega,\cdot)$ is equicontinuous with
respect $n\in\mathbb{Z}$ for each fixed $\omega\in\Omega$. When
$D^{N}F(\omega,x)$ is Hölder continuous with a tempered Hölder constant, then
$D^{N}F(\omega,\cdot)$ is $\epsilon$-slowly continuous; when the system is
deterministic, the uniform continuity of $D^{N}F$ implies that it is
$\epsilon$-slowly continuous.

We assume that ${F}(n,\omega,0)=0$ for any $n\in\mathbb{Z}$ and any
$\omega\in\Omega$, and we use the notation $DF(\omega,\cdot)$ (instead of
$D^{1}F(\omega,\cdot)$) for the first order derivative of $F$ with respect to
$x$.

Consider a linear random diffeomorphism
$\Lambda(\omega):=\Lambda(\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$. Then
it generates a linear random dynamical system

$\displaystyle{\Lambda}(n,\omega):=\begin{cases}\Lambda(\theta^{n-1}\omega)\cdots\Lambda(\omega),&n>0,\\\
{\rm id},&n=0,\\\
\Lambda^{-1}(\theta^{n}\omega)\cdots\Lambda^{-1}(\theta^{-1}\omega),&n<0.\end{cases}$
(2.4)

Assume that $\Lambda(\omega)$ satisfies the conditions of the Multiplicative
Ergodic Theorem, that is,

$\max\\{0,\log\|\Lambda(\cdot)\|\\}\in
L^{1}(\Omega,\mathcal{F},\mathbb{P})\quad\hbox{and}\quad\max\\{0,\log\|\Lambda^{-1}(\cdot)\|\\}\in
L^{1}(\Omega,\mathcal{F},\mathbb{P}).$ (2.5)

and $\mathbb{P}$ is an ergodic invariant measure for $\theta$. Then, by the
Multiplicative Ergodic Theorem ([1]), there exists a $\theta$-invariant set
$\tilde{\Omega}\subset\Omega$ of full measure such that for each
$\omega\in\tilde{\Omega}$, the linear random dynamical system
$\Lambda(n,\omega)$ has $p$ Lyapunov exponents: $\lambda_{j}$ with the
multiplicity $d_{j}$, $j=1,...,p$, and the corresponding Oseledets spaces
$E_{j}(\omega)$ with dimension $d_{j}$ form an invariant splitting of the
phase space $\mathbb{R}^{d}$:

$\mathbb{R}^{d}=E_{1}(\omega)\oplus\cdots\oplus E_{{p}}(\omega).$

Since $\mathbb{P}$ is ergodic, all $\lambda_{i}$, $d_{i}$, and $p$ are
constant. For the remainder of this article, we use $\Omega$ to denote
$\tilde{\Omega}$ and assume that these statements are true for all
$\omega\in\Omega$.

Assume that the linear random dynamical system $\Lambda(n,\omega)$ is
hyperbolic, i.e., there is no zero Lyapunov exponent. We arrange the Lyapunov
exponents as

$\displaystyle\lambda_{1}>\cdots>\lambda_{\tau}>0>\lambda_{\tau+1}>\cdots>\lambda_{{p}}$
(2.6)

for a certain $\tau\in\\{1,...,{p}-1\\}$. It follows from [42, Lemma 2.8] that
$\Lambda(n,\omega)$ is conjugate to a block diagonal linear random dynamical
system by a random isomorphism and the corresponding Oseledets spaces are
given by

$E_{j}(\omega)=X_{j}:=\\{0\\}\times\cdots\times\\{0\\}\times\mathbb{R}^{d_{j}}\times\\{0\\}\times\cdots\times\\{0\\}\subset\mathbb{R}^{d}.$

Moreover, for any small $\epsilon>0$ one has

$\displaystyle\begin{split}&\|\Lambda(n,\omega)|_{X_{j}}\|\leq
K(\omega)e^{(\lambda_{j}-\epsilon)n},\qquad\forall n<0,\\\
&\|\Lambda(n,\omega)|_{X_{j}}\|\leq
K(\omega)e^{(\lambda_{j}+\epsilon)n},\qquad\forall n\geq 0,\end{split}$ (2.7)

for each $j=1,...,p$, where $K(\omega)>0$ is a random variable tempered from
above. In particular, it satisfies the exponential dichotomy

$\displaystyle\begin{split}&\|\Lambda(n,\omega)|_{X_{u}}\|\leq
K(\omega)e^{(\lambda_{\tau}-\epsilon)n},\qquad\forall n<0,\\\
&\|\Lambda(n,\omega)|_{X_{s}}\|\leq
K(\omega)e^{(\lambda_{\tau+1}+\epsilon)n},\qquad\forall n\geq 0,\end{split}$
(2.8)

where $\lambda_{\tau}>0>\lambda_{\tau+1}$ and $X_{u}:=X_{1}\oplus\cdots\oplus
X_{\tau},$ $X_{s}:=X_{{\tau}+1}\oplus\cdots\oplus X_{{p}}.$ Notice that this
exponential dichotomy is nonuniform since $K(\omega)$ may be arbitrarily large
as $\omega$ varies. However, along each orbit
$(\theta^{n}\omega)_{n\in\mathbb{Z}}$, $K(\omega)$ can increase only at a
subexponential rate which is one of the intrinsic features of RDS. The
following definition on random conjugacy can be found in [1, 42, 44].

######  Definition 2.1.

Two tempered random diffeomporhisms $F(\omega,\cdot)$ and $G(\omega,\cdot)$
are said to be $C^{k,\beta}$ conjugate, where $k\geq 0$ is an integer and
$\beta\in[0,1]$ is a real, if there exists a $C^{k,\beta}$ random
diffeomorphism $\Phi(\omega,\cdot):V(\omega)\to\mathbb{R}^{d}$ such that

$\displaystyle\Phi(\theta\omega,F(\omega,x))=G(\omega,\Phi(\omega,x)),\qquad\forall
x\in V(\omega),$ (2.9)

where $V(\omega)$ is a tempered ball. In particular, as mentioned in the
introduction, $F$ is said to be $C^{k,\beta}$ linearized if $G$ is a linear
random diffeomporhism.

Now, we restate our main theorem of this paper for the sake of completeness of
this section.

######  Theorem 2.1.

Let $F:\Omega\times V(\omega)\to\mathbb{R}^{d}$ be a tempered $C^{2,\alpha}$
$(\alpha\in[0,1])$ random diffeomorphism with an $\epsilon$-slowly continuous
second order derivative when $\alpha=0$. Assume that its linearization
$\Lambda(\omega)=DF(\omega,0)$ at the hyperbolic fixed point $x=0$ satisfies
the condition (2.5). If the Lyapunov exponents satisfy a partial second order
non-resonant condition

$\displaystyle\lambda_{j}\neq\lambda_{i}+\lambda_{\kappa}$ (2.10)

for $i=1,...,{\tau}$, $\kappa={\tau}+1,...,{{p}}$ and $j=1,...,{{p}}$, then
the following assertions are true:

  * (i)

Foliations: $F$ has $C^{1,\beta_{\alpha}}$ random stable and unstable
foliations with $C^{2,\alpha}$ leaves in a tempered ball $U(\omega)\subset
V(\omega)$, where $\beta_{\alpha}=0$ when $\alpha=0$ and
$\beta_{\alpha}\in(0,\alpha]$ when $\alpha\in(0,1]$.

  * (ii)

Conjugacy: $F(\omega,\cdot)$ is conjugate to its linear part $\Lambda(\omega)$
in a tempered ball $U(\omega)\subset V(\omega)$ by a $C^{1,\beta_{\alpha}}$
random diffeomorphism $\Phi:\Omega\times U(\omega)\to\mathbb{R}^{d}$
satisfying $\Phi(\omega,0)=0$ and $D\Phi(\omega,0)={\rm id}.$

Remark that the assumption of $\epsilon$-slowly continuity is only necessary
for random dynamical systems in the case of $\alpha=0$ since, when
$\alpha\in(0,1]$ or $F$ is deterministic, the second order derivative of
$F(\omega,\cdot)$ is automatically $\epsilon$-slowly continuous. Moreover,
when $F$ is deterministic, the relationship between Lyapunov exponent
$\lambda$ and its corresponding eigenvalue $r$ is $|r|=e^{\lambda}$. Thus
condition (2.10) becomes (1.5). As a result, we have the following $C^{1}$
linearization theorem.

######  Corollary 2.1.

Let $F:V\to\mathbb{R}^{d}$ be a $C^{2,\alpha}$ ($\alpha\in[0,1]$)
diffeomorphism over a neighborhood $V\subset\mathbb{R}^{d}$ of $x=0$. Suppose
the eigenvalues of its linearization $\Lambda:=DF(0)$ at hyperbolic fixed
point $x=0$ satisfy (1.5). Then $F$ is conjugate to its linear part $\Lambda$
in a small neighborhood $U\subset V$ by a $C^{1,\beta_{\alpha}}$
diffeomorphism $\Phi:U\to\mathbb{R}^{d}$ satisfying $\Phi(0)=0$ and
$D\Phi(0)={\rm id},$ where $\beta_{\alpha}=0$ when $\alpha=0$ and
$\beta_{\alpha}\in(0,\alpha]$ when $\alpha\in(0,1]$.

This corollary extends Belitskii’s $C^{1}$ linearization theorem to
$C^{1,\beta_{\alpha}}$ linearization, where $\beta_{\alpha}>0$ whenever
$\alpha>0$. Remark that $C^{1,\beta_{\alpha}}$ linearization with
$\beta_{\alpha}>0$ has important applications in problems such as Lorenz
attractors ([26]) and topological entropy ([30]).

##  3\. Random invariant distribution

In this section, we study the Hölder continuity of random invariant
distributions. We first extend the local random diffeomorphism
$F(\omega,x)=\Lambda(\omega)x+f(\omega,x)$ defined on a tempered ball
$V(\omega)=\\{x\in\mathbb{R}^{d}:\|x\|\leq\rho(\omega)\\}$ to a global one. In
what follows, for a random variable (for example $K:\Omega\to(0,\infty)$), we
use $K(\omega)^{n}$ to denote $K(\omega)$ to the power of $n\in\mathbb{Z}$ and
let $U(\omega):=\\{x\in\mathbb{R}^{d}:\|x\|\leq\rho(\omega)/2\\}$, where

$\|x\|:=\max\\{\|x_{1}\|,...,\|x_{p}\|\\}\quad{\rm
for}~{}~{}x=x_{1}+\cdots+x_{p}\in X_{1}\oplus\cdots\oplus X_{p}.$

Consider a random smooth cut-off function
$u(\omega,\cdot):\mathbb{R}^{d}\to(0,1)$ such that

$\displaystyle u(\omega,x)=\left\\{\begin{array}[]{lll}1,&\forall x\in
U(\omega),\vspace{1ex}\\\ 0,&\forall x\in\mathbb{R}^{d}\backslash
V(\omega),\end{array}\right.\quad\rho(\omega)^{r}\|D^{r}u(\omega,x)\|\leq
C_{u},~{}\forall r=1,2,3,$ (3.3)

where $C_{u}$ is a positive constant. The construction of $u$ is given in
Appendix.

Let $\tilde{F}(\omega,x)$ be an extension of $F(\omega,x)$ given by

$\displaystyle\tilde{F}(\omega,x)=\begin{cases}\Lambda(\omega)x+u(\omega,x)f(\omega,x),&\quad\text{for
all}\;x\in V(\omega)\\\ \Lambda(\omega)x,&\quad\text{for
all}\;x\in\mathbb{R}^{d}\backslash V(\omega).\end{cases}$

Then, by elementary estimations, we have the following result for the
extension $\tilde{F}(\omega,x)$.

######  Lemma 3.1.

The following properties hold for $\tilde{F}(\omega,x)$:

  * (i)

$\tilde{F}(\omega,x)=F(\omega,x)$ for all $x\in U(\omega)$.

  * (ii)

The derivatives of $\tilde{F}$ satisfy the following estimates.

$\displaystyle\begin{split}&\|D\tilde{F}(\omega,x)-\Lambda(\omega)\|\leq\delta(\omega),\qquad\|D^{2}\tilde{F}(\omega,x)\|\leq
M_{\epsilon}(\omega),\\\
&\|D^{2}\tilde{F}(\omega,x)-D^{2}\tilde{F}(\omega,y)\|\leq
L_{\epsilon,\rho}(\omega)\|x-y\|^{\alpha},\quad\forall
x,y\in\mathbb{R}^{d},\end{split}$

where $\delta(\omega):=M_{\epsilon}(\omega)\rho(\omega)$,
$M_{\epsilon}(\omega)$ and $L_{\epsilon,\rho}(\omega)$ are $\epsilon$-slowly
varying random variables for any $\epsilon>0$ such that
$M_{\epsilon}(\omega)\geq(3C_{u}+1)M(\omega)$ and
$L_{\epsilon,\rho}(\omega)\geq 7C_{u}M(\omega)/\rho(\omega)+L(\omega)$. Here
$M(\omega)$ and $L(\omega)$ are given in (2.3) with $N=2$.

In what follows, we still let $F(\omega,\cdot)$ denote the global mapping
$\tilde{F}(\omega,\cdot)$, and therefore
$F(\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies that

$\displaystyle\begin{split}&\|DF(\omega,x)-\Lambda(\omega)\|\leq\delta(\omega),\qquad\|D^{2}F(\omega,x)\|\leq
M_{\epsilon}(\omega),\\\ &\|D^{2}F(\omega,x)-D^{2}F(\omega,y)\|\leq
L_{\epsilon,\rho}(\omega)\|x-y\|^{\alpha},\quad\forall
x,y\in\mathbb{R}^{d},\end{split}$ (3.4)

and $D^{2}F(\omega,\cdot)$ is $\epsilon$-slowly continuous when $\alpha=0$.
Note that the third inequality of (3.4) holds for every Hölder exponents
$\tilde{\alpha}\in[0,\alpha]$ because $F(\omega,x)-\Lambda(\omega)x$ is equal
to $0$ outside the small neighborhood $V(\omega)$.

Throughout this paper, we choose $\epsilon>0$ being sufficiently small such
that

$\displaystyle\epsilon<\frac{1}{100}\min\Big{\\{}$ $\displaystyle
1,~{}\min_{i=1,...,p-1}\frac{\lambda_{j}-\lambda_{j+1}}{2\lambda_{\max}}\min\\{\lambda_{\tau},-\lambda_{\tau+1}\\},\min_{\mbox{\tiny$\begin{array}[]{c}i\in\\{1,...,\tau\\},\kappa\in\\{\tau+1,...,{p}\\},\\\
j\in\\{1,...,{p}\\}\end{array}$}}|\lambda_{i}+\lambda_{\kappa}-\lambda_{j}|\Big{\\}},$
(3.7)

where $\lambda_{\rm max}:=\max\\{2\lambda_{1},-2\lambda_{{p}}\\}>0$. Note that
the choice of $\epsilon$ depends on the Lyapunov exponents only. Let the
diameter of $V(\omega)$ be defined by

$\rho(\omega):=1/(M_{\epsilon}(\omega)\mathfrak{M}_{\epsilon}(\omega))>0$

for a given $\epsilon$-slowly varying tempered random variable
$\mathfrak{M}_{\epsilon}(\omega)>0$ satisfying

$\displaystyle\begin{split}\mathfrak{M}_{\epsilon}(\omega)\geq\max\bigg{\\{}&4,~{}\delta_{\lambda}^{-1}K_{\epsilon}(\theta\omega),~{}8C_{\lambda}K_{\epsilon}(\omega)^{3},~{}(2M_{\epsilon}(\omega)K_{\epsilon}(\omega)^{5})^{2},~{}(2K_{\epsilon}(\omega)^{2})^{4},\\\
&\big{\\{}12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\big{\\}}^{4(2\lambda_{\max}-\lambda_{j+1}-\epsilon)/(\lambda_{j}-\lambda_{j+1}-3\epsilon)},\\\
&2C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega),2e^{\lambda_{\max}}C^{3}_{\lambda}K_{\epsilon}(\omega)K_{\epsilon}(\theta\omega)\bigg{\\}}\end{split}$
(3.8)

where $K_{\epsilon}(\omega)\geq\max\\{1,K_{\lambda}K(\omega)\\}>0$ and
$C_{\epsilon}(\omega):=4(K_{\epsilon}(\omega))^{2}M_{\epsilon}(\omega)>0$ are
$\epsilon$-slowly varying random variables, $\delta_{\lambda}>0$ is a small
constant and $C_{\lambda},K_{\lambda}\geq 1$ are constants. Note that the
constants $\delta_{\lambda},C_{\lambda},K_{\lambda}$ are determined by the
Lyapunov exponents only, which will be given in Lemma 3.2 and (3.20). Then, as
seen in Lemma 3.1,

$\displaystyle\delta(\omega):=M_{\epsilon}(\omega)\rho(\omega)=1/\mathfrak{M}_{\epsilon}(\omega)>0$
(3.9)

is small and tempered.

Next, we obtain from (2.1) that

$\displaystyle
D{F}(n,\omega,x)=\begin{cases}D{F}(\theta^{n-1}\omega,{F}(n-1,\omega,x))\circ\cdots\circ
D{F}(\omega,{F}(0,\omega,x)),&n>0,\\\ {\rm id},&n=0,\\\
\big{\\{}D{F}(\theta^{n}\omega,D{F}(n,\omega,x))\big{\\}}^{-1}\circ\cdots\circ\big{\\{}D{F}(\theta^{-1}\omega,D{F}(-1,\omega,x))\big{\\}}^{-1},&n<0.\end{cases}$
(3.10)

For every $\omega\in\Omega$ and $x\in\mathbb{R}^{d}$, we set

$\varpi:=(\omega,x)\in{\bf\Omega}:=\Omega\times\mathbb{R}^{d},\qquad\vartheta\varpi:=(\theta\omega,F(\omega,x)),\qquad{\bf\Lambda}(\varpi):=DF(\omega,x).$

Then, $D{F}(n,\omega,x)$ is a cocycle generated by ${\bf\Lambda}(\varpi)$
driven by $\vartheta^{n}\varpi:=(\theta^{n}\omega,F(n,\omega,x))$. The first
inequality of (3.4) implies that ${\bf\Lambda}(\varpi)$ is a small
perturbation of $\Lambda(\omega)$ with the difference

$\|{\bf\Lambda}(\varpi)-\Lambda(\omega)\|\leq\delta(\omega).$

Note that $\Lambda(\omega)=\Lambda(\varpi)$ since $\Lambda(\omega)$ does not
depend on $x$. Using (2.7) and applying the roughness of the tempered
exponential dichotomy ([64, Theorem 1]) to the system
$e^{-(\lambda_{j}+\lambda_{j+1})n/2}\Lambda(n,\omega)$ for each
$j=1,\cdots,p-1$ with the splitting

$\mathbb{R}^{d}=\big{(}X_{1}\oplus\cdots\oplus
X_{j}\big{)}\oplus\big{(}X_{j+1}\oplus\cdots\oplus X_{p}\big{)},$

we have the following lemma.

######  Lemma 3.2.

There are positive constants $\delta_{\lambda}$ and $K_{\lambda}$ depending on
the Lyapunov exponents only such that if
$\delta(\omega)\leq\delta_{\lambda}/K_{\epsilon}(\theta\omega)$, then there is
an invariant splitting for the cocycle ${\bf\Lambda}(n,\varpi)$ generated by
${\bf\Lambda}(\varpi)$

$\mathbb{R}^{d}=E_{1}(\omega,x)\oplus\cdots\oplus E_{{p}}(\omega,x)$

such that for each $j=1,\cdots,p$,

$\displaystyle\begin{split}&\|{\bf\Lambda}(n,\varpi)|_{E_{j}}\|\leq
K_{\lambda}K(\omega)e^{(\lambda_{j}-2\epsilon)n},\qquad\forall n<0,\\\
&\|{\bf\Lambda}(n,\varpi)|_{E_{j}}\|\leq
K_{\lambda}K(\omega)e^{(\lambda_{j}+2\epsilon)n},\qquad\forall n\geq
0.\end{split}$

The invariance means that
${\bf\Lambda}(n,\varpi)E_{j}(\varpi)=E_{j}(\vartheta^{n}\varpi)$. Without loss
of generality, we still let $\epsilon$ denote $2\epsilon$. Note that

$D{F}(n,\omega,x)={\bf\Lambda}(n,\varpi)\;\text{ and
}F(\omega,x)=F(1,\omega,x).$

Thus, we have

$\displaystyle
DF(\omega,x)\\{E_{j}(\omega,x)\\}=E_{j}(\theta\omega,F(\omega,x)),\qquad\forall
j=1,...,{p},$ (3.11)

and

$\displaystyle\begin{split}&\|DF(n,\omega,x)|_{E_{j}(\omega,x)}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n},\qquad\forall n<0,\\\
&\|DF(n,\omega,x)|_{E_{j}(\omega,x)}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}+\epsilon)n},\qquad\forall n\geq
0,\end{split}$ (3.12)

where $K_{\epsilon}(\omega)\geq K_{\lambda}K(\omega)$ is an $\epsilon$-slowly
varying random variable. We point out that the measurability of
$E_{j}(\omega,x)$ with respect to $\omega$ follows from that $E_{j}(\omega,x)$
is the tangent spaces of an intermediate foliation whose partial derivative is
measurable with respect to $\omega$ (see (4.38) below in the next section). As
in the deterministic case (see [50, Sections 2 and 6]), for each $j=1,...,p$,

$E_{j}(\omega):=\\{E_{j}(\omega,x)\subset\mathbb{R}^{d}:x\in\mathbb{R}^{d}\\}$

is called a random intermediate distribution and $x$ is the base point of the
distribution. In particular,

$E_{u}(\omega):=\\{E_{u}(\omega,x)\subset\mathbb{R}^{d}:x\in\mathbb{R}^{d}\\}\mbox{
and
}E_{s}(\omega):=\\{E_{s}(\omega,x)\subset\mathbb{R}^{d}:x\in\mathbb{R}^{d}\\}$

are called the random unstable distribution and random stable distribution
respectively, with fibers

$\displaystyle E_{u}(\omega,x):=E_{1}(\omega,x)\oplus\cdots\oplus
E_{\tau}(\omega,x),$ $\displaystyle
E_{s}(\omega,x):=E_{\tau+1}(\omega,x)\oplus\cdots\oplus E_{{p}}(\omega,x)$

satisfying the following:

  * (i)

$\mathbb{R}^{d}=E_{u}(\omega,x)\oplus E_{s}(\omega,x);$

  * (ii)

The splitting is invariant, i.e.,

$\displaystyle\begin{split}&DF(\omega,x)E_{u}(\omega,x)=E_{u}(\theta\omega,F(\omega,x)),\\\
&DF(\omega,x)E_{s}(\omega,x)=E_{s}(\theta\omega,F(\omega,x));\end{split}$
(3.13)

  * (ii)

For each $\varrho_{0}\in(0,\,\min\\{\lambda_{\tau},-\lambda_{\tau+1}\\})$,
there exists a random variable $K_{\epsilon}(\omega)>0$ tempered from above
such that

$\displaystyle\begin{split}\|DF(n,\omega,x)|_{E_{u}(\omega,x)}\|&\leq
K_{\epsilon}(\omega)e^{\varrho_{0}n},\qquad\forall n<0,\\\
\|DF(n,\omega,x)|_{E_{s}(\omega,x)}\|&\leq
K_{\epsilon}(\omega)e^{-\varrho_{0}n},\qquad\forall n\geq 0.\end{split}$
(3.14)

Finally, we show that each $E_{j}(\omega,x)$ is Hölder continuous with respect
to $x$. Recall that the distance between two subspaces $E$ and $\tilde{E}$ of
$\mathbb{R}^{d}$ is given by

$\displaystyle{\rm dist}(E,\tilde{E}):=\max\Big{(}\max_{v_{1}\in
E,\,\|v_{1}\|=1}{\rm
dist}(v_{1},\tilde{E}),\max_{v_{2}\in\tilde{E},\,\|v_{2}\|=1}{\rm
dist}(v_{2},E)\Big{)},$ (3.15)

where ${\rm dist}(v,E):=\min_{w\in E}\|v-w\|$.

######  Theorem 3.1.

For all $x,y\in\mathbb{R}^{d}$, $\omega\in\Omega$ and all $j=1,...,{p}$,

$\displaystyle\begin{split}&{\rm
dist}(E_{j}(\omega,x),E_{j}(\omega,y))\leq\|x-y\|^{\beta_{E}},\\\ &{\rm
dist}(E_{j}(\omega,x),E_{j}(\omega,y))\leq\delta_{E}(\omega),\end{split}$
(3.16)

where
$\beta_{E}:=\min_{\ell=1,...,p-1}(\lambda_{\ell}-\lambda_{\ell+1}-3\epsilon)/(6\lambda_{\max})\in(0,1)$
and $\delta_{E}(\omega):=\delta(\omega)^{\beta_{E}}.$

Proof. First of all, we define the subspaces

$\displaystyle E_{+}(\omega,x):=E_{1}(\omega,x)\oplus\cdots\oplus
E_{j}(\omega,x),$ $\displaystyle
E_{-}(\omega,x):=E_{j+1}(\omega,x)\oplus\cdots\oplus E_{{p}}(\omega,x)$

and claim that for all integers $m\geq 0$

$\displaystyle\begin{split}\|DF(m,\omega,x)\zeta_{-}\|&\leq
2K_{\epsilon}(\omega)e^{(\lambda_{j+1}+\epsilon)m}\|\zeta_{-}\|,\\\
\|DF(m,\omega,x)\zeta_{-}^{\bot}\|&\geq(2K_{\epsilon}(\omega))^{-1}e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|\end{split}$
(3.17)

for $\zeta_{-}\in E_{-}(\omega,x)$ and
$\zeta_{-}^{\bot}\in\\{E_{-}(\omega,x)\\}^{\bot}$, the orthogonal complement
of $E_{-}(\omega,x)$. In fact, we have from (3.12) that

$\displaystyle\begin{split}&\|DF(n,\omega,x)\zeta_{+}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n}\|\zeta_{+}\|,\qquad\forall
n<0,\\\ &\|DF(n,\omega,x)\zeta_{-}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j+1}+\epsilon)n}\|\zeta_{-}\|,\qquad\forall
n\geq 0,\end{split}$ (3.18)

for $\zeta_{+}\in E_{+}(\omega,x)$ and $\zeta_{-}\in E_{-}(\omega,x)$. Then
the first inequality of (3.17) follows from the second one of (3.18)
immediately.

Now, we show that the second inequality of (3.17) holds. We first notice that
for $m\geq 0$

$\displaystyle\big{(}DF(-m,\omega,x)\big{)}^{-1}=DF(m,\theta^{-m}\omega,F(-m,\omega,x))$

and therefore the first inequality of (3.18) yields

$\displaystyle\|DF(m,\theta^{-m}\omega,F(-m,\omega,x))\tilde{\zeta}_{+}\|\geq(1/K_{\epsilon}(\omega))e^{(\lambda_{j}-\epsilon)m}\|\tilde{\zeta}_{+}\|$

for $\tilde{\zeta}_{+}\in E_{+}(\theta^{-m}\omega,F(-m,\omega,x))$. Replacing
$F(-m,\omega,x)$ and $\theta^{-m}\omega$ with $x$ and $\omega$ respectively,
we obtain

$\displaystyle\|DF(m,\omega,x)\zeta_{+}\|\geq(1/K_{\epsilon}(\theta^{m}\omega))e^{(\lambda_{j}-\epsilon)m}\|\zeta_{+}\|\geq(1/K_{\epsilon}(\omega))e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{+}\|.$
(3.19)

Let $\Pi_{\pm}(\omega,x):\mathbb{R}^{d}\to E_{\pm}(\omega,x)$ be the
projections associated with $\mathbb{R}^{d}=E_{+}(\omega,x)\oplus
E_{-}(\omega,x)$ and let $\pi_{\pm}:\mathbb{R}^{d}\to X_{\pm}$ be the
projections associated with $\mathbb{R}^{d}=X_{+}\oplus X_{-}$, where

$\displaystyle X_{+}:=X_{1}\oplus\cdots\oplus X_{j},\quad
X_{-}:=X_{j+1}\oplus\cdots\oplus X_{{p}}.$

Then, using the arguments as in [64, (3.41)-(3.44) and (3.16)], we have

$\displaystyle\|\Pi_{+}(\omega,x)-\pi_{+}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega),\quad\|\Pi_{-}(\omega,x)-\pi_{-}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega),$ (3.20)

where $C_{\lambda}>0$ is a constant depending only on the Lyapunov exponents.

Using the second inequality of (3.20), when $\delta(\omega)$ is small enough,
there are $p-j$ vectors

$\displaystyle
v_{1}^{-}:=(v_{1,1},\cdots,v_{1,j},1,\cdots,0)^{T},~{}...,~{}v_{p-j}^{-}:=(v_{p-j,1},\cdots,v_{p-j,j},0,\cdots,1)^{T}\in
E_{-}(\omega,x)$

with $v_{i,j}\in\mathbb{R}$ ($T$ means the transpose of a vector) such that

$\displaystyle\max\\{|v_{i,1}|,\cdots,|v_{i,j}|\\}=\|v_{i}^{-}-\pi_{-}v_{i}^{-}\|=\|(\Pi_{-}(\omega,x)-\pi_{-})v_{i}^{-}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\|v_{i}^{-}\|,$

which implies that $\|v_{i}^{-}\|=\max\\{|v_{i,1}|,\cdots,|v_{i,j}|,1\\}=1$
since $\delta(\omega)>0$ is sufficiently small. Therefore

$\displaystyle\max\\{|v_{i,1}|,\cdots,|v_{i,j}|\\}=\|v_{i}^{-}-\pi_{-}v_{i}^{-}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega).$ (3.21)

For any vector
$\zeta_{-}^{\bot}:=(\zeta_{1},...\zeta_{j},\zeta_{j+1},...,\zeta_{p})\in(E_{-}(\omega,x))^{\bot},$
since $\zeta_{-}^{\bot}$ is perpendicular to $v_{1}^{-},\cdots,v_{p-j}^{-}$,
we have $B\zeta_{-}^{\bot}=0,$ where

$\displaystyle B:=\begin{pmatrix}v_{1,1}&\cdots&v_{1,j}&1&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
v_{p-j,1}&\cdots&v_{p-j,j}&0&\cdots&1\end{pmatrix}.$

Thus, $B\zeta_{-}^{\bot}=0$ is equivalent to

$\displaystyle\begin{pmatrix}\zeta_{j+1}\\\ \vdots\\\
\zeta_{p}\end{pmatrix}=\begin{pmatrix}-v_{1,1}&\cdots&-v_{1,j}\\\
\vdots&\ddots&\vdots\\\
-v_{p-j,1}&\cdots&-v_{p-j,j}\end{pmatrix}\begin{pmatrix}\zeta_{1}\\\ \vdots\\\
\zeta_{j}\end{pmatrix},$

which together with (3.21) indicates that

$\displaystyle\|\pi_{-}\zeta_{-}^{\bot}\|\leq
jC_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\|\zeta_{-}^{\bot}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\|\zeta_{-}^{\bot}\|.$

Here, we omit $j\leq p$ (the number of the Lyapunov exponents) since
$C_{\lambda}>0$ denotes a sufficiently large constant depending on the
Lyapunov exponents. It implies that

$\|\pi_{+}\zeta_{-}^{\bot}\|\geq\|\zeta_{-}^{\bot}\|-\|\pi_{-}\zeta_{-}^{\bot}\|\geq(1-C_{\lambda}\delta(\omega)K_{\epsilon}(\omega))\|\zeta_{-}^{\bot}\|.$

Therefore,

$\displaystyle\|\Pi_{+}(\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\geq\|\pi_{+}\zeta_{-}^{\bot}\|-\|\Pi_{+}(\omega,x)\zeta_{-}^{\bot}-\pi_{+}\zeta_{-}^{\bot}\|$
$\displaystyle\geq\\{1-2C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\\}\|\zeta_{-}^{\bot}\|$
(3.22)

and

$\displaystyle\|\Pi_{-}(\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\leq\|\pi_{-}\zeta_{-}^{\bot}\|+\|\Pi_{-}(\omega,x)\zeta_{-}^{\bot}-\pi_{-}\zeta_{-}^{\bot}\|$
$\displaystyle\leq
2C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\|\zeta_{-}^{\bot}\|.$ (3.23)

It follows from (3.18)-(3.23) that

$\displaystyle\|DF(m,\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\geq\|DF(m,\omega,x)\Pi_{+}(\omega,x)\zeta_{-}^{\bot}\|-\|DF(m,\omega,x)\Pi_{-}(\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\geq(1/K_{\epsilon}(\omega))e^{(\lambda_{j}-2\epsilon)m}\|\Pi_{+}(\omega,x)\zeta_{-}^{\bot}\|-K_{\epsilon}(\omega)e^{(\lambda_{j+1}+\epsilon)m}\|\Pi_{-}(\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\geq(1/K_{\epsilon}(\omega))\\{1-2C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\\}e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|$
$\displaystyle\quad-2C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)^{2}e^{(\lambda_{j+1}+\epsilon)m}\|\zeta_{-}^{\bot}\|$
$\displaystyle\geq(1/K_{\epsilon}(\omega))e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|-4C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)^{2}e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|$
$\displaystyle\geq(2K_{\epsilon}(\omega))^{-1}e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|$
(3.24)

since $\mathfrak{M}_{\epsilon}(\omega)\geq
8C_{\lambda}K_{\epsilon}(\omega)^{3}$ (i.e.,
$0<\delta(\omega)\leq(8C_{\lambda}K_{\epsilon}(\omega)^{3})^{-1}$) by (3.8).
This shows the second inequality of (3.17) and (3.17) is proved.

Next, we know from the first inequality of (3.4) that

$\displaystyle\|DF(\omega,x)-DF(\omega,y)\|\leq\|DF(\omega,x)-\Lambda(\omega)\|+\|DF(\omega,y)-\Lambda(\omega)\|\leq
2\delta(\omega),\quad\forall x,y\in\mathbb{R}^{d},$

and from (3.12) that

$\displaystyle\|DF(m,\omega,x)\|\leq\max_{j=1,...,p}\|DF(m,\omega,x)|_{E_{j}(\omega,x)}\|\leq
K_{\epsilon}(\omega)e^{\lambda_{\max}m},\qquad\forall m\geq 0,$ (3.25)

where $\lambda_{\rm max}:=\max\\{2\lambda_{1},-2\lambda_{{p}}\\}>0.$ Then, we
have

$\displaystyle\|DF(\theta^{m}\omega,F(m,\omega,x))-DF(\theta^{m}\omega,F(m,\omega,y))\|$
$\displaystyle=\|DF(\theta^{m}\omega,F(m,\omega,x))-DF(\theta^{m}\omega,F(m,\omega,y))\|^{1/2+1/2}$
$\displaystyle\leq(2\delta(\theta^{m}\omega))^{1/2}\\{M_{\epsilon}(\theta^{m}\omega)\|F(m,\omega,x)-F(m,\omega,x)\|\\}^{1/2}$
$\displaystyle\leq(2\delta(\theta^{m}\omega))^{1/2}\\{M_{\epsilon}(\theta^{m}\omega)K_{\epsilon}(\omega)e^{\lambda_{\max}m}\|x-y\|\\}^{1/2}$
$\displaystyle\leq\\{2\delta(\omega)M_{\epsilon}(\omega)K_{\epsilon}(\omega)\\}^{1/2}e^{(1/2)(\lambda_{\max}+2\epsilon)m}\|x-y\|^{1/2}$
(3.26)

by the second inequality of (3.4) and

$\displaystyle\|DF(\theta^{m}\omega,F(m,\omega,x))-DF(\theta^{m}\omega,F(m,\omega,y))\|\leq
2\delta(\theta^{m}\omega)\leq 2e^{\epsilon m}\delta(\omega).$ (3.27)

In the above (3.26)-(3.27), we know that
$e^{-\epsilon|k|}\delta(\theta^{k}\omega)\leq\delta(\omega),$ as known from
(3.9).

Notice the fact that

$\displaystyle a_{m-1}...a_{0}-b_{m-1}...b_{0}$
$\displaystyle=a_{m-1}...a_{1}\,(a_{0}-b_{0})+(a_{m-1}-b_{m-1})\,b_{m-2}...b_{0}$
$\displaystyle=\sum_{i=0}^{m-1}a_{m-1}...a_{i+1}\,(a_{i}-b_{i})\,b_{i-1}...b_{1}.$

It follows from (3.26) that

$\displaystyle\|DF(m,\omega,x)-DF(m,\omega,y)\|$
$\displaystyle=\|D{F}(\theta^{m-1}\omega,{F}(m-1,\omega,x))\circ\cdots\circ
D{F}(\omega,{F}(0,\omega,x))$
$\displaystyle\quad-D{F}(\theta^{m-1}\omega,{F}(m-1,\omega,y))\circ\cdots\circ
D{F}(\omega,{F}(0,\omega,y))\|$
$\displaystyle\leq\sum_{i=0}^{m-1}\|DF(m-i-1,\theta^{i+1}\omega,F(i+1,\omega,x))\|$
$\displaystyle\quad\cdot\|DF(\theta^{i}\omega,F(i,\omega,x))-DF(\theta^{i}\omega,F(i,\omega,y))\|\,\|DF(i,\omega,y)\|$
$\displaystyle\leq\sum_{i=0}^{m-1}K_{\epsilon}(\theta^{i+1}\omega)e^{\lambda_{\max}(m-i-1)}\\{2\delta(\omega)M_{\epsilon}(\omega)K_{\epsilon}(\omega)\\}^{1/2}e^{(1/2)(\lambda_{\max}+2\epsilon)i}\|x-y\|^{1/2}K_{\epsilon}(\omega)e^{\lambda_{\max}i}$
$\displaystyle\leq\sum_{i=0}^{m-1}e^{\epsilon(i+1)}e^{\lambda_{\max}(m-1)}e^{(1/2)(\lambda_{\max}+2\epsilon)i}\\{2\delta(\omega)M_{\epsilon}(\omega)K_{\epsilon}(\omega)^{5}\\}^{1/2}\|x-y\|^{1/2}$
$\displaystyle\leq e^{2\lambda_{\max}m}\delta(\omega)^{1/4}\|x-y\|^{1/2}$
(3.28)

since
$\mathfrak{M}_{\epsilon}(\omega)\geq(2M_{\epsilon}(\omega)K_{\epsilon}(\omega)^{5})^{2}$
(i.e.,
$\delta(\omega)\leq(2M_{\epsilon}(\omega)K_{\epsilon}(\omega)^{5})^{-2}$) by
(3.8), where

$me^{\epsilon(i+1)}e^{(1/2)(\lambda_{\max}+2\epsilon)i}\leq
e^{\lambda_{\max}m}$

as $0\leq i\leq m-1$ and $\epsilon>0$ is small. Similarly, it follows from
(3.27) that

$\displaystyle\|DF(m,\omega,x)-DF(m,\omega,y)\|$
$\displaystyle\leq\sum_{i=0}^{m-1}\|DF(m-i-1,\theta^{i+1}\omega,F(i+1,\omega,x))\|$
$\displaystyle\quad\cdot\|DF(\theta^{i}\omega,F(i,\omega,x))-DF(\theta^{i}\omega,F(i,\omega,y))\|\,\|DF(i,\omega,y)\|$
$\displaystyle\leq\sum_{i=0}^{m-1}K_{\epsilon}(\theta^{i+1}\omega)e^{\lambda_{\max}(m-i-1)}2e^{\epsilon
m}\delta(\omega)K_{\epsilon}(\omega)e^{\lambda_{\max}i}$
$\displaystyle\leq\sum_{i=0}^{m-1}e^{\epsilon(i+1)}e^{\lambda_{\max}(m-1)}2e^{\epsilon
m}\delta(\omega)K_{\epsilon}(\omega)^{2}\leq
e^{2\lambda_{\max}m}\delta(\omega)^{3/4}$ (3.29)

since $\mathfrak{M}_{\epsilon}(\omega)\geq(2K_{\epsilon}(\omega)^{2})^{4}$
(i.e., $\delta(\omega)\leq(2K_{\epsilon}(\omega)^{2})^{-4}$) by (3.8).

Hence, using (3.17) and (3.28)-(3.29), we apply [50, Lemma 3.10] for every
fixed $\omega\in\Omega$, which is given exactly as follows (in notations used
in [50]).

######  Lemma 3.3.

Let $(A_{n})_{n\in\mathbb{N}}$ and $(B_{n})_{n\in\mathbb{N}}$ be two sequences
of real $N\times N$ matrices such that for some $\Delta\in(0,1)$ and $a>1$,

$\|A_{n}-B_{n}\|\leq\Delta a^{n}$

for all positive integers $n$. Assume that there exist subspaces
$E_{A},E_{B}\in\mathbb{R}^{N}$ and numbers $0<\lambda<\mu$ and $C>1$ such that
$\lambda<a$ and for each $n\geq 0$,

$\displaystyle\|A_{n}v\|\leq C\lambda^{n}~{}if~{}v\in
E_{A};\qquad\|A_{n}w\|\geq C^{-1}\mu^{n}\|w\|~{}if~{}w\in{E_{A}}^{\bot};$
$\displaystyle\|B_{n}v\|\leq C\lambda^{n}~{}if~{}v\in
E_{B};\qquad\|B_{n}w\|\geq C^{-1}\mu^{n}\|w\|~{}if~{}w\in{E_{B}}^{\bot}.$

Then

${\rm dist}(E_{A},E_{B})\leq
3C^{2}\frac{\mu}{\lambda}\Delta^{\frac{\log\mu-\log\lambda}{\log
a-\log\lambda}}.$

Notice that if $\|x-y\|<1$ then the notations of [50, Lemma 3.10] and ours
have the correspondence:

$\displaystyle n:=m,\quad A_{m}:=DF(m,\omega,x),\quad B_{m}:=DF(m,\omega,y),$
$\displaystyle\Delta:=\delta(\omega)^{1/4}\|x-y\|^{1/2}~{}{\rm
or}~{}\delta(\omega)^{3/4},\quad a:=e^{2\lambda_{\max}},\quad\mbox{(see
\eqref{DFmm-2} or \eqref{DFmm-1})}$ $\displaystyle
E_{A}:=E_{-}(\omega,x),\quad E_{B}:=E_{-}(\omega,y),\quad\mbox{(see \eqref{ED-
md})}$ $\displaystyle
C:=2K_{\epsilon}(\omega),\quad\lambda:=e^{\lambda_{j+1}+\epsilon},\quad\mu:=e^{\lambda_{j}-2\epsilon}.\quad\mbox{(see
\eqref{ED-md})}$

Then, all conditions of Lemma 3.3 are verified and we conclude that for all
$x,y$ satisfying $\|x-y\|<1$

$\displaystyle{\rm dist}(E_{-}(\omega,x),E_{-}(\omega,y))$ $\displaystyle\leq
12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\big{(}\delta(\omega)^{1/4}\|x-y\|^{1/2}\big{)}^{(\lambda_{j}-\lambda_{j+1}-3\epsilon)/(2\lambda_{\max}-\lambda_{j+1}-\epsilon)}$
$\displaystyle\leq\|x-y\|^{(1/2)(\lambda_{j}-\lambda_{j+1}-3\epsilon)/(2\lambda_{\max}-\lambda_{j+1}-\epsilon)}$

since
$\mathfrak{M}_{\epsilon}(\omega)\geq\big{\\{}12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\big{\\}}^{4(2\lambda_{\max}-\lambda_{j+1}-\epsilon)/(\lambda_{j}-\lambda_{j+1}-3\epsilon)}$
by (3.8), i.e.,

$0<\delta(\omega)\leq\big{\\{}12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\big{\\}}^{-4(2\lambda_{\max}-\lambda_{j+1}-\epsilon)/(\lambda_{j}-\lambda_{j+1}-3\epsilon)},$

and that

$\displaystyle{\rm dist}(E_{-}(\omega,x),E_{-}(\omega,y))$ $\displaystyle\leq
12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\delta(\omega)^{(3/4)(\lambda_{j}-\lambda_{j+1}-3\epsilon)/(2\lambda_{\max}-\lambda_{j+1}-\epsilon)}$
$\displaystyle\leq\delta(\omega)^{(1/2)(\lambda_{j}-\lambda_{j+1}-3\epsilon)/(2\lambda_{\max}-\lambda_{j+1}-\epsilon)}.$

A similar conclusion holds in the subspaces $E_{+}(\omega,x)$. Then, by
choosing different $j\in\\{1,...,p\\}$ and intersecting appropriate subspaces,
when $\|x-y\|<1$ we prove (3.16) with

$\beta_{E}:=\min_{\ell=1,...,p-1}\frac{\lambda_{\ell}-\lambda_{\ell+1}-3\epsilon}{6\lambda_{\max}},\qquad\delta_{E}(\omega):=\delta(\omega)^{\beta_{E}},$

where $\beta_{E}\in(0,1)$, as known from (3.7). When $\|x-y\|\geq 1$, the
first inequality of (3.16) can be indicated by the second inequality of (3.16)
as $\delta_{E}(\omega)<1$. This completes the proof of Theorem 3.1. $\Box$

The original idea of the above proof comes from [9] for the invariant
distributions of Anosov diffeomorphims, i.e., hyperbolic diffeomorphims
defined on manifolds, from which we see that the classical method only gives
Hölder continuity of distributions in the base point $x$. In fact, it is
widely believed that the distributions are not $C^{1}$ (not even Lipschitzian)
smooth with respect to $x$ in general without the bunching condition (see e.g.
[21, 33, 50]). However, in the following Theorem 9.1, we will show that in the
local case the stable and unstable distributions are $C^{1,\beta_{\alpha}}$
under some mild assumptions (i.e., the assumptions of Theorem 2.1).

In order to prove Theorem 9.1, we need to study the intermediate foliations in
section 4 and then decompose a cohomological equation for the distribution in
every intermediate foliation in section 5, where two lemmas concerning random
normal form and the decomposed cohomological equation will be proved in
sections 6-8.

##  4\. Random invariant foliation

In this section, we present the results on various invariant foliations for
random dynamical systems according to the splits of Lyapunov exponents. Some
of results are borrowed from [1, 42, 43, 56].

First, from [1, pp.362-363] and [42, Theorem 3.8 and 3.9], there are the
random unstable foliation
$\\{{\mathcal{W}}_{u}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ and the random stable
foliation $\\{{\mathcal{W}}_{s}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ of
$\mathbb{R}^{d}$ with leaves ${\mathcal{W}}_{u}(\omega,x)$ and
${\mathcal{W}}_{s}(\omega,x)$, which are measurable with respect to $\omega$
and continuous with respect to $x$, such that
$\mathbb{R}^{d}=\cup_{x\in\mathbb{R}^{d}}{\mathcal{W}}_{u}(\omega,x)=\cup_{x\in\mathbb{R}^{d}}{\mathcal{W}}_{s}(\omega,x)$
and

$\displaystyle
F(\omega,{\mathcal{W}}_{u}(\omega,x))\subset{\mathcal{W}}_{u}(\theta\omega,F(\omega,x)),\quad
F(\omega,{\mathcal{W}}_{s}(\omega,x))\subset{\mathcal{W}}_{s}(\theta\omega,F(\omega,x)).$
(4.1)

Notice that (4.1) means the invariance of foliations under the action of
$F(\omega,\cdot)$. Let $\pi_{u}:\mathbb{R}^{d}\to X_{u}$ and
$\pi_{s}:\mathbb{R}^{d}\to X_{s}$ be the projections associated with
$\mathbb{R}^{d}=X_{u}\oplus X_{s}$ and let

$\displaystyle\Lambda_{*}(\omega):=\Lambda(\omega)|_{X_{*}},\quad\Lambda_{*}(n,\omega):=\Lambda(n,\omega)|_{X_{*}}\qquad{\rm
for}~{}~{}*=u,s.$

Consider the following random Lyapunov-Perron equations (abbreviated as the
L-P equations) for constructing the unstable and stable foliations
respectively.

$\displaystyle\begin{split}p_{n}(\omega,x,y_{u})&=\Lambda_{u}(n,\omega)(y_{u}-\pi_{u}x)-\sum_{k=n}^{-1}\Lambda_{u}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{u}(p_{k})(\omega,x,y_{u})\\\
&\quad+\sum_{k=-\infty}^{n-1}\Lambda_{s}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{s}(p_{k})(\omega,x,y_{u}),\qquad\forall
n<0,\\\
p_{0}(\omega,x,y_{u})&=y_{u}-\pi_{u}x+\sum_{k=-\infty}^{-1}\Lambda_{s}({-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{s}(p_{k})(\omega,x,y_{u}),\qquad
n=0,\end{split}$ (4.2)

where
$\Xi_{k}^{*}(p_{k})(\omega,x,y_{u}):=\pi_{*}f\big{(}\theta^{k}\omega,p_{k}(\omega,x,y_{u})+F(k,\omega,x)\big{)}-\pi_{*}f\big{(}\theta^{k}\omega,F(k,\omega,x)\big{)}$
for $*=u,s$, and

$\displaystyle\begin{split}q_{n}(\omega,x,y_{s})&=\Lambda_{s}(n,\omega)(y_{s}-\pi_{s}x)+\sum_{k=0}^{n-1}\Lambda_{s}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{s}(q_{k})(\omega,x,y_{s})\\\
&\quad-\sum_{k=n}^{\infty}\Lambda_{u}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{u}(q_{k})(\omega,x,y_{s}),\qquad\forall
n>0,\\\
q_{0}(\omega,x,y_{s})&=y_{s}-\pi_{s}x-\sum_{k=0}^{\infty}\Lambda_{u}({-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{u}(q_{k})(\omega,x,y_{s}),\qquad
n=0,\end{split}$ (4.3)

where
$\Xi_{k}^{*}(q_{k})(\omega,x,y_{s}):=\pi_{*}f\big{(}\theta^{k}\omega,q_{k}(\omega,x,y_{s})+F(k,\omega,x)\big{)}-\pi_{*}f\big{(}\theta^{k}\omega,F(k,\omega,x)\big{)}$
for $*=u,s$. It follows from [42, Theorem 3.8 and 3.9] that there is a
constant $C_{\lambda}>0$ depending only on the Lyapunov exponents such that if

$\delta(\omega)\leq C_{\lambda}/(K(\theta\omega)K(\omega)),$ (4.4)

then both equation (4.2) and equation (4.3) have continuous solutions
$(p_{n}(\omega,\cdot,\cdot))_{n\leq 0}$ and
$(q_{n}(\omega,\cdot,\cdot))_{n\geq 0}$ respectively. Notice that the
condition (4.4) holds since (3.8) and (3.9), which insures that the associated
Lyapunov-Perron operators are contraction. Furthermore, the leaves of random
unstable foliation and random stable foliation are given by

$\displaystyle\begin{split}{\mathcal{W}}_{u}(\omega,x):=\\{x+p_{0}(\omega,x,y_{u}):y_{u}\in
X_{u}\\},\quad{\mathcal{W}}_{s}(\omega,x):=\\{x+q_{0}(\omega,x,y_{s}):y_{s}\in
X_{s}\\},\end{split}$ (4.5)

respectively and pass through the point $x\in\mathbb{R}^{d}$. This point $x$
is called the base point of the leaves.

Next, we consider random pseudo-unstable foliations and pseudo-unstable
foliations. As we did in section 3, we decompose $\mathbb{R}^{d}=X_{+}\oplus
X_{-}$ for $1\leq j<d$, where

$\displaystyle X_{+}:=X_{1}\oplus\cdots\oplus X_{j},\quad
X_{-}:=X_{j+1}\oplus\cdots\oplus X_{{p}}.$

Let $\pi_{\pm}$ be the associated projections. Set

$\displaystyle\Lambda_{\pm}(\omega):=\Lambda(\omega)|_{X_{\pm}},\quad\Lambda_{\pm}(n,\omega):=\Lambda(n,\omega)|_{X_{\pm}},\quad
f_{\pm}:=\pi_{\pm}f.$

Consider the following Lyapunov-Perron equations associated with the pseudo-
unstable and pseudo-stable invariant foliations respectively,

$\displaystyle\begin{split}p_{n,j}(\omega,x,y_{+})&=\Lambda_{+}(n,\omega)(y_{+}-\pi_{+}x)-\sum_{k=n}^{-1}\Lambda_{+}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{+}(p_{k,j})(\omega,x,y_{+})\\\
&\quad+\sum_{k=-\infty}^{n-1}\Lambda_{-}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{-}(p_{k,j})(\omega,x,y_{+}),\qquad\forall
n<0,\\\
p_{0,j}(\omega,x,y_{+})&=y_{+}-\pi_{+}x+\sum_{k=-\infty}^{-1}\Lambda_{-}({-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{-}(p_{k,j})(\omega,x,y_{+}),\qquad
n=0,\end{split}$ (4.6)

where
$\Xi_{k}^{\pm}(p_{k,j})(\omega,x,y_{+}):=f_{\pm}\big{(}\theta^{k}\omega,p_{k,j}(\omega,x,y_{+})+F(k,\omega,x)\big{)}-f_{\pm}\big{(}\theta^{k}\omega,F(k,\omega,x)\big{)},$
and

$\displaystyle\begin{split}q_{n,j}(\omega,x,y_{-})&=\Lambda_{-}(n,\omega)(y_{-}-\pi_{-}x)+\sum_{k=0}^{n-1}\Lambda_{-}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{-}(q_{k,j})(\omega,x,y_{-})\\\
&\quad-\sum_{k=n}^{\infty}\Lambda_{+}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{+}(q_{k,j})(\omega,x,y_{-}),\qquad\forall
n\geq 0,\\\
q_{0,j}(\omega,x,y_{-})&=y_{-}-\pi_{-}x-\sum_{k=0}^{\infty}\Lambda_{+}({-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{+}(q_{k,j})(\omega,x,y_{-}),\qquad
n=0,\end{split}$ (4.7)

where
$\Xi_{k}^{\pm}(q_{k,j})(\omega,x,y_{-}):=f_{\pm}\big{(}\theta^{k}\omega,q_{k,j}(\omega,x,y_{-})+F(k,\omega,x)\big{)}-f_{\pm}\big{(}\theta^{k}\omega,F(k,\omega,x)\big{)}.$
Using the same arguments used in [56, Lemma 4.5], one has that equations (4.6)
and (4.7) have continuous solutions

$(p_{n,j}(\omega,\cdot,\cdot))_{n\leq 0}\in
C^{-}_{\varrho}:=\big{\\{}\\{p_{n}\\}_{n\leq
0}:\;p_{n}\in\mathbb{R}^{d},\;\;\sup_{n\leq 0}\\{e^{-\varrho
n}\|p_{n}\|\\}<\infty\big{\\}}$

and

$(q_{n,j}(\omega,\cdot,\cdot))_{n\geq 0}\in
C^{+}_{\varrho}:=\big{\\{}\\{q_{n}\\}_{n\geq
0}:\;q_{n}\in\mathbb{R}^{d},\;\;\sup_{n\geq 0}\\{e^{-\varrho
n}\|q_{n}\|\\}<\infty\big{\\}}$

respectively, where
$\varrho\in(\lambda_{j+1}+3\epsilon,\lambda_{j}-3\epsilon)$ and condition
(4.4) is used to show that the associated Lyapunov-Perron operators are
contraction in the Banach spaces $C^{-}_{\varrho}$ and $C^{+}_{\varrho}$
respectively.

Using these solutions, one obtains the following:

  * $\bullet$

The strong-unstable (including only unstable directions) foliations
$\\{{\mathcal{W}}_{\leq j}^{uu}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ for
$j\leq\tau$.

  * $\bullet$

The pseudo-unstable (including unstable directions and stable ones with larger
Lyapunov exponents) foliations $\\{{\mathcal{W}}_{\leq
j}^{pu}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ for $j>\tau$.

  * $\bullet$

The pseudo-stable (including stable directions and unstable ones with smaller
Lyapunov exponents) foliations $\\{{\mathcal{W}}_{\geq
j+1}^{ps}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ for $j<\tau$.

  * $\bullet$

The strong-stable (including only stable directions) foliations
$\\{{\mathcal{W}}_{\geq j+1}^{ss}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ for
$j\geq\tau$.

The leaves of these foliations are given by

$\displaystyle\begin{split}&{\mathcal{W}}_{\leq
j}^{uu}(\omega,x):=\\{x+p_{0,j}(\omega,x,y_{+})\in\mathbb{R}^{d}:y_{+}\in
X_{+}\\},\quad\forall j\leq\tau,\\\ &{\mathcal{W}}_{\leq
j}^{pu}(\omega,x):=\\{x+p_{0,j}(\omega,x,y_{+})\in\mathbb{R}^{d}:y_{+}\in
X_{+}\\},\quad\forall j>\tau,\\\ &{\mathcal{W}}_{\geq
j+1}^{ps}(\omega,x):=\\{x+q_{0,j}(\omega,x,y_{-})\in\mathbb{R}^{d}:y_{-}\in
X_{-}\\},\quad\forall j<\tau,\\\ &{\mathcal{W}}_{\geq
j+1}^{ss}(\omega,x):=\\{x+q_{0,j}(\omega,x,y_{-})\in\mathbb{R}^{d}:y_{-}\in
X_{-}\\},\quad\forall j\geq\tau,\end{split}$ (4.8)

respectively. Note that the case $j=\tau$ are the random unstable and stable
foliations. The results on random strong unstable foliations and strong stable
foliations can also be found in [43].

The following theorem is on the regularity of the leaves and their continuity
with respect to the base point $x$.

######  Theorem 4.1.

Suppose that the random diffeomorphism
$F:\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies (3.4). Then, the
measurable mappings $p_{0,j}(\omega,x,\cdot)$ ($j\leq\tau$) and
$q_{0,j}(\omega,x,\cdot)$ ($j\geq\tau$) are tempered $C^{2,\alpha}$
${\rm(}\alpha\in[0,1]{\rm)}$ and $p_{0,j}(\omega,x,\cdot)$ ($j>\tau$) and
$q_{0,j}(\omega,x,\cdot)$ ($j<\tau$) are tempered $C^{1,\beta}$ with a
constant $\beta\in(0,1]$. All the above derivatives and Hölder coefficients
are uniformly bounded with respect to $x,y_{-},y_{+}$, and

$\displaystyle\begin{split}&\|\partial_{y_{+}}p_{0,j}(\omega,x,y_{+})-{\rm
id}_{+}\|\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega),\\\
&\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-{\rm id}_{-}\|\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega)\end{split}$
(4.9)

for all $j=1,...,p$ and all
$(\omega,x,y_{\pm})\in\Omega\times\mathbb{R}^{d}\times X_{\pm}$, where
$C_{\lambda}>0$ is a constant, ${\rm id}_{\pm}$ is the identity in $X_{\pm}$,
and $C_{\epsilon}(\omega):=4(K_{\epsilon}(\omega))^{2}M_{\epsilon}(\omega)$.
Moreover, the following results hold:

  * (A1)

For every $\omega\in\Omega$, the mappings $x\mapsto p_{0,j}(\omega,x,\cdot)$
and $x\mapsto q_{0,j}(\omega,x,\cdot)$ are uniformly continuous in the
$C^{1,\beta}$-topology on $U_{1}$;

  * (A2)

the mappings $x\mapsto\partial^{2}_{y_{+}^{2}}p_{0,j}(\omega,x,y_{+})$
($j\leq\tau$) and $x\mapsto\partial^{2}_{y_{-}^{2}}q_{0,j}(\omega,x,y_{-})$
($j\geq\tau$) are Hölder continuous with uniformly bounded Hölder coefficients
with respect to $x,y_{+},y_{-}$ on $U_{1}$,

where $U_{1}:=\\{x\in\mathbb{R}^{d}:\|x\|<1/2\\}$.

######  Remark 4.1.

(i) In the theorem above, we used the following simpler notations:

$\partial_{y_{+}}p_{0,j}(\omega,x,y_{+}):=\frac{\partial
p_{0,j}(\omega,x,y_{+})}{\partial{y_{+}}},\qquad\partial^{2}_{y_{+}^{2}}p_{0,j}(\omega,x,y_{+}):=\frac{\partial^{2}p_{0,j}(\omega,x,y_{+})}{\partial
y_{+}^{2}}.$

  * (ii)

The most part of this theorem for pseudo-stable foliations was established in
[56] except estimate (4.9) and the uniform boundedness of derivatives with
respect to $x,y_{+},y_{-}$.

  * (iii)

In both pseudo-unstable and pseudo-stable cases, this theorem gives that every
leaf of the foliations is $C^{1,\tilde{\beta}}$ for all
$\tilde{\beta}\in(0,\beta]$, where $\beta\in(0,1]$ is determined by the
spectral gap between $\lambda_{j}$ and $\lambda_{j+1}$. In fact, according to
[56, Theorem 3.1], the $\beta$ can be chosen as

$\displaystyle\beta=\min\Bigg{\\{}\min_{j=1,...,p-1}\frac{\lambda_{j}-\lambda_{j+1}}{\lambda_{\max}},~{}\min_{\mbox{\tiny$\begin{array}[]{c}i\in\\{1,...,\tau\\},\kappa\in\\{\tau+1,...,{p}\\},\\\
j\in\\{1,...,{p}\\}\end{array}$}}\frac{|\lambda_{i}+\lambda_{\kappa}-\lambda_{j}|}{2\lambda_{\max}}\Bigg{\\}}$
(4.12)

with $\lambda_{\max}:=\max\\{2\lambda_{1},-2\lambda_{p}\\}$.

  * (iv)

Mapping $x\mapsto h(x,\cdot)$ is continuous in the $C^{1,\beta}$-topology
means that $h(x,y)$ is $C^{1}$ with respect to $y$ such that

$\displaystyle\|h(x,\cdot)-h(\tilde{x},\cdot)\|_{C^{1,\beta}}$
$\displaystyle:=\|h(x,\cdot)-h(\tilde{x},\cdot)\|_{C^{1}}+\sup_{y,\tilde{y}\in\mathbb{R}^{d}}\frac{\|\\{\partial_{y}h(x,y)-\partial_{y}h(\tilde{x},y)\\}-\\{\partial_{y}h(x,\tilde{y})-\partial_{y}h(\tilde{x},\tilde{y})\\}\|}{\|y-\tilde{y}\|^{\beta}}\to
0$

as $\|x-\tilde{x}\|\to 0$.

Proof of Theorem 4.1. We prove the theorem only for $q_{0,j}$ since the proof
for $p_{0,j}$ are almost the same. First, using (2.7), we have the following
exponential dichotomy:

$\displaystyle\begin{split}&\|\Lambda(n,\omega)|_{X_{+}}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n},~{}~{}\forall n<0,\\\
&\|\Lambda(n,\omega)|_{X_{-}}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j+1}+\epsilon)n},~{}~{}\forall n\geq
0,\end{split}$ (4.13)

where $K_{\epsilon}(\omega)\geq K(\omega)$. Then, using [56, Lemma 4.5], we
obtain that equation (4.7) has a unique solution
$(q_{n,j}(\omega,x,y_{-}))_{n\geq 0}$, measurable with respect to $\omega$ and
tempered $C^{2,\alpha}$ with respect to $y_{-}$ when $j\geq\tau$ (or tempered
$C^{1,\beta}$ when $j<\tau$), such that

$\displaystyle\sup_{n\geq 0}\\{e^{-\varrho
n}\|q_{n,j}(\omega,x,y_{-})\|\\}<\infty,\qquad\sup_{n\geq 0}\\{e^{-\varrho
n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\|\\}<\infty,$
$\displaystyle\sup_{n\geq 0}\Bigg{\\{}e^{-\varrho
n}\frac{\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{n,j}(\omega,x,\tilde{y}_{-})\|}{\|y_{-}-\tilde{y}_{-}\|^{\beta}}\Bigg{\\}}<\infty.$

Furthermore, differentiating both sides of equation (4.7) with respect to
$y_{-}$, we get

$\displaystyle\begin{split}&\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\\\
&=\Lambda_{-}(n,\omega)+\sum_{k=0}^{n-1}\Lambda_{-}(n-k-1,\theta^{k+1}\omega)Df_{-}(\theta^{k}\omega,q_{k,j}(\omega,x,y_{-})+F(k,\omega,x))\,\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\\\
&\quad-\sum_{k=n}^{\infty}\Lambda_{+}(n-k-1,\theta^{k+1}\omega)Df_{+}(\theta^{k}\omega,q_{k,j}(\omega,x,y_{-})+F(k,\omega,x))\,\partial_{y_{-}}q_{k,j}(\omega,x,y_{-}),\quad\forall
n>0,\\\ &\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})\\\ &={\rm
id}_{-}-\sum_{k=0}^{\infty}\Lambda_{+}(-k-1,\theta^{k+1}\omega)Df_{+}(\theta^{k}\omega,q_{k,j}(\omega,x,y_{-})+F(k,\omega,x))\,\partial_{y_{-}}q_{k,j}(\omega,x,y_{-}),\quad
n=0.\end{split}$ (4.14)

Then, for an arbitrarily given constant
$\varrho\in(\lambda_{j+1}+3\epsilon,\lambda_{j}-3\epsilon)$, we have

$\displaystyle e^{-\varrho n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\|$
$\displaystyle\leq e^{-\varrho
n}\|\Lambda_{-}(n,\omega)\|+e^{-\varrho}\sum_{k=0}^{n-1}e^{-\varrho(n-k-1)}\|\Lambda_{-}(n-k-1,\theta^{k+1}\omega)\|\,\delta(\theta^{k}\omega)e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|$
$\displaystyle\quad-e^{-\varrho}\sum_{k=n}^{\infty}e^{-\varrho(n-k-1)}\|\Lambda_{+}(n-k-1,\theta^{k+1}\omega)\|\delta(\theta^{k}\omega)e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|$ $\displaystyle\leq
K_{\epsilon}(\omega)+C_{\lambda}K_{\epsilon}(\theta^{k+1}\omega)\delta(\theta^{k}\omega)\sup_{k\geq
0}\big{\\{}e^{-\varrho k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|\big{\\}}$
$\displaystyle\leq K_{\epsilon}(\omega)+\frac{1}{2}\sup_{k\geq
0}\big{\\{}e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|\big{\\}},$

where (3.4) and (4.13), the fact that
$C_{\lambda}K_{\epsilon}(\theta^{k+1}\omega)\delta(\theta^{k}\omega)\leq 1/2$
(following from (3.8)) are used. Hence,

$\sup_{n\geq 0}\big{\\{}e^{-\varrho
n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\|\big{\\}}\leq
2K_{\epsilon}(\omega).$

Similarly, we have that

$\sup_{n\geq 0}\big{\\{}e^{-\varrho n}\|q_{n,j}(\omega,x,y_{-})\|\big{\\}}\leq
2K_{\epsilon}(\omega)(\|x\|+\|y_{-}\|)$

and

$\sup_{n\geq
0}\big{\\{}e^{-(\varrho+2\epsilon)n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{n,j}(\omega,x,\tilde{y}_{-})\|\big{\\}}\leq
4(K_{\epsilon}(\omega))^{2}M_{\epsilon}(\omega)\|y_{-}-\tilde{y}_{-}\|^{\beta}.$

Summarizing the above discussion gives

$\displaystyle\begin{split}&q_{0,j}(\omega,x,\pi_{-}x)=0,\qquad\partial_{y_{-}}q_{0,j}(\omega,0,0)={\rm
id}_{-},\\\ &\sup_{n\geq 0}\big{\\{}e^{-\varrho
n}\|q_{n,j}(\omega,x,y_{-})\|\big{\\}}\leq
C_{\epsilon}(\omega)(\|x\|+\|y_{-}\|),\quad\sup_{n\geq 0}\big{\\{}e^{-\varrho
n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\|\big{\\}}\leq
C_{\epsilon}(\omega),\\\
&\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})\|\leq
C_{\epsilon}(\omega)\|y_{-}-\tilde{y}_{-}\|^{\beta},\end{split}$ (4.15)

where
$C_{\epsilon}(\omega):=4(K_{\epsilon}(\omega))^{2}M_{\epsilon}(\omega)>0$.

To see that (4.9) holds, we consider the case of $n=0$ in (4.14). It follows
from (4.13), (3.4) and (4.15) that

$\displaystyle\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-{\rm
id}_{-}\|=e^{(\varrho+2\epsilon)\,0}\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,{{0}},{{0}})\|$
$\displaystyle\leq
e^{-(\varrho+\epsilon)}\sum_{k=0}^{\infty}e^{-\epsilon(k+1)}e^{(\varrho+2\epsilon)(k+1)}\|\Lambda_{+}(-k-1,\theta^{k+1}\omega)\|e^{-\epsilon
k}\delta(\theta^{k}\omega)e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|$ $\displaystyle\leq
e^{-(\varrho+\epsilon)}\sum_{k=0}^{\infty}e^{-\epsilon(k+1)}K_{\epsilon}(\theta^{k+1}\omega)e^{(\varrho-\lambda_{j}+3\epsilon)(k+1)}e^{-\epsilon
k}\delta(\theta^{k}\omega)e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|$ $\displaystyle\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega),$

where the facts that $\varrho-\lambda_{j}+3\epsilon<0$ and
$e^{-\epsilon|k|}\delta(\theta^{k}\omega)\leq\delta(\omega)$ (by (3.9)) are
used. This proves the second inequality of (4.9).

Next, we prove (A1) for the mapping $x\mapsto q_{0,j}(\omega,x,\cdot)$. For
the purpose, we claim that

$\displaystyle\sup_{n\geq
0}\Big{\\{}e^{-\lambda_{\max}n}\|q_{n,j}(\omega,x,y_{-})-q_{n,j}(\omega,\tilde{x},y_{-})\|\Big{\\}}\leq
L_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}$ (4.16)

for all $x,\tilde{x},y_{-},\tilde{y}_{-}\in U_{1}$, where
$L_{\epsilon}(\omega)>0$ is an $\epsilon$-slowly varying random variable and
$\sigma\in(0,1)$ is a constant such that
$\sigma<(\lambda_{j}-\lambda_{j+1})/(4\lambda_{\max})$. In order to prove
(4.16), we notice from (3.25) that

$\displaystyle e^{-\lambda_{\max}k}\|F(k,\omega,x)-F(k,\omega,\tilde{x})\|$
$\displaystyle\leq
e^{-\lambda_{\max}k}\sup_{\xi\in\mathbb{R}^{d}}\|DF(k,\omega,\xi)\|\,\|x-\tilde{x}\|$
$\displaystyle\leq K_{\epsilon}(\omega)\|x-\tilde{x}\|,\qquad\forall k\geq 0.$
(4.17)

Using the notation $[h(\cdot)]_{\tilde{x}}^{x}:=h(x)-h(\tilde{x})$ for short,
we have

$\displaystyle\big{[}f(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))-f(\theta^{k}\omega,F(k,\omega,\cdot))\big{]}_{\tilde{x}}^{x}$
$\displaystyle=\big{[}f(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,x))\big{]}_{\tilde{x}}^{x}+\big{[}f(\theta^{k}\omega,q_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))\big{]}_{\tilde{x}}^{x}$
$\displaystyle\quad-\big{[}f(\theta^{k}\omega,F(k,\omega,\cdot))\big{]}_{\tilde{x}}^{x}$
$\displaystyle=\big{[}f(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,x))\big{]}_{\tilde{x}}^{x}$
$\displaystyle\quad+\int_{0}^{1}\big{[}Df(\theta^{k}\omega,tq_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))q_{k,j}(\omega,\tilde{x},y_{-})\big{]}_{\tilde{x}}^{x}dt.$
(4.18)

Then, setting

${\tilde{\lambda}}:=\lambda_{j+1}+(\lambda_{j}-\lambda_{j+1})/2,\qquad\|h(\cdot)\|_{\tilde{x}}^{x}:=\|h(x)-h(\tilde{x})\|$

for short, we obtain from (4.18), (3.4), (4.15) and (4.17) that

$\displaystyle
e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}f(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))-f(\theta^{k}\omega,F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}$
$\displaystyle\leq\sup_{\xi\in\mathbb{R}^{d}}\|Df(\theta^{k}\omega,\xi)\|\,e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}$
$\displaystyle\quad+e^{-(\lambda_{\max}\sigma+\epsilon)k}\Big{(}\big{\|}Df(\theta^{k}\omega,tq_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}\Big{)}^{\sigma}e^{-{\tilde{\lambda}}k}\|q_{k,j}(\omega,\tilde{x},y_{-})\|$
$\displaystyle\leq\delta(\theta^{k}\omega)e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}+e^{-\epsilon
k}M_{\epsilon}(\theta^{k}\omega)^{\sigma}\Big{(}e^{-\lambda_{\max}k}\|F(k,\omega,\cdot)\big{\|}_{\tilde{x}}^{x}\Big{)}^{\sigma}C_{\epsilon}(\omega)(\|\tilde{x}\|+\|y_{-}\|)$
$\displaystyle\leq\delta(\theta^{k}\omega)e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\,+\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}$
(4.19)

for all $k\geq 0$ and all $\tilde{x},y_{-}\in U_{1}$, where
$\tilde{M}_{\epsilon}(\omega):=2M_{\epsilon}(\omega)^{\sigma}K_{\epsilon}(\omega)^{\sigma}C_{\epsilon}(\omega)>0$
and the following estimate is used:

$\big{\|}Df(\theta^{k}\omega,tq_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}\leq\Big{(}\big{\|}Df(\theta^{k}\omega,tq_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}\Big{)}^{\sigma}$

for $\sigma\in(0,1)$ since
$\|Df(\theta^{k}\omega,x)\|\leq\delta(\theta^{k}\omega)<1/2$ due to (3.4).

Notice that
${\tilde{\lambda}}+\lambda_{\max}\sigma+2\epsilon\in(\lambda_{j+1}+3\epsilon,\lambda_{j}-3\epsilon)$
with small $\epsilon$ due to the above choices of ${\tilde{\lambda}}$ and
$\sigma$, which implies that

$\sup_{k\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\Big{\\}}<\infty$

by (4.15). Hence, we see from (4.7), (4.13) and (4.19) that for $n\geq 0$

$\displaystyle
e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\|q_{n,j}(\omega,x,y_{-})-q_{n,j}(\omega,\tilde{x},y_{-})\|$
$\displaystyle\leq
e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\|\Lambda_{-}(n,\omega)\|\|x-\tilde{x}\|$
$\displaystyle\quad+e^{-{\tilde{\lambda}}-(\lambda_{\max}\sigma+2\epsilon)}\sum_{k=0}^{n-1}e^{-{\tilde{\lambda}}(n-k-1)-(\lambda_{\max}\sigma+2\epsilon)(n-k-1)}\|\Lambda_{-}(n-k-1,\theta^{k+1}\omega)\|$
$\displaystyle\quad\cdot
e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}f_{-}(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))-f_{-}(\theta^{k}\omega,F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}$
$\displaystyle\quad+e^{-{\tilde{\lambda}}-(\lambda_{\max}\sigma+2\epsilon)}\sum_{k=n}^{\infty}e^{-{\tilde{\lambda}}(n-k-1)-(\lambda_{\max}\sigma+2\epsilon)(n-k-1)}\|\Lambda_{+}(n-k-1,\theta^{k+1}\omega)\|$
$\displaystyle\quad\cdot
e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}f_{+}(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))-f_{+}(\theta^{k}\omega,F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}$
$\displaystyle\leq
K_{\epsilon}(\omega)e^{(\lambda_{j+1}-{\tilde{\lambda}}-\lambda_{\max}\sigma-\epsilon)n}\|x-\tilde{x}\|^{\sigma}$
$\displaystyle\quad+e^{-{\tilde{\lambda}}-(\lambda_{\max}\sigma+2\epsilon)}\bigg{(}\sum_{k=0}^{n-1}K_{\epsilon}(\theta^{k+1}\omega)e^{(\lambda_{j+1}-{\tilde{\lambda}}-\lambda_{\max}\sigma-\epsilon)(n-k-1)}$
$\displaystyle\quad+\sum_{k=n}^{\infty}K_{\epsilon}(\theta^{k+1}\omega)e^{(\lambda_{j}-{\tilde{\lambda}}-\lambda_{\max}\sigma-3\epsilon)(n-k-1)}\bigg{)}$
$\displaystyle\quad\cdot\sup_{k\geq
0}\Big{\\{}\delta(\theta^{k}\omega)e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\,+e^{-\epsilon
k}\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}\Big{\\}}$
$\displaystyle\leq K_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}$
$\displaystyle\quad+C_{\lambda}K_{\epsilon}(\theta^{k+1}\omega)\sup_{k\geq
0}\Big{\\{}\delta(\theta^{k}\omega)e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\,+e^{-\epsilon
k}\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}\Big{\\}}$
$\displaystyle\leq
2C_{\lambda}K_{\epsilon}(\omega)\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}+\frac{1}{2}\sup_{k\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\Big{\\}}$
(4.20)

when $\|x-\tilde{x}\|<1$. Here, the facts $\mathfrak{M}_{\epsilon}(\omega)\geq
2C_{\lambda}K_{\epsilon}(\theta\omega)$ (i.e.,
$\delta(\theta^{k}\omega)\leq(2C_{\lambda}K_{\epsilon}(\theta^{k+1}\omega))^{-1}$
by (3.8)) and
$\lambda_{j+1}-{\tilde{\lambda}}-\lambda_{\max}\sigma-\epsilon<0$,
$\lambda_{j}-{\tilde{\lambda}}-\lambda_{\max}\sigma-3\epsilon>0$ (see the
above choices of ${\tilde{\lambda}}$ and $\sigma$) are used. Hence, we have

$\displaystyle\sup_{n\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\|q_{n,j}(\omega,x,y_{-})-q_{n,j}(\omega,\tilde{x},y_{-})\|\Big{\\}}\leq
4C_{\lambda}K_{\epsilon}(\omega)\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma},$

which proves (4.16) by setting an $\epsilon$-slowly varying random variable
$L_{\epsilon}(\omega)\geq
4C_{\lambda}K_{\epsilon}(\omega)\tilde{M}_{\epsilon}(\omega)>0$ since
$e^{-\lambda_{\max}n}\leq
e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}$.

Next, we prove that

$\displaystyle\sup_{n\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\big{\|}\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{n,j}(\omega,\tilde{x},y_{-})\big{\|}\Big{\\}}\leq\tilde{L}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma^{2}},$
(4.21)

where
$\tilde{L}_{\epsilon}(\omega):=2C_{\lambda}K_{\epsilon}(\omega)M_{\epsilon}(\omega)(L_{\epsilon}(\omega)+K_{\epsilon}(\omega))^{\sigma}C_{\epsilon}(\omega)>0$.
In fact, notice that by (4.16) and (4.17) we have

$\displaystyle
e^{-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}Df(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))\big{\|}^{x}_{\tilde{x}}$
$\displaystyle=\Big{(}\big{\|}Df(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))\big{\|}^{x}_{\tilde{x}}\Big{)}^{1-\sigma}$
$\displaystyle\quad\cdot e^{-\epsilon
k}\Big{(}e^{-\lambda_{\max}k}\big{\|}Df(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))\big{\|}^{x}_{\tilde{x}}\Big{)}^{\sigma}$
$\displaystyle\leq(2\delta(\theta^{k}\omega))^{1-\sigma}e^{-\epsilon
k}M_{\epsilon}(\theta^{k}\omega)^{\sigma}\Big{(}e^{-\lambda_{\rm
max}k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}^{x}_{\tilde{x}}+e^{-\lambda_{\rm
max}k}\big{\|}F(k,\omega,\cdot)\big{\|}^{x}_{\tilde{x}}\Big{)}^{\sigma}$
$\displaystyle\leq
M_{\epsilon}(\omega)(L_{\epsilon}(\omega)+K_{\epsilon}(\omega))^{\sigma}\|x-\tilde{x}\|^{\sigma^{2}}$
(4.22)

when $\|x-\tilde{x}\|<1$. Then, using similar arguments to (4.20), we obtain
from (4.14) that

$\displaystyle
e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{n,j}(\omega,\tilde{x},y_{-})\|$
$\displaystyle=C_{\lambda}K_{\epsilon}(\omega)\sup_{k\geq
0}\bigg{\\{}e^{-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}Df(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))\big{\|}^{x}_{\tilde{x}}$
$\displaystyle\quad\cdot
e^{-{\tilde{\lambda}}k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|+\|Df(\theta^{k}\omega,q_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\tilde{x}))\|$
$\displaystyle\quad\cdot
e^{-{\tilde{\lambda}}k-(\lambda_{\max}+2\epsilon)k}\big{\|}\partial_{y_{-}}q_{k,j}(\omega,\cdot,y_{-})\big{\|}^{x}_{\tilde{x}}\bigg{\\}}$
$\displaystyle\leq
C_{\lambda}K_{\epsilon}(\omega)M_{\epsilon}(\omega)(L_{\epsilon}(\omega)+K_{\epsilon}(\omega))^{\sigma}C_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma^{2}}$
$\displaystyle\quad+\frac{1}{2}\sup_{k\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}k-(\lambda_{\max}+2\epsilon)k}\big{\|}\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{k,j}(\omega,\tilde{x},y_{-})\big{\|}\Big{\\}},$
(4.23)

which yields (4.21).

Setting $n=0$ in (4.21) we obtain that

$\displaystyle\big{\|}\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})\big{\|}\leq\tilde{L}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma^{2}}.$
(4.24)

Therefore for any small constant $\varepsilon>0$, we obtain

$\displaystyle\frac{\big{\|}\\{\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})\\}-\\{\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},\tilde{y}_{-})\\}\big{\|}}{\|y_{-}-\tilde{y}_{-}\|^{(1-\varepsilon)\beta}\|x-\tilde{x}\|^{\varepsilon\sigma^{2}}}$
$\displaystyle\leq\bigg{(}\frac{\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})\|+\|\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},\tilde{y}_{-})\|}{\|y_{-}-\tilde{y}_{-}\|^{\beta}}\bigg{)}^{1-\varepsilon}$
$\displaystyle\quad\cdot\bigg{(}\frac{\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})\|+\|\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},\tilde{y}_{-})\|}{\|x-\tilde{x}\|^{\sigma^{2}}}\bigg{)}^{\varepsilon}$
$\displaystyle\leq 2C_{\epsilon}(\omega)\tilde{L}_{\epsilon}(\omega),$ (4.25)

due to (4.15). Combining (4.16) (setting $n=0$), (4.24) with (4.25) together
gives that

$\displaystyle\|q_{0,j}(\omega,x,\cdot)-q_{0,j}(\omega,\tilde{x},\cdot)\|_{C^{1,(1-\varepsilon)\beta}}$
$\displaystyle=\|q_{0,j}(\omega,x,\cdot)-q_{0,j}(\omega,\tilde{x},\cdot)\|_{C^{1}}$
$\displaystyle\quad+\sup_{y_{-},\tilde{y}_{-}\in
X_{-}}\frac{\big{\|}\\{\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})\\}-\\{\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},\tilde{y}_{-})\\}\big{\|}}{\|y_{-}-\tilde{y}_{-}\|^{(1-\varepsilon)\beta}}$
$\displaystyle\leq\\{L_{\epsilon}(\omega)+\tilde{L}_{\epsilon}(\omega)+2C_{\epsilon}(\omega)\tilde{L}_{\epsilon}(\omega)\\}\|x-\tilde{x}\|^{\varepsilon\sigma^{2}}$

when $\|x-\tilde{x}\|<1$. Since $\beta$ is in an open interval determined by
the spectral gaps (see (4.12)) and $\varepsilon>0$ can be sufficiently small,
it is reasonable to use $\beta$ to denote $(1-\varepsilon)\beta$ and therefore
the mapping $x\mapsto q_{0,j}(\omega,x,\cdot)$ is uniformly continuous in the
$C^{1,\beta}$-topology on $U_{1}$. This proves (A1) for the mapping $x\mapsto
q_{0,j}(\omega,x,\cdot)$.

Similarly to (4.24), in the case of $j\geq\tau$, we show that the mapping
$x\mapsto\partial^{2}_{y_{-}^{2}}q_{0,j}(\omega,x,y_{-})$ is Hölder continuous
with uniformly bounded Hölder coefficients with respect to $x,y_{-}$ on
$U_{1}$. This proves (A2) for the mapping $x\mapsto q_{0,j}(\omega,x,\cdot)$
by using a similar discussion to the one given below (4.24). The proof of this
theorem is completed. $\Box$

For the sake of convenient representations, we use notations

$\displaystyle{\mathcal{W}}_{\leq
j}^{u}(\omega,x):=\left\\{\begin{array}[]{lll}{\mathcal{W}}_{\leq
j}^{uu}(\omega,x),&\\!j\\!<\\!\tau,\vspace{1ex}\\\
{\mathcal{W}}_{u}(\omega,x),&\\!j\\!=\\!\tau,\vspace{1ex}\\\
{\mathcal{W}}_{\leq
j}^{pu}(\omega,x),&\\!j\\!>\\!\tau,\end{array}\right.\qquad{\mathcal{W}}_{\geq
j+1}^{s}(\omega,x):=\left\\{\begin{array}[]{lll}{\mathcal{W}}_{\geq
j+1}^{ps}(\omega,x),&\\!j\\!<\\!\tau,\vspace{1ex}\\\
{\mathcal{W}}_{s}(\omega,x),&\\!j\\!=\\!\tau,\vspace{1ex}\\\
{\mathcal{W}}_{\geq j+1}^{ss}(\omega,x),&\\!j\\!>\\!\tau.\end{array}\right.$
(4.32)

Notice that these leaves are graphs of the mappings

$y_{+}\mapsto p_{0,j}(\omega,x,y_{+})+\pi_{+}x-y_{+},\qquad y_{-}\mapsto
q_{0,j}(\omega,x,y_{-})+\pi_{-}x-y_{-}$

with Lipschitz constants $<1$ by (4.9). Then, corresponding to each $X_{j}$ we
introduce the leaf

$\displaystyle{\mathcal{W}}_{j}(\omega,x):=\left\\{\begin{array}[]{lll}{\mathcal{W}}_{\leq
1}^{u}(\omega,x),&\quad\mbox{if }j=1,\vspace{1ex}\\\ {\mathcal{W}}_{\leq
j}^{u}(\omega,x)\cap{\mathcal{W}}_{\geq j}^{s}(\omega,x),&\quad\mbox{if
}j=2,...,{{p}}-1,\vspace{1ex}\\\
{\mathcal{W}}_{\geq{{p}}}^{s}(\omega,x),&\quad\mbox{if
}j={{p}}.\end{array}\right.$ (4.36)

Clearly, for each $j\in\\{1,...,p\\}$ it defines a foliation
${\mathcal{W}}_{j}(\omega):=\\{{\mathcal{W}}_{j}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$,
called the random intermediate foliation, and it is the graph of a mapping
$\check{q}_{j}(\omega,x,\cdot):X_{j}\to X_{1}\oplus\cdots\oplus X_{j-1}\oplus
X_{j+1}\oplus\cdots\oplus X_{p}$ such that
$\mathbb{R}^{d}=\cup_{x\in\mathbb{R}^{d}}{\mathcal{W}}_{j}(\omega,x)$ and the
following corollary holds.

######  Corollary 4.1.

Suppose that the random diffeomorphism
$F:\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies (3.4). Then for each
$j=1,...,p$ the measurable mapping $\check{q}_{j}(\omega,x,\cdot)$ is tempered
$C^{1,\beta}$ with a constant $\beta\in(0,1]$ such that

$\displaystyle\|\partial_{y_{j}}\check{q}_{j}(\omega,x,y_{j})-{\rm
id}_{j}\|\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega)$ (4.37)

for all $(\omega,x,y_{j})\in\Omega\times\mathbb{R}^{d}\times X_{j}$, where
${\rm id}_{j}$ is the identity in $X_{j}$. Moreover, the mapping
$x\mapsto\check{q}_{j}(\omega,x,\cdot)$ is uniformly continuous in the
$C^{1,\beta}$-topology on $U_{1}:=\\{x\in\mathbb{R}^{d}:\|x\|<1/2\\}$.

Concerning the relation between the intermediate distribution $E_{j}(\omega)$
and the intermediate foliation ${\mathcal{W}}_{j}(\omega)$, we further show
that

$\displaystyle E_{j}(\omega,x)$
$\displaystyle=\\{\partial_{y_{j}}\check{q}(\omega,x,\pi_{j}x)\,z_{j}\in\mathbb{R}^{d}:~{}z_{j}\in
X_{j}\\},\qquad\forall x\in\mathbb{R}^{d},$ (4.38)

(see (9.2) below in the stable case for more details), where
$\pi_{j}:\mathbb{R}^{d}\to X_{j}$ is defined by
$\pi_{j}(x_{1}+\cdots+x_{p}):=x_{j}$. It means that every fiber of
$E_{j}(\omega)$ at $x$ is the tangent space of the leaf of
${\mathcal{W}}_{j}(\omega)$ at $x$.

##  5\. Decomposition of a cohomological equation

In order to show that the stable and unstable distributions obtained in
section 3 are $C^{1,\beta_{\alpha}}$ with a constant
$\beta_{\alpha}\in[0,\alpha]$, we will prove that there are
$C^{1,\beta_{\alpha}}$ vectors fields $v^{s}_{j}(\omega,x)$ and
$v^{u}_{j}(\omega,x)$ that span $E_{s}(\omega,x)$ and $E_{u}(\omega,x)$ at
every point $x\in\mathbb{R}^{d}$ respectively (see [61, p.41]). This will be
done in sections 5-9 mainly for $E_{s}(\omega,x)$, as the proof for
$E_{u}(\omega,x)$ is similar.

In this section, we first derive a cohomological equation for the vector field
$v^{s}_{j}(\omega,x)$. Then, we solve the cohomological equation in 6 steps to
show that, for any $\bar{x}\in\mathbb{R}^{d}$ and $i=1,...,\tau$, there is a
basis (depending on $\bar{x}$ and $i$) of $E_{s}(\omega,x)$ which is
differentiable at $\bar{x}$ along every leaf
${\mathcal{W}}_{i}(\omega,\bar{x})$ of the intermediate foliation with a
(Hölder) continuous differential. Letting
$T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})$ be the tangent space of
${\mathcal{W}}_{i}(\omega,\bar{x})$ at $\bar{x}$, we have the following
theorem.

######  Theorem 5.1.

Suppose that the random diffeomorphism
$F:\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies (3.4) and that the
Lyapunov exponents satisfy (2.10). Then, for any point
$\bar{x}\in\mathbb{R}^{d}$ and $i=1,...,\tau$, the fibers $E_{s}(\omega,x)$
have bases
$\\{{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)\\}_{\kappa={\tau}+1,...,{p},~{}\iota=1,...,d_{\kappa}}$
for all $x\in{\mathcal{W}}_{i}(\omega,\bar{x})$ near $\bar{x}$, whose
differential (with respect to $x$)

$\displaystyle
dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x}):T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})\to\mathbb{R}^{d}~{}~{}(=T_{v_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})}\mathbb{R}^{d}\,)$
(5.1)

at the point $\bar{x}$ is $\beta_{v}$-Hölder continuous with respect to
$\bar{x}$ for every $\kappa={\tau}+1,...,{p}$ and $\iota=1,...,d_{\kappa}$,
where $\beta_{v}:=\min\\{\beta_{E},\epsilon/\lambda_{\max},\alpha\\}$.
Moreover, the differential and the Hölder coefficient are measurable with
respect to $\omega$ and are uniformly bounded with respect to $\bar{x}$ and
$i$.

######  Remark 5.1.

(i) The $\beta_{v}$-Hölder continuity of
$dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})$ with respect to $\bar{x}$
means that

$\displaystyle\|dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})\circ\hbar(\omega,\bar{x})-dv_{\bar{y},i}^{\kappa,\iota}(\omega,\bar{y})\circ\hbar(\omega,\bar{y})\|\leq
L(\omega)\|\bar{x}-\bar{y}\|^{\beta_{v}}$

when $\|\bar{x}-\bar{y}\|<1$, where $\hbar(\omega,\cdot)$ is a homeomorphism
such that $\hbar^{\pm 1}(\omega,\cdot)$ are $\beta_{v}$-Hölder continuous and
$\hbar^{-1}(\omega,\cdot)$ trivializes the bundle
${\mathcal{E}}:=\bigcup_{\bar{x}\in\mathbb{R}^{d}}T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})$0
with the base $\mathbb{R}^{d}$. If $\beta_{v}=0$ (when $\alpha=0$) then
$\beta_{v}$-Hölder continuity merely means continuity, and the differential is
uniformly bounded with respect to $\bar{x}$ and $i$.

(ii) The bases of $E_{s}(\omega,x)$ are not well defined for all
$x\in\mathbb{R}^{d}$ since
$\\{{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)\\}_{\kappa={\tau}+1,...,{p},~{}\iota=1,...,d_{\kappa}}$
depends on $\bar{x}$ and $i$. Therefore, it is possible that
$x\in{\mathcal{W}}_{i}(\omega,\bar{x})\cap{\mathcal{W}}_{j}(\omega,\bar{y})$
for certain $\bar{x},\bar{y}\in\mathbb{R}^{d}$ and $i,j\in\\{1,...\tau\\}$,
but $v_{\bar{x},i}^{\kappa,\iota}(\omega,x)\neq
v_{\bar{y},j}^{\kappa,\iota}(\omega,x)$. We will solve this problem in section
9 by finding a “canonical” basis of $E_{s}(\omega,x)$ not depending on
$\bar{x},i$ and giving the relation between the “canonical” basis and
$\\{{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)\\}_{\kappa={\tau}+1,...,{{p}},~{}\iota=1,...,d_{\kappa}}$.

Proof of Theorem 5.1. We consider the random cohomological equation for
$v(\omega,x)\in E_{s}(\omega,x)$, i.e.,

$\displaystyle
DF(\omega,x)v(\omega,x)=v(\theta\omega,F(\omega,x)),\quad\forall\omega\in\Omega,\quad\forall
x\in\mathbb{R}^{d}.$ (5.2)

In what follows, our goal is to find out the solutions $v(\omega,x)$ of
equation (5.2), which form the basis of $E_{s}(\omega,x)$ and are
differentiable at every point $x\in\mathbb{R}^{d}$ with (Hölder) continuous
derivatives. We will show this in 6 steps.

Step 1. Given any point $\bar{x}\in\mathbb{R}^{d}$, transform the orbit
$(F(n,\omega,\bar{x}))_{n\in\mathbb{Z}}$ to the origin $0$ to get a new system
(5.7), and then reduce to its normal form (5.10).

In order to study the differentiability of $v(\omega,\cdot)$ along the orbit
$(F(n,\omega,\bar{x}))_{n\in\mathbb{Z}}$, we let

$\varpi:=(\omega,\bar{x})\in\bar{\bf\Omega}:=\Omega\times\mathbb{R}^{d},\quad\vartheta\varpi:=(\theta\omega,F(\omega,\bar{x})),\quad{\bf\Lambda}(\varpi):=DF(\omega,\bar{x}).$

Then, we have

$\vartheta^{n}\varpi=(\theta^{n}\omega,F(n,\omega,\bar{x})),\quad{\bf\Lambda}(\vartheta^{n}\varpi)=DF(\theta^{n}\omega,F(n,\omega,\bar{x})),\quad\forall
n\in\mathbb{Z}.$

Note that the random matrix ${\bf\Lambda}(\varpi)$ generates the linear
cocycle

$\displaystyle{\bf\Lambda}(n,\varpi):=\begin{cases}{\bf\Lambda}(\vartheta^{n-1}\varpi)\cdots{\bf\Lambda}(\varpi),&n>0,\\\
{\rm id},&n=0,\\\
{\bf\Lambda}^{-1}(\vartheta^{n}\varpi)\cdots{\bf\Lambda}^{-1}(\vartheta^{-1}\varpi),&n<0.\end{cases}$
(5.3)

It is clear that ${\bf\Lambda}(n,\varpi)=DF(n,\omega,\bar{x})$. Therefore, by
(3.12) we have

$\displaystyle\begin{split}&\|{\bf\Lambda}(n,\varpi)|_{E_{j}(\varpi)}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n},\qquad\forall n<0,\\\
&\|{\bf\Lambda}(n,\varpi)|_{E_{j}(\varpi)}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}+\epsilon)n},\qquad\forall n\geq
0.\end{split}$

Then, we use [1, Corollary 4.3.12] to block-diagonalize
${\bf\Lambda}(n,\varpi)$, namely, there is a random matrix $P(\varpi)\in
Gl(d,\mathbb{R})$ such that

$\displaystyle\bar{\bf\Lambda}(\varpi):=P(\vartheta\varpi){\bf\Lambda}(\varpi)P^{-1}(\varpi)={\rm
diag}(\bar{\bf\Lambda}_{1}(\varpi),...,\bar{\bf\Lambda}_{p}(\varpi)),$ (5.4)

and

$\displaystyle\begin{split}&\|\bar{\bf\Lambda}_{j}(n,\varpi)\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n},\qquad\forall n<0,\\\
&\|\bar{\bf\Lambda}_{j}(n,\varpi)\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}+\epsilon)n},\qquad\forall n\geq
0.\end{split}$ (5.5)

Moreover precisely,
$P(\varpi):=(\vec{e}_{1,1}(\varpi),...,\vec{e}_{p,d_{p}}(\varpi))$ with
measurable unit vectors $\vec{e}_{j,\iota}(\varpi)$ for all $j=1,...,p$ and
$\iota=1,...,d_{j}$ is a transition matrix from the standard basis
$\mathfrak{C}$ of $\mathbb{R}^{d}$ to a basis
$\mathfrak{C}(\varpi):=\\{\vec{e}_{j,\iota}(\varpi)\\}_{j=1,...,p,\,\iota=1,...,d_{j}}$
associated with the measurable decompositions

$\mathbb{R}^{d}=E_{1}(\varpi)\oplus\cdots\oplus E_{p}(\varpi).$

By the second inequality of (3.16), we can choose
$\vec{e}_{j,\iota}(\varpi)\in\mathfrak{C}(\varpi)$ that is uniformly (with
respect to $\varpi$) close to the corresponding standard unit vector
$\vec{e}_{j,\iota}\in\mathfrak{C}$. Thus, for all $\varpi:=(\omega,\bar{x})$
and $\tilde{\varpi}:=(\omega,\bar{y})\in\bar{\bf\Omega}$,

$\displaystyle\|\vec{e}_{j,\iota}(\varpi)-\vec{e}_{j,\iota}(\tilde{\varpi})\|\leq
2{\rm dist}(E_{j}(\varpi),E_{j}(\tilde{\varpi}))$

by (3.15). Then, it follows from Theorem 3.1 that

$\displaystyle\begin{split}&\|P^{\pm 1}(\varpi)-P^{\pm
1}(\tilde{\varpi})\|\leq\|\bar{x}-\bar{y}\|^{\beta_{E}},\quad\forall\varpi:=(\omega,\bar{x}),~{}\tilde{\varpi}:=(\omega,\bar{y})\in\bar{\bf\Omega},\\\
&\|P^{\pm 1}(\varpi)-{\rm id}\|=\|P^{\pm 1}(\omega,\bar{x})-P^{\pm
1}(\omega,0)\|\leq\delta_{E}(\omega),\end{split}$ (5.6)

where the constant $2$ is omitted without loss of generality.

Now, for every $\varpi=(\omega,\bar{x})$ set

$\displaystyle\hat{\bf
F}(\varpi,x):=P(\vartheta\varpi)F(\omega,P^{-1}(\varpi)x+\bar{x})-P(\vartheta\varpi)F(\omega,\bar{x}),\quad
x\in\mathbb{R}^{d}.$ (5.7)

Then,

$\displaystyle\hat{\bf F}(\vartheta^{n}\varpi,x)$
$\displaystyle=P(\theta^{n+1}\omega,F(n+1,\omega,\bar{x}))F(\theta^{n}\omega,P^{-1}(\theta^{n}\omega,F(n,\omega,\bar{x}))x+F(n,\omega,\bar{x}))$
$\displaystyle\quad-P(\theta^{n+1}\omega,F(n+1,\omega,\bar{x}))F(\theta^{n}\omega,F(n,\omega,\bar{x})),$

which means that the problem of $F(\theta^{n}\omega,\cdot)$ near its orbit
$F(n,\omega,\bar{x})$ is converted to the problem of $\hat{\bf F}(\varpi,x)$
near the origin $0$. It is straightforward to compute that

$\displaystyle\hat{\bf F}(\varpi,{{0}})={{0}},\qquad D\hat{\bf
F}(\varpi,{{0}})=P(\vartheta\varpi){\bf\Lambda}(\varpi)P^{-1}(\varpi)=\bar{\bf\Lambda}(\varpi),$

and

$\displaystyle\begin{split}\|D^{2}\hat{\bf F}(\varpi,x)\|\leq
8M_{\epsilon}(\omega),\quad\forall x\in\mathbb{R}^{d},\end{split}$ (5.8)

due to (3.4). To show (5.8), we also need to note that

$\displaystyle\|P^{\pm 1}(\varpi)\|\leq
1+\delta_{E}(\omega)=1+\delta(\omega)^{\beta_{E}}=1+1/\mathfrak{M}_{\epsilon}(\omega)^{\beta_{E}}\leq
2$ (5.9)

by (5.6) and (3.8). For the sake of simplicity, we still use
$M_{\epsilon}(\omega)$ to denote $8M_{\epsilon}(\omega)$.

Next, we state a result on the normal form of $\hat{\bf F}(\varpi,x)$, which
will be proved in section 6.

######  Proposition 5.1.

There is a $C^{2,\alpha}$ random diffeomorphism
$N(\varpi,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ transforming $\hat{\bf
F}(\varpi,\cdot)$ to $\bar{\bf F}(\varpi,\cdot)$, i.e.,

$\displaystyle\bar{\bf F}(\varpi,x):=N(\vartheta\varpi,\cdot)\circ\hat{\bf
F}(\varpi,\cdot)\circ N^{-1}(\varpi,x),$ (5.10)

such that

$\displaystyle\bar{\bf F}(\varpi,{{0}})={{0}},\quad D\bar{\bf
F}(\varpi,{{0}})=\bar{\bf\Lambda}(\varpi),\quad\partial^{2}_{x_{u}x_{s}}\bar{\bf
F}(\varpi,{{0}})={\bf 0}~{}~{}~{}\mbox{\rm(the zero mapping)},$ (5.11)

and

$\displaystyle\begin{split}&N^{\pm 1}(\varpi,0)=0,\qquad DN^{\pm
1}(\varpi,0)={\rm id},\quad\|DN^{\pm 1}(\varpi,x)-{\rm
id}\|\leq\delta(\omega)/K_{\epsilon}(\omega),\\\ &\|D^{2}N^{\pm
1}(\varpi,x)\|\leq\check{M}_{\epsilon}(\omega),\qquad\|D^{2}N^{\pm
1}(\varpi,x)-D^{2}N^{\pm
1}(\varpi,y)\|\leq\check{L}_{\epsilon,\delta}(\omega)\|x-y\|^{\alpha}\end{split}$
(5.12)

for all $x,y\in\mathbb{R}^{d}$, where $\check{M}_{\epsilon}(\omega)>0$ is an
$\epsilon$-slowly varying random variable depending on
$K_{\epsilon}(\omega),M_{\epsilon}(\omega)$ and
$\check{L}_{\epsilon,\delta}(\omega)>0$ is an $\epsilon$-slowly varying random
variable depending on $K_{\epsilon}(\omega)$, $M_{\epsilon}(\omega)$,
$\delta(\omega)$.

Moreover, $D^{2}N^{\pm 1}(\varpi,0)=D^{2}N^{\pm 1}(\omega,\bar{x},0)$ are
measurable with respect to $\omega$ and are continuous with respect to
$\bar{x}$ such that

$\displaystyle\|D^{2}N^{\pm 1}(\omega,\bar{x},0)-D^{2}N^{\pm
1}(\omega,\bar{y},0)\|\leq
C_{\lambda}K_{\epsilon}(\omega)^{3}\hat{L}_{\epsilon,\rho}(\omega)\|\bar{x}-\bar{y}\|^{\beta_{N}},$
(5.13)

when $\|\bar{x}-\bar{y}\|<1$, where
$\beta_{N}:={\min\\{\epsilon/\lambda_{\max},\alpha\\}}$ and
$\hat{L}_{\epsilon,\rho}(\omega)>0$ is an $\epsilon$-slowly varying random
variable depending on $K_{\epsilon}(\omega)$, $M_{\epsilon}(\omega)$ and
$L_{\epsilon,\rho}(\omega)$.

Remark that in the above definition of $\beta_{N}$, the choice of $\epsilon>0$
depends on the Lyapunov exponents only (see (3.7)). Moreover, a key part of
the above result is to obtain the third equality of (5.11), which will be used
to estimate the nonlinear part of a cohomological equation deduced from (5.2),
as will be seen in (7.17)-(7.20) below.

Using (5.7) and (5.10), we have that

$\displaystyle\bar{\bf F}(\varpi,x)={\Psi}(\vartheta\varpi,\cdot)\circ
F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x),\quad\forall x\in\mathbb{R}^{d},$
(5.14)

where ${\Psi}(\varpi,x):=N(\varpi,\cdot)\circ P(\varpi)(x-\bar{x}).$ It
follows from (3.4) and (5.12) that

$\displaystyle\begin{split}&\|D\bar{\bf
F}(\varpi,x)-\bar{\bf\Lambda}(\varpi)\|\leq
C_{\lambda}\delta(\omega),\qquad\|D^{2}\bar{\bf
F}(\varpi,x)\|\leq\bar{M}_{\epsilon}(\omega),\\\ &\|D^{2}\bar{\bf
F}(\varpi,x)-D^{2}\bar{\bf
F}(\varpi,y)\|\leq\bar{L}_{\epsilon,\delta}(\omega)\|x-y\|^{\alpha},\quad\forall
x,y\in\mathbb{R}^{d},\end{split}$ (5.15)

and $D^{2}\bar{\bf F}(\varpi,\cdot)$ is $\epsilon$-slowly continuous when
$\alpha=0$, where $\bar{M}_{\epsilon}(\omega)>0$ is an $\epsilon$-slowly
varying random variable depending on
$K_{\epsilon}(\omega),M_{\epsilon}(\omega)$, and
$\bar{L}_{\epsilon,\delta}(\omega)>0$ is an $\epsilon$-slowly varying random
variable depending on
$K_{\epsilon}(\omega),M_{\epsilon}(\omega),\delta(\omega)$. As we saw in
section 3 (below (3.4)), the third inequality of (5.15) holds for all Hölder
exponents $\tilde{\alpha}\in[0,\alpha]$.

Step 2. Convert equation (5.2) to a new equation (5.17) with respect to
$\bar{\bf F}(\varpi,x)$, whose solution is represented by $\bar{\bf
v}(\varpi,x)$, and establish the relation between $v(\omega,x)$ and $\bar{\bf
v}(\varpi,x)$.

For any $v(\omega,x)$ satisfying equation (5.2), we define $\bar{\bf
v}(\varpi,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ by

$\displaystyle\begin{split}\bar{\bf
v}(\varpi,x):=D{\Psi}(\varpi,{\Psi}^{-1}(\varpi,x))\,v(\omega,{\Psi}^{-1}(\varpi,x)).\end{split}$
(5.16)

Then, $\bar{\bf v}(\varpi,x)$ satisfies

$\displaystyle D\bar{\bf F}(\varpi,x)\bar{\bf v}(\varpi,x)=\bar{\bf
v}(\vartheta\varpi,\bar{\bf F}(\varpi,x)),\quad\forall x\in\mathbb{R}^{d}.$
(5.17)

In fact, using (5.2) and (5.14), we obtain that

$\displaystyle D\bar{\bf F}(\varpi,x)\bar{\bf v}(\varpi,x)$
$\displaystyle=D\bar{\bf
F}(\varpi,x)\,D{\Psi}(\varpi,{\Psi}^{-1}(\varpi,x))\,v(\omega,{\Psi}^{-1}(\varpi,x))$
$\displaystyle=D{\Psi}(\vartheta\varpi,F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x))\,DF(\omega,{\Psi}^{-1}(\varpi,x))\,D{\Psi}^{-1}(\varpi,x)$
$\displaystyle\quad\cdot
D{\Psi}(\varpi,{\Psi}^{-1}(\varpi,x))\,v(\omega,{\Psi}^{-1}(\varpi,x))$
$\displaystyle=D{\Psi}(\vartheta\varpi,F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x))\,DF(\omega,{\Psi}^{-1}(\varpi,x))\,v(\omega,{\Psi}^{-1}(\varpi,x))$
$\displaystyle=D{\Psi}(\vartheta\varpi,F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x))\,v(\theta\omega,F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x))$
$\displaystyle=D{\Psi}(\vartheta\varpi,{\Psi}^{-1}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}(\varpi,x))\,v(\theta\omega,{\Psi}^{-1}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}(\varpi,x))$ $\displaystyle=\bar{\bf v}(\vartheta\varpi,\bar{\bf
F}(\varpi,x)).$

The converse is also true. Thus, the problem of finding the solution
$v(\omega,x)$ of equation (5.2) is equivalent to finding the solution
$\bar{\bf v}(\varpi,x)$ of equation (5.17).

Let ${\mathcal{W}}_{i}(\varpi):={\mathcal{W}}_{i}(\omega,\bar{x})$ be the leaf
of an intermediate foliation of $F(\omega,x)$ (see (4.36)) passing through
$\bar{x}$ for every $i=1,...,\tau$. Then,

$\displaystyle\bar{\mathcal{N}}_{i}(\varpi):=\Psi(\varpi,{\mathcal{W}}_{i}(\varpi)),\quad\forall
n\in\mathbb{Z},$ (5.18)

is an invariant manifold (i.e., the leaf passing through $0$) of $\bar{\bf
F}(\varpi,x)$ corresponding to the subspace $X_{i}$. In fact, using (5.14), we
have that

$\displaystyle\bar{\bf F}(\varpi,\bar{\mathcal{N}}_{i}(\varpi))$
$\displaystyle=\bar{\bf
F}(\varpi,\cdot)\circ\Psi(\varpi,{\mathcal{W}}_{i}(\varpi))=\Psi(\vartheta\varpi,\cdot)\circ
F(\omega,{\mathcal{W}}_{i}(\varpi))$
$\displaystyle\subset\Psi(\vartheta\varpi,{\mathcal{W}}_{i}(\vartheta\varpi))=\bar{\mathcal{N}}_{i}(\vartheta\varpi).$

Step 3. Linearize $\bar{\bf F}(\varpi,x)$ along
$\bar{\mathcal{N}}_{i}(\varpi)$ for $i=1,...,\tau$.

By Corollary 4.1 (using (5.15) instead of (3.4)), we see that
$\bar{\mathcal{N}}_{i}(\varpi)$ is the graph of a mapping, i.e,

$\displaystyle\bar{\mathcal{N}}_{i}(\varpi)=\\{x_{i}+{\gamma}_{i}(\varpi,x_{i}):x_{i}\in
X_{i}\\},\quad i=1,...,\tau,$ (5.19)

where ${\gamma}_{i}(\varpi,\cdot):X_{i}\to X_{1}\oplus\cdots\oplus
X_{i-1}\oplus X_{i+1}\oplus\cdots\oplus X_{p}$ is a $C^{1,{\beta}}$ mapping
such that

$\displaystyle\begin{split}&{\gamma}_{i}(\varpi,{{0}})={{0}},\quad
D{\gamma}_{i}(\varpi,{{0}})={\bf 0},\quad\|D{\gamma}_{i}(\varpi,x_{i})\|\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega),\\\
&\|D{\gamma}_{i}(\varpi,x_{i})-D{\gamma}_{i}(\varpi,y_{i})\|\leq
L_{\epsilon,{\gamma}}(\omega)\|x_{i}-y_{i}\|^{\beta},\quad\forall
x_{i},y_{i}\in X_{i},\end{split}$ (5.20)

with an $\epsilon$-slowly varying tempered random variable
$L_{\epsilon,\gamma}(\omega)>0$. Using the transformation

$\varphi_{i}(\varpi,x):=x-{\gamma}_{i}(\varpi,x_{i}),$

we straighten up the invariant manifold $\bar{\mathcal{N}}_{i}({\varpi})$.
Then, we have a $C^{1,{\beta}}$ mapping $\bar{\bf
F}_{i}(\varpi,\cdot):X_{i}\to X_{i}$ by

$\displaystyle\bar{\bf
F}_{i}(\varpi,x_{i}):=\varphi_{i}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}(\varpi,\cdot)\circ\varphi^{-1}_{i}(\varpi,x_{i}),$ (5.21)

which satisfies that

$\displaystyle\begin{split}&\bar{\bf F}_{i}(\varpi,{{0}})={{0}},\quad
D\bar{\bf F}_{i}(\varpi,{{0}})=\bar{\bf\Lambda}_{i}(\varpi),\quad\|D\bar{\bf
F}_{i}(\varpi,x_{i})-\bar{\bf\Lambda}_{i}(\varpi)\|\leq\delta^{*}(\omega),\\\
&\|D\bar{\bf F}_{i}^{\pm 1}(\varpi,x_{i})-D\bar{\bf F}_{i}^{\pm
1}(\varpi,y_{i})\|\leq
L_{\epsilon}^{*}(\omega)\|x_{i}-y_{i}\|^{\beta},\qquad\forall x_{i},y_{i}\in
X_{i},\end{split}$ (5.22)

by (5.15) and (5.20). Here, $\bar{\bf\Lambda}_{i}(\varpi)$ is given in (5.4),
$\delta^{*}(\omega):=C_{\lambda}K_{\epsilon}(\omega)^{2}C_{\epsilon}(\omega)\delta(\omega)>0$
and $L_{\epsilon}^{*}(\omega)>0$ depends on $L_{\epsilon,{\gamma}}(\omega)$
and $\bar{M}_{\epsilon}(\omega)$.

In order to solve equation (5.17), we will use the Lyapunov norms to overcome
the nonuniformity (see Lemma 7.1 below together with the remark given at the
end of section 7). Since the Lyapunov norms is applied only to the linear
system $\bar{\bf\Lambda}_{i}(\varpi)$, we need to linearize $\bar{\bf
F}_{i}(\varpi,x_{i})$.

The following lemma states that $\bar{\bf F}_{i}(\varpi,x_{i})$ is
$C^{1,\beta}$ with $\beta\in(0,1]$ given in (4.12) conjugate to its linear
part, which will be proved in section 7.

######  Lemma 5.1.

There is a $C^{1}$ diffeomorphism $\varphi_{i,*}(\varpi,\cdot):X_{i}\to X_{i}$
such that

$\displaystyle\bar{\bf\Lambda}_{i}(\varpi)\varphi_{i,*}(\varpi,x_{i})=\varphi_{i,*}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}_{i}(\varpi,x_{i}),\quad\forall i=1,...,\tau,$ (5.23)

and

$\displaystyle\begin{split}D\varphi_{i,*}^{\pm 1}(\varpi,{{0}})={\rm
id}_{i},\quad\|D\varphi_{i,*}^{\pm 1}(\varpi,x_{i})-{\rm id}_{i}\|\leq
C^{*}_{4\epsilon}(\omega)\|x_{i}\|^{\beta},\quad\forall x_{i}\in
X_{i},\end{split}$ (5.24)

where $C^{*}_{4\epsilon}(\omega)>0$ is a $4\epsilon$-slowly varying tempered
random variable and ${\rm id}_{i}$ the identity mapping in $X_{i}$.

Step 4. Decompose equation (5.17), obtained in step 2, along every
$\bar{\mathcal{N}}_{i}(\varpi)$ for $i=1,...,\tau$ by using the conjugacy
obtained in step 3, and solve the decomposed equation (5.26).

Recall that $\varphi_{i}(\varpi,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$
straightens up $\bar{\mathcal{N}}_{i}({\varpi})$ which is tangent to $X_{i}$
at ${{0}}$. Let

$\displaystyle\bar{\varphi}_{i}(\varpi,\cdot):=\varphi_{i,*}(\varpi,\cdot)\circ\varphi_{i}(\varpi,\cdot)|_{\bar{\mathcal{N}}_{i}({\varpi})}:\bar{\mathcal{N}}_{i}({\varpi})\to
X_{i}.$ (5.25)

Then, we have

$\displaystyle\bar{\varphi}_{i}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}(\varpi,\cdot)\circ\bar{\varphi}^{-1}_{i}(\varpi,x_{i})=\bar{\bf\Lambda}_{i}(\varpi)$

by (5.21) and (5.23). Replacing $x$ in (5.17) with
$\bar{\varphi}^{-1}_{i}(\varpi,x_{i})$, we obtain an equation

$\displaystyle D\bar{\bf
F}(\varpi,\bar{\varphi}_{i}^{-1}(\varpi,x_{i}))\,\bar{\bf
v}_{i}(\varpi,x_{i})=\bar{\bf
v}_{i}(\vartheta\varpi,\bar{\bf\Lambda}_{i}(\varpi)x_{i})$ (5.26)

for $i=1,...,\tau$, where $\bar{\bf
v}_{i}(\varpi,\cdot):X_{i}\to\mathbb{R}^{d}$ is defined by

$\displaystyle\bar{\bf v}_{i}(\varpi,x_{i}):=\bar{\bf
v}(\varpi,\bar{\varphi}_{i}^{-1}(\varpi,x_{i})).$ (5.27)

For $\kappa={\tau}+1,...,{p}$ and $\iota=1,...,d_{\kappa}$, where $d_{\kappa}$
is the dimension of $X_{\kappa}$, let
$\vec{e}_{\kappa,\iota}:=(0,...,1,...,0)^{T}$ be the standard unit vectors in
$X_{\kappa}$, i.e., the $(d_{1}+\cdots+d_{\kappa-1}+\iota)$-th component is 1
and others are 0.

The following proposition is on the existence of solutions of equation (5.26),
which will be proved in sections 7 and 8 for the cases of $\alpha\in(0,1]$ and
$\alpha=0$ respectively.

######  Proposition 5.2.

Given a $\kappa={\tau}+1,...,{p}$ and $d_{\kappa}$ standard unit vectors
$\vec{e}_{\kappa,\iota}\in X_{\kappa}$ ($\iota=1,...,d_{\kappa}$), for every
$\varpi\in\bar{\bf\Omega}$, there exist sets $\\{{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,x_{i})\\}_{n\in\mathbb{Z}}$
($i=1,...,\tau$) of the solutions of equation (5.26) such that ${\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,{{0}})=\bar{\bf\Lambda}_{\kappa}(n,\varpi)\,\vec{e}_{\kappa,\iota}\in
X_{\kappa},$

$\displaystyle\frac{\|{\bar{\bf v}_{i}^{\kappa,\iota}}(\varpi,x_{i})-{\bar{\bf
v}_{i}^{\kappa,\iota}}(\varpi,{{0}})\|}{\|x_{i}\|}\to 0\qquad{\rm
as}~{}\|x_{i}\|\to 0,$ (5.28)

and

$\displaystyle\sup_{n\in\mathbb{Z}}\sup_{x_{i}\in
X_{i}\backslash\\{{{0}}\\}}\bigg{\\{}e^{-\lambda_{\kappa}n-12\epsilon|n|}\frac{\|{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,x_{i})-{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,{{0}})\|}{\|x_{i}\|^{\varsigma}}\bigg{\\}}\leq
M_{v}(\omega),$ (5.29)

where $M_{v}(\omega)>0$ is a tempered random variable and $\varsigma\in(0,1)$
is a constant satisfying

$\displaystyle\begin{split}0<\varsigma<\frac{1}{10}\min\Bigg{\\{}\frac{\lambda_{\tau}}{\lambda_{\max}},\frac{-\lambda_{\tau+1}}{\lambda_{\max}},\min_{\mbox{\tiny$\begin{array}[]{c}{\kappa\in\\{\tau+1,...,p\\},j\in\\{1,...,p\\}}\\\
{\rm such~{}that}~{}\lambda_{\kappa}-\lambda_{j}\neq
0\end{array}$}}\frac{|\lambda_{\kappa}-\lambda_{j}|}{\lambda_{1}}\Bigg{\\}}\end{split}$
(5.30)

with $\lambda_{\rm max}:=\max\\{2\lambda_{1},-2\lambda_{p}\\}>0$.

Step 5. Use $\\{\bar{\bf
v}_{i}^{\kappa,\iota}(\theta^{n}\varpi,x_{i})\\}_{n\in\mathbb{Z}}$ to get a
set $\\{v_{\bar{x},i}^{\kappa,\iota}(\theta^{n}\omega,x)\\}_{n\in\mathbb{Z}}$
of solutions of equation (5.2) for every $\omega\in\Omega$ on the leaves
$\\{{\mathcal{W}}_{i}(\theta^{n}\omega,F(n,\omega,\bar{x}))\\}_{n\in\mathbb{Z}}$
through (5.16). Then, show that $v_{\bar{x},i}^{\kappa,\iota}(\omega,x)$ is
differentiable at $\bar{x}$ along ${\mathcal{W}}_{i}(\omega,\bar{x})$ such
that the differential is (Hölder) continuous with respect to $\bar{x}$.

For each point $x\in{\mathcal{W}}_{i}(\varpi)$ for $i=1,...,\tau$, by the
invariance of foliation, we have that
$F(n,\omega,x)\in{\mathcal{W}}_{i}(\vartheta^{n}\varpi)$. Then, using (5.18)
and the fact that $\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)$ straightens
up $\bar{\mathcal{N}}_{i}(\vartheta^{n}\varpi)$, we get

$\displaystyle\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)\circ\Psi(\vartheta^{n}\varpi,F(n,\omega,x))\in
X_{i},\qquad\forall n\in\mathbb{Z},$ (5.31)

It implies that

${\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)\circ{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))$

is well-defined. Then, by (5.16) (implying
$v(\omega,x)=\\{D{\Psi}(\varpi,x)\\}^{-1}\bar{\bf
v}(\varpi,{\Psi}(\varpi,x))$) and by (5.27), we obtain that

$\displaystyle{v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n}\omega,F(n,\omega,x))$
$\displaystyle:=\\{D{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))\\}^{-1}\,{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)\circ{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))),\quad\forall
n\in\mathbb{Z},$ (5.32)

satisfies (5.2) at $F(n,\omega,x)$, i.e.,

$\displaystyle
DF(\theta^{n}\omega,F(n,\omega,x))\,{v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n}\omega,F(n,\omega,x))={v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n+1}\omega,F(n+1,\omega,x)).$
(5.33)

Since ${\Psi}^{-1}(\varpi,x)=P^{-1}(\varpi)\circ N^{-1}(\varpi,x)+\bar{x}$ due
to the definition of ${\Psi}$ given below (5.14), we have

${\Psi}^{-1}(\vartheta^{n}\varpi,x)=P^{-1}(\vartheta^{n}\varpi)\circ
N^{-1}(\vartheta^{n}\varpi,x)+F(n,\omega,\bar{x})$

by replacing $\varpi=(\omega,\bar{x})$ and $\bar{x}$ with
$\vartheta^{n}\varpi=(\theta^{n}\omega,F(n,\omega,\bar{x}))$ and
$F(n,\omega,\bar{x})$ respectively. Therefore,

$\displaystyle\\{D{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))\\}^{-1}$
$\displaystyle=D{\Psi}^{-1}(\vartheta^{n}\varpi,{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x)))$
$\displaystyle=P^{-1}(\vartheta^{n}\varpi)\,DN^{-1}(\vartheta^{n}\varpi,{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))),$

which together with (5.32) yields

$\displaystyle{v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n}\omega,F(n,\omega,x))$
$\displaystyle=P^{-1}(\vartheta^{n}\varpi)\,DN^{-1}(\vartheta^{n}\varpi,{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x)))\,{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)\circ{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))).$
(5.34)

Next, we show the differentiability of
${v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)$ at $x=\bar{x}$ along
${\mathcal{W}}_{i}(\omega,\bar{x})={\mathcal{W}}_{i}(\varpi)$. In view of
(5.31) with $n=0$, we let

$\displaystyle x_{i}:=\bar{\varphi}_{i}(\varpi,\cdot)\circ\Psi(\varpi,x)\in
X_{i}.$

Thus, $x\mapsto x_{i}(x)$ is a $C^{1}$ diffeomorphism from
${\mathcal{W}}_{i}(\varpi)$ to $X_{i}$ with the inverse $x_{i}\mapsto
x(x_{i}):=\Psi^{-1}(\varpi,\cdot)\circ\bar{\varphi}_{i}^{-1}(\varpi,x_{i})$
such that $x({{0}})=\bar{x}$. Then, in view of (5.34) with $n=0$ and with $x$
being replaced by
$x(x_{i}):=\Psi^{-1}(\varpi,\cdot)\circ\bar{\varphi}_{i}^{-1}(\varpi,x_{i})$,
we have that

$\displaystyle{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x(x_{i}))=P^{-1}(\varpi)\,DN^{-1}(\varpi,\bar{\varphi}_{i}^{-1}(\varpi,x_{i}))\,{\bar{\bf
v}_{i}^{\kappa,\iota}}(\varpi,x_{i}),\quad\forall x_{i}\in X_{i}.$

Therefore,

$\displaystyle\frac{dv_{\bar{x},i}^{\kappa,\iota}(\omega,x(x_{i}))}{dx_{i}}\Big{|}_{x_{i}={{0}}}$
$\displaystyle=P^{-1}(\varpi)\\{D^{2}N^{-1}(\varpi,{{0}}){\rm
id}_{i}\\}\,\vec{e}_{\kappa,\iota}$
$\displaystyle=P^{-1}(\omega,\bar{x})\\{D^{2}N^{-1}(\omega,\bar{x},{{0}}){\rm
id}_{i}\\}\,\vec{e}_{\kappa,\iota}\in{\mathcal{L}}(X_{i},\mathbb{R}^{d})$
(5.35)

since $D\bar{\varphi}_{i}^{-1}(\varpi,{{0}})={\rm id}_{i}$ (see (5.20) and
(5.24)), ${\bar{\bf
v}_{i}^{\kappa,\iota}}(\varpi,{{0}})=\vec{e}_{\kappa,\iota}$ (see Proposition
5.2) and $D{\bar{\bf v}_{i}^{\kappa,\iota}}(\varpi,{{0}})=0$ (see (5.28)).
Then (5.35) determines the differential (5.1) of
$v_{\bar{x},i}^{\kappa,\iota}(\omega,\cdot):{\mathcal{W}}_{i}(\omega,\bar{x})\to\mathbb{R}^{d}$
along ${\mathcal{W}}_{i}(\omega,\bar{x})$ at $\bar{x}$ ([61, p.16]).

More precisely, since

$\frac{dx({{0}})}{dx_{i}}=\frac{\Psi^{-1}(\varpi,\cdot)\circ\bar{\varphi}_{i}^{-1}(\varpi,x_{i})}{dx_{i}}\Big{|}_{x_{i}=0}=P^{-1}(\varpi)\partial_{x_{i}}N^{-1}(\varpi,0)=P^{-1}(\omega,\bar{x}){\rm
id}_{i}\in{\mathcal{L}}(X_{i},\mathbb{R}^{d})$

by (5.12), we have that
$\\{P^{-1}(\omega,\bar{x})\vec{e}_{i,\hat{\iota}}\\}_{\hat{\iota}=1,...,d_{i}}$
is a basis of $T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})$, where
$\\{\vec{e}_{i,\hat{\iota}}\\}_{\hat{\iota}=1,...,d_{i}}$ is the standard
basis of $X_{i}$. Notice that every
$P^{-1}(\omega,\bar{x})\vec{e}_{i,\hat{\iota}}$ corresponds (one-to-one) to a
linear mapping

$\displaystyle
h_{\mathcal{W}_{i}}\mapsto\frac{\partial(h_{\mathcal{W}_{i}}\circ
x(x_{i}))}{\partial
x_{i,\hat{\iota}}}\Big{|}_{x_{i}=0}=Dh_{\mathcal{W}_{i}}(\bar{x})P^{-1}(\omega,\bar{x}){\rm
id}_{i,\hat{\iota}}\in{\mathcal{L}}(X_{i,\hat{\iota}},\mathbb{R})$ (5.36)

for all smooth functions
$h_{\mathcal{W}_{i}}:{\mathcal{W}}_{i}(\omega,\bar{x})\to\mathbb{R}$, where
${\rm id}_{i,\hat{\iota}}$ denotes the identity in the 1-dimensional space
$X_{i,\hat{\iota}}$. This observation enables us to use
$P^{-1}(\omega,\bar{x})\vec{e}_{i,\hat{\iota}}$ here to denote the tangent
vector for convenience, although (5.36) is exactly the definition of tangent
vector given in [61, Definition 1.19]. Then, we have that for any given ${\bf
a}_{\omega,\bar{x}}\in T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})~{}(\cong
X_{i})$, there are $c_{1},\cdots,c_{d_{i}}\in\mathbb{R}$ such that

$\displaystyle{\bf
a}_{\omega,\bar{x}}=\sum_{\hat{\iota}=1,...,d_{i}}c_{\hat{\iota}}P^{-1}(\omega,\bar{x})\vec{e}_{i,\hat{\iota}}.$
(5.37)

This enables us to compute the differential of
$v_{\bar{x},i}^{\kappa,\iota}(\omega,\cdot)$ along
${\mathcal{W}}_{i}(\omega,\bar{x})$ at $\bar{x}$ by (5.35), that is,

$\displaystyle dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})\,{\bf
a}_{\omega,\bar{x}}$
$\displaystyle=\sum_{\hat{\iota}=1,...,d_{i}}c_{\hat{\iota}}P^{-1}(\omega,\bar{x})\\{D^{2}N^{-1}(\omega,\bar{x},{{0}})\vec{e}_{i,\hat{\iota}}\\}\,\vec{e}_{\kappa,\iota}$
$\displaystyle=\sum_{\hat{\iota}=1,...,d_{i}}c_{\hat{\iota}}P^{-1}(\omega,\bar{x})D^{2}N^{-1}(\omega,\bar{x},{{0}})(\vec{e}_{i,\hat{\iota}},\vec{e}_{\kappa,\iota})$
(5.38)

(see [61, p.17, 1.23(a)]). Note that
$dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x}):T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})\to\mathbb{R}^{d}~{}(=T_{v_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})}\mathbb{R}^{d})$.

Regarding the Hölder continuity of
$dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})$ with respect to $\bar{x}$, we
define the bundle

${\mathcal{E}}:=\bigcup_{\bar{x}\in\mathbb{R}^{d}}T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x}),$

which is a subbundle of $T\mathbb{R}^{d}(=\mathbb{R}^{2d})$ (therefore, in
${\mathcal{E}}$, one can use the same norm as the one in $\mathbb{R}^{2d}$).
In order to trivialize ${\mathcal{E}}$, let ${\bf
a}:=\sum_{\hat{\iota}=1,...,d_{i}}c_{\hat{\iota}}\vec{e}_{i,\hat{\iota}}\in
X_{i}$ and define a mapping $\hbar:\Omega\times\mathbb{R}^{d}\times
X_{i}\to{\mathcal{E}}$ as

$\displaystyle\hbar(\omega,\bar{x},{\bf a})={\bf a}_{\omega,\bar{x}}.$

Since $\mathbb{R}^{d}\times X_{i}$ is a subspace of $\mathbb{R}^{2d}$, we
regard $\hbar(\omega,\cdot)$ as a mapping from a subset of $\mathbb{R}^{2d}$
into $\mathbb{R}^{2d}$, which is clearly one-to-one. Thus, in view of (5.37),
we see that $\hbar(\omega,\cdot)$ and $\hbar^{-1}(\omega,\cdot)$ are
measurable with respect to $\omega$, $\beta_{E}$-Hölder continuous with
respect to $\bar{x}$ with a uniformly bounded Hölder coefficient with respect
to $\bar{x}$ and $i$ by (5.6). Moreover, we have that
$\hbar(\omega,\bar{x},{\bf a})$ and $\hbar^{-1}(\omega,\bar{x},{\bf a})$ are
linear with respect to ${\bf a}$ and that $\hbar^{-1}(\omega,\cdot)$
trivializes $T_{\mathbb{R}^{d}}{\mathcal{W}}_{i}(\omega,\bar{x})$.

In what follows, we rewrite $\hbar(\omega,\bar{x},{\bf a})$ as
$\hbar(\omega,\bar{x}){\bf a}$ for convenience. Then, for any ${\bf a}\in
X_{i}$ such that $\|{\bf a}\|=1$, using (5.38), we have that

$\displaystyle\|dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})\circ\hbar(\omega,\bar{x}){\bf
a}-dv_{\bar{y},i}^{\kappa,\iota}(\omega,\bar{y})\circ\hbar(\omega,\bar{y}){\bf
a}\|=\|dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x}){\bf
a}_{\omega,\bar{x}}-dv_{\bar{y},i}^{\kappa,\iota}(\omega,\bar{y}){\bf
a}_{\omega,\bar{y}}\|$
$\displaystyle\leq\sum_{\hat{\iota}=1,...,d_{i}}\big{\|}c_{\hat{\iota}}P^{-1}(\omega,\bar{x})D^{2}N^{-1}(\omega,\bar{x},{{0}})(\vec{e}_{i,\hat{\iota}},\vec{e}_{\kappa,\iota})-c_{\hat{\iota}}P^{-1}(\omega,\bar{y})D^{2}N^{-1}(\omega,\bar{y},{{0}})(\vec{e}_{i,\hat{\iota}},\vec{e}_{\kappa,\iota})\big{\|}$
$\displaystyle\leq
4\check{M}_{\epsilon}(\omega)\|\bar{x}-\bar{y}\|^{\beta_{E}}+4C_{\lambda}K_{\epsilon}(\omega)^{3}\hat{L}_{\epsilon,\rho}(\omega)\|\bar{x}-\bar{y}\|^{\beta_{N}}$
$\displaystyle\leq
4\\{\check{M}_{\epsilon}(\omega)+C_{\lambda}K_{\epsilon}(\omega)^{3}\hat{L}_{\epsilon,\rho}(\omega)\\}\|\bar{x}-\bar{y}\|^{\beta_{v}}$

when $\|\bar{x}-\bar{y}\|<1$. Here, (5.6) and (5.12)-(5.13) are also used to
obtain the above estimates, and

$\beta_{v}:=\min\\{\beta_{E},\beta_{N}\\}=\min\\{\beta_{E},\epsilon/\lambda_{\max},\alpha\\}\geq
0.$

Hence,

$\displaystyle\|dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})\circ\hbar(\omega,\bar{x})-dv_{\bar{y},i}^{\kappa,\iota}(\omega,\bar{y})\circ\hbar(\omega,\bar{y})\|\leq
4\\{\check{M}_{\epsilon}(\omega)+C_{\lambda}K_{\epsilon}(\omega)^{3}\hat{L}_{\epsilon,\rho}(\omega)\\}\|\bar{x}-\bar{y}\|^{\beta_{v}},$

which means that the differential
$dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})$ is $\beta_{v}$-Hölder
continuous with respect to $\bar{x}$ with a uniformly bounded Hölder
coefficient with respect to $\bar{x}$ and $i$.

Step 6. Show that when $x\in{\mathcal{W}}_{i}(\omega,\bar{x})$ is close to
$\bar{x}$, $v_{\bar{x},i}^{\kappa,\iota}(\omega,x)$ belong to
$E_{s}(\omega,x)$ for all $\kappa={\tau}+1,...,{p}$ and
$\iota=1,...,d_{\kappa}$, and form a basis of $E_{s}(\omega,x)$.

Next, we show that for every $x\in{\mathcal{W}}_{i}(\omega,\bar{x})$ near
$\bar{x}$ the set
$\\{{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)\\}_{\kappa={\tau}+1,...,{p},~{}\iota=1,...,d_{\kappa}}$
is a basis of $E_{s}(\omega,x)$. Notice that (5.34) gives

$\displaystyle{v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n}\omega,F(n,\omega,\bar{x}))$