text:
\section{Introduction and setting}\label{SEC:1}
       \subsection{Introduction}
	   In smooth ergodic theory of dynamical systems, SRB (Sinan-Ruelle-Bowen) measures are the invariant measures most compatible with volume when volume is not preserved \cite{Young02}.
	   Roughly speaking, they are invariant measures with smooth density in the expanding, or unstable, directions of the dynamical systems.
	   A principal result concerning SRB measures is the relationship between entropy, volume growth, and Lyapunov exponents.
	   The results say that, for a $C^2$ diffeomorphism on a finite-dimensional compact Riemannian manifold, SRB measures are characterized by invariant measures satisfying the Pesin entropy formula, in which entropy is equal to volume growth on unstable manifolds, and equivalently the sum of positive Lyapunov exponents.
	   These results are proved by Ledrappier, Strelcyn and Ledrappier, Young in \cite{Ledrappier82} and \cite{Young85} respectively.

	   The random dynamical system considers the composition of sequences of random maps.
	   This is a more general set-up than the iteration of a certain map and covers the most important examples of dynamical systems, such as products of random maps, stochastic and random ordinary and partial differential equations.
	   We recommend the reader to \cite{Arnold98} and \cite{Liu95} for more detailed about random dynamical systems.
	   Bahnm\"{u}ller, Liu \cite{Liu98} and Liu \cite{Liu95} extend the results that SRB measures are characterized by invariant measures satisfying the Pesin entropy formula to the random case on finite-dimensional compact Riemannian manifold.

	   The smooth ergodic theory extended to infinite dimensional dynamical systems.
	   The multiplicative ergodic theorem \cite{Oseledec68}, which plays a fundamental role in the smooth ergodic theory and nonuniform hyperbolic theory (see \cite{Pesin07} for more details of this theory) in finite-dimensional dynamical systems, be extended to the infinite setting.
	   Ruelle \cite{Ruelle82} first proved the multiplicative ergodic theorem for compact linear operators on a separable Hilbert space.
	   Man\'{e} \cite{M83} overcame the lack of inner product structure and extended the multiplicative ergodic theorem to compact operators on Banach spaces.
	   Later, Thieullen \cite{Thieullen87} obtained the multiplicative ergodic theorem for continuous bounded linear operator cocycles on Banach spaces.
	   Lian and Lu \cite{Lian10} established the multiplicative ergodic theorem for strong measurable operator cocycles on separable Banach spaces.
	   Froyland \cite{Froyland13} extended it to a continuous semi-invertible cocycle on Banach spaces, and Tokman \textit{et al} \cite{Quas12} set up the multiplicative ergodic theorem for a strong measurable case on Banach spaces.
	   The nonuniform hyperbolic theory in infinite dimensional dynamical systems also be studied,
	   Lian, Lu \cite{Lian10} and Varzaneh, Riedel \cite{Varzaneh21} construct the unstable and stable manifolds on random dynamical systems on a certain Banach space and a fields of Banach spaces.
	   The more results about infinite dimensional dynamical systems can refer to \cite{Blumenthal16, Young16, Doan09, Lian11, Lian20, Lucas17}.

	   The SRB measures for infinite dimensional dynamical systems to be studied.
	   Li and Shu \cite{Li12} proved the Ruelle's inequality in Banach space and \cite{Li13} proved the equivalence of SRB measures and invariant measures satisfying the Pesin's entropy formula in Hilbert space for random dynamical systems.
	   The main difficulty in the extent of this result to Banach space is that there is no significant definition of volume and determination on Banach setting.
	   But recently,  Blumenthal and Young \cite{Young17} overcome this difficulty, and proved that the equivalence of SRB measure and invariant measures satisfy the Pesin's entropy formula for a certain mapping on Banach spaces.
	   The more results about SRB measures for infinite dimensional dynamical systems can refer to \cite{Lian17, Lian16}.

	   The main topic in this paper concerns the relations between SRB measures and the invariant measures satisfying the Pesin's entropy formula in random dynamical systems on Banach space.
	   The main difficulty in this topic is the discussion of measurability and, the random dynamical systems are more non-uniformly than certain mapping cases.
	   For example, the attractor is random and the continuity of certain Oseledets is not uniformly under the different mappings.

	   This paper is organized as follows:
	   In Section \ref{SEC:1}, we state the main setting and the main results, also introduce some notations roughly.
	   In Section \ref{SEC:3}, we build the technical tools called Lyapunov norms and prove the continuity of certain Oseledets.
	   In Section \ref{SEC:4}, we discussed some properties of nonuniformly hyperbolic theory.
	   In Section \ref{SEC:5}, we give the proof of the main theorem.

	   \subsection{Setting and results}\label{setting}
	   Let $\Omega$ be a Polish space.
	   We consider the probability space $(\Omega,\cF,\bP)$, where $\cF$ is the completion of the Borel $\sigma$-algebra of $\Om$ with respect to $\bP$.
	   Let $\theta$ be an invertible ergodic transformation with respect to the probability measure $\bP$.

	   Let $(\cB,|\cdot|)$ be a separable Banach space with a Borel $\sigma$-algebra $\mathscr{B}$.
	   We denote by $B(\cB)$ the space of all bounded linear operators on $\cB$ and also use $|\cdot|$ to refer to the operator norm on $B(\cB)$.
	   The \textit{strong operator topology} $SOT(\cB)$ on $B(\cB)$ is the topology generated by the sub-base
	   $\{V_{x,y,\epsilon}=\{T:|T(x)-y|<\epsilon\}:x,y\in\cB, \epsilon>0\}$.
	   A sequence of bounded linear operators $T_n:n>0$ converges to $T\in B(\cB)$ on $SOT(\cB)$ if and only if $T_n(x)\rightarrow T(x)$ as $n\rightarrow \infty$ for every $x\in \cB$.
	   The \textit{strong $\sigma$-algebra} on $B(\cB)$ is the Borel $\sigma$-algebra of the strong operator topology $SOT(\cB)$.
	   For a function $L:\Omega\rightarrow B(\cB)$, we call $L$ is \textit{strongly measurable}, if $\omega \mapsto L(\om)v$ is measurable for every $v\in \cB$.
	   We refer that $L$ is strongly measurable if and only if it is measurable with respect to the strong $\sigma$-algebra on $B(\cB)$ (see \cite[Lemma A.4]{Quas12}).

	   For each $x\in \cB$, we write $\cB_x$ for the tangent space to the point $x$, which we regard as $\cB+\{x\}$.

	   A \textit{random dynamical system} can be represented as a measure-preserving skew product map.
	   Let $(\Omega \times \cB, \cF\otimes \mathscr{B})$ be the product measurable space.
	   A \textit{random transformation} over $\te$ is a measurable transformation of the form
	   \begin{align*}
	   	\Phi:\Omega \times \cB &\rightarrow \Omega \times \cB \\
	   	(\omega,x)&\mapsto (\theta(\omega),f_{\omega}(x)).
	   \end{align*}
	   where $f_{\omega}$ is a map from $\cB$ to itself.
	   Note that $\Phi$ is measurable implies that $f_{\omega}:\cB \rightarrow \cB$ is measurable for each $\om\in \Om$ and the mapping $\omega \mapsto f_{\omega}(x)$ is measurable for each $x\in\cB$.

	   Let $\mu$ be a $\Phi$-invariant probability measure projection to $\bP$.
	   We consider $(\Phi,\mu)$.
	   For every $n\in \N$ and $\om \in \Om$, define the nonlinear cocycle $\{f^{n}_{\omega}\}$ by
	   \begin{equation*}
	   	f^{n}_{\omega}=
	   	\begin{cases}
	   		f_{\theta^{n-1}(\omega)}\circ \cdots \circ f_{\omega} & n>0;\\
	   		Id &n=0.
	   	\end{cases}
	   \end{equation*}
	   We call the random dynamical system corresponding $(\Phi,\mu)$ is the compositions of sequences of maps $\{f^{n}_{\omega}:\omega\in \Omega,n\in\N\}$.

	   We call a set $\Lambda \subset \Omega \times \cB$ is \textit{random compact} if the $\omega$-section $\Lambda_{\omega}:=\{x\in \cB:(\omega,x)\in \Lambda\}$ is compact for every $\om\in \Om$ and the mapping $\omega\mapsto d(x,\Lambda_{\omega})$ is measurable for every $x\in\cB$, where $d(x,\Lambda_{\omega}):=\inf_{y\in \Lambda_{\omega} }|x-y|$ is the distance between a point and a compact set.
	   Note that all the random compact subsets of $\Om\times \cB$ are measurable (see Proposition \ref{Prop:random compact}).

	   \textbf{Basic assumptions}
	   To guarantee the multiplicative ergodic theorem and basic nonuniform hyperbolic theory, the following properties are assumed:
	   \begin{enumerate}
	   	\item[(H1)]
	   	\begin{enumerate}
	   		\item[(i)]   the map $f_{\omega}$ is $C^2$ Fr\'{e}chet differentiable and injective;
	   		\item[(ii)]  the derivative of $f_{\omega}$ at $x\in \cB$, denote by $Df_{\omega}(x)$, is also injective;
	   		\item[(iii)] the mapping $\omega \mapsto f_{\omega}(x)$ is measurable for every $x\in \cB$.
	   	\end{enumerate}
	   	\item[(H2)]
	   	\begin{enumerate}
	   		\item[(i)]   the subset $A\subset \Omega\times\cB$ is random compact and satisfied $\Phi(A)=A$;
	   		\item[(ii)]  $\mu$ is supported on $A$;
	   		\item[(iii)] there exists some $r_0>0$ such that
	   		$$\int \log^{+}|(f_{\omega}|_{B(A_{\omega},r_0)})|_{C^2}d\bP(\om)<\infty,$$
	   		where $B(A_{\omega},r_0)=\{y\in \cB: d(y,A_{\omega})< r_0\}$.
	   	\end{enumerate}
	   	\item[(H3)]
	   	we assume for $\mu$-almost every $(\omega,x)$,
	   	$$l_{\alpha}(\omega,x):=\lim_{n\rightarrow \infty}\frac{1}{n} \log |Df^n_{\omega}(x)|_{\alpha}<0.$$
	   	Here $|Df_{\omega}(x)|_{\alpha}$ is the Kuratowski measure of the set $Df_{\omega}(x)(B)$, where $B$ is the closed unit ball in $\cB$.
	   \end{enumerate}

	   	\textbf{Remarks on basic assumptions}
	   	Condition (H2) implies that $f_{\omega}(A_{\omega})=A_{\theta(\omega)}$ for every $\om\in \Om$.
	   	By (H1), one can defined $(f_{\omega}|_{A_{\om}})^{-1}$ on $A_{\theta(\omega)}$ which is also continuous.
	   	Therefore, for every $n>0$ we define the map $f_{\omega}^{-n}$ on $A_{\omega}$ by
	   	$$f_{\omega}^{-n}:=(f_{\theta^{-1}\omega}\circ \cdots \circ f_{\theta^{-n}\omega}|A_{\theta^{-n}\omega})^{-1},$$
	   	and denote by $\Phi^{-n}(\omega,x):=(\theta^{-n}\omega,f_{\omega}^{-n}(x) )$ for every $(\omega,x)\in A$.
	   	Since $\Phi$ is measurable, Lusin's theorem yields that there exists an increasing sequence $A_n: n\in\N$ of compact subsets of $A$, satisfying $\mu(\cup_{n\geq 0}A_n)=1$ and $\Phi|_{A_n}$ is continuous for each $n$.
	   	Denote by $B_n:=\Phi(A_n), n\geq 0$ which also is a compact subset of $A$ and satisfied $\mu(\cup_{n\geq 0}B_n)=1$.
	   	Note that $\Phi|_{A_n}$ is a homeomorphism from $A_n$ onto $B_n$ since $\Phi|_{A_n}$ is injective and continuous.
	   	So we have $(\Phi|_{A_n})^{-1}$ is continuous on $B_n$ for each $n\geq 0$.
	   	Hence, it follows from Lusin's theorem that $(\Phi|_{A})^{-1}$ is also measurable on $A$.

	   	The function for $\omega$ which we integrated at (H2) (iii) is measurable by (H1) (iii) and (H2) (i).
	   	In fact, choose $\{x_n\}_{n>0}$ be a countable dense subset of $\cB$, then we have
	   	$$\sup_{x\in B(A_{\omega},r_0)}| \fo(x)|=\sup_{n>0}|f_{\omega}(x_n)|\cdot \chi_{B(A_{\omega},r_0)}(x_n),$$
	   	where $\chi_{E}$ is a characteristic function of the set $E$.
	   	Since $A$ is random compact, one has $\{\omega: d(x_n,A_{\omega})<r_0\}$ is a measurable subset for each $n$.
	   	This implies that $\omega \mapsto \chi_{B(A_{\omega},r_0)}(x_n)$ is measurable.
	   	By assumption (H1) (iii), $\om\mapsto |f_{\omega}(x_n)|$ is also measurable for each $n$.
	   	So, the $C^0$ norm is measurable.

	   	For the $C^1$ norm, fix $x\in \cB$ and $u\in \cB_x$, by Fr\'{e}chet differentiability
	   	$$Df_{\omega}(x)u=\lim_{k\rightarrow \infty} \dfrac{\fo(x+t_ku)-{\fo(x)}}{t_k},$$
	   	where $\{t_k\}_{k>0}$ is a sequence of positive numbers which converges to zero.
	   	Then, by (H1) (iii) we have the map $\omega \mapsto Df_{\omega}(x)u$ is measurable for each $x\in \cB$ and each $u\in \cB_x$.
	   	The $C^1$ norm is measurable follows from the fact that $\omega \mapsto |Df_{\omega}(x_n)|$ is measurable for every $n$, and that in turn is a direct consequence of $\omega\mapsto Df_{\omega}(x_n)u$ is measurable for any $u\in \cB_{x_n}$ and $\cB$ is separable.
	   	The $C^2$ norm is measurable can be deduced in the same fashion.

	   	The limit in condition (H3) exists by the sub-additive ergodic theorem, guaranteeing that there are at most finitely many positive and zero Lyapunov exponents of $(\Phi,\mu)$ and, each of finite multiplicity.

	   \textbf{Two other relevant conditions}
	   \begin{enumerate}
	   	\item[(H4)]	$(\Psi,\mu)$ has no zero Lyapunov exponents.
	   	\item[(H5)] the set $A$ in (H2) is \textit{visible}, means that there exists a compact subset $E\subset \cB$ with finite box-counting dimension such that $\{\omega:A_{\omega}\subset E\}$ has positive $\bP$-measure.
	   \end{enumerate}
	   We state below a provisional definition of SRB measures, and the formal one we will state in Section \ref{SEC:5}.
	   The definition of induced volume will state in Section \ref{SEC:Induced volumes}.
	   \begin{definition}\label{Def:SRB}
	   	We say $\mu$ is a random SRB measure if
	   	(i)  it has a positive exponent for $\mu$-almost every $(\omega,x)$, and
	   	(ii) the conditional measures of $\mu$ on unstable manifolds are absolutely continuous with respect to the induced volume.
	   \end{definition}

	   By the multiplicative ergodic theorem (see Section \ref{SEC:3}), for $\mu$-almost every points we let $\lambda_1(\omega,x)>\lambda_2(\omega,x)>\cdots$ with multiplicities $m_1(\omega,x),m_2(\omega,x),\cdots,$ be the distinct Lyapunov exponents of $(\Phi,\mu)$.
	   Denote $a^{+}=\max\{a,0\}$, and let $h(\mu|\bP)$ be the conditional entropy of $(\Phi,\mu)$, which the definition we will be given later.

	   The main results of this paper are the following:
	   \begin{maintheorem}\label{Thm:A}
	   Suppose $(\Phi,\mu)$ satisfies (H1)-(H3) above, and assume $\mu$ is a random SRB measure, then we have
	   \begin{equation}\label{eq:entropy formula}
	   		h(\mu|\bP)=\int \sum_{i} m_i(\omega,x)\lambda^{+}_{i}(\omega,x)d\mu.
	   \end{equation}
	   \end{maintheorem}

	   \begin{maintheorem}\label{Thm:B}
	   Suppose $(\Phi,\mu)$ satisfies (H1)-(H5).
	   If $\lambda_1>0$ for $\mu$-almost everywhere, and the entropy formula \eqref{eq:entropy formula} holds, then $\mu$ is a random SRB measure.
	   \end{maintheorem}

	   In the remainder of this section, we explain what is meant by conditional entropy.

	   \textbf{Measure-theoretic entropies of RDS}
	   Let $(\Phi,\mu)$ be as above.
	   The partition $\eta_0:=\{\{\omega\}\times \cB\}_{\omega\in \Omega}$ is a measurable partition of $\Om \times \cB$ since $\Om$ is a Polish space.
	   By Rokhlin's disintegration theorem, there exists a family of measures $\{\mu_{\omega}\}_{\omega\in \Omega}$ on $\cB$ such that for every measurable set $B \subset \Omega\times \cB$,
	   $$\mu(B)=\int \mu_{\omega}(B_{\omega}) d\bP(\omega).$$
	   For every measurable partition $\eta$ of $\Omega \times \cB$, we define the conditional entropy of $\eta$ by
	   \begin{align*}
	   	h(\mu|\bP,\eta):=\lim_{n\rightarrow \infty} \frac{1}{n} H_{\mu}(\vee_{i=0}^{n-1}\Phi^{-i}\eta |\eta_0),
	   \end{align*}
	   where $H_\mu(\eta_1|\eta_2)$ is the conditional entropy of $\eta_1$ given by $\eta_2$.
	   The conditional entropy of $(\Phi,\mu)$ is given by
	   $$h(\mu|\bP):=\sup\{h(\mu|\bP,\eta):\eta \ \text{is a measurable partition such that} \ h(\mu|\bP,\eta)<\infty\}.$$

	   \textbf{Guide to notation} The following notations are used throughout the paper.
	   \begin{itemize}
	   	\item $\Om$ is a Polish space, $\cF$ is the completion of the Borel $\sigma$-algebra of $\Om$ with respect to the probability measure $\bP$.
	   	\item $\cB$ is a separable Banach space with norm $|\cdot|$, and $\mathscr{B}$ is the Borel $\sigma$-algebra of $\cB$.
	   	\item $\cF\otimes \mathscr{B}$ is the product $\sigma$-algebra of the product space $\Om \times \cB$.
	   	\item $E_{\om}=\{x:(\ox)\in E\}$ is the $\om$-section of the subset $E$ of $\Om \times \cB$.
	   	\item $B(x,r)$ is the closed ball of radius $r$ with the center $x$, for $x\in\cB$ and $r>0$.
	   	\item $B(\cB)$ is the space of all bounded linear operators on $\cB$, we also use $|\cdot|$ to denote the operator norm, and denote by $SOT(\cB)$ the strong operator topology on $\cB$.
	   	\item $\cG(\cB)$ is the Grassmannian of all closed subspaces of $\cB$, $\cG_{k}(\cB)$ and $\cG^{k}(\cB)$ are the subsets of all closed k-dimensional subspaces and $k$-codimensional subspaces respectively.
	   	\item $d_{H}$ is the Hausdorff distance on $\cG(\cB)$ and $G(\cdot,\cdot)$ is the gap function, defined in Section \ref{SEC:Gaps}.
	   	\item For $E\in \cG_{k}(\cB)$ and $T\in B(\cB)$, $m_{E}$ is the induced volume and $\det(T|E)$ is the determinate function.
	   	      Those notations are defined in Section \ref{SEC:Induced volumes}.
	   	\item $\nu_{W}$ is the induced volume on a $C^1$ embedded (or injectively immersed) finite-dimensional submanifold $W\subset \cB$, defined in the end of Section \ref{SEC:Induced volumes}.
	   	\item $\cB_x$ is the tangent space to a point $x\in \cB$ which we regard as $\{x\}+\cB$, and the exponential map $\exp_x:\cB_x\rightarrow \cB$ is the affine linear map sending $v\in \cB_x$ to $v+x\in \cB$.
	   	\item $\Gamma$ is the $\Phi$-invariant and full $\mu$-measure subset of $\Om\times \cB$ as in Theorem \ref{MET}.
	   	\item For $(\ox)\in \Gamma$, we decomposition $\cB_x=E^{u}(\ox)\oplus E^{cs}(\ox)$ in the beginning of Section \ref{SEC:Lyapunov norms}, and denote by $\pi^{\tau}_{(\ox)}$ the projection from $\cB_{x}$ onto $E^{\tau}(\ox)$ via the splitting for $\tau=u,cs$.
	   	\item For $(\ox)\in \Gamma$, $\Lnorm{\cdot}{(\ox)}$ is the Lyapunov norm which defined in Section \ref{SEC:Lyapunov norms}.
	   	\item $\Lnorm{\cdot}{}$ is the operator norm of bounded linear operators with respect to the Lyapunov norms.
	   	      For example, let  $L:(\cB_{x}, \Lnorm{\cdot}{(\ox)}) \rightarrow (\cB_{f_{\om}(x)}, \Lnorm{\cdot}{\Phi(\ox)})$ and $T:E^{u}(\ox) \rightarrow E^{cs}(\ox)$ are two bounded linear operators, then $\Lnorm{L}{}=\sup\{\Lnorm{Lv}{\Phi(\ox)}:v\in \cB_{x}, \Lnorm{v}{(\ox)}=1\}$ and $\Lnorm{T}{}=\sup\{\Lnorm{Tu}{(\ox)}:u\in E^{u}(\ox), \Lnorm{u}{(\ox)}=1\}$.
	   	\item The constants $\lambda$, $\varepsilon_0$ as in Section \ref{SEC:Lyapunov norms}, and the constant $\delta_1$ as in \eqref{eq:delta1}.
	   	\item For $(\ox)\in \Gamma$, the connecting map $\wf_{(\ox)}=\exp^{-1}_{f_{\omega}x}\circ f_{\om}\circ \exp_{x}$.
	   	\item The function $\ell:\Gamma\rightarrow [1,\infty)$ and the constant $\varepsilon_1$ are as in \eqref{eq:functionl}.
	   	\item The uniformly set $\Gamma_{\ell}$ defined in the beginning of Section \ref{SEC:CCS}.
	   	\item For $r>0$, $\tau=u,cs$ and $(\ox)\in \Gamma$, $\widetilde{B}_{(\omega,x)}(r)=\{v\in \cB_x:\Lnorm{v}{(\ox)}\leq r\}$,
	   	      $\widetilde{B}^{\tau}_{(\omega,x)}(r)=\{v\in E^{\tau}(\ox):\Lnorm{v}{(\ox)}\leq r\}$ and $B^{\tau}_{(\omega,x)}(r)=\{v\in E^{\tau}(\ox):|v|\leq r\}$.
	   	\item For $(\ox)\in \Gamma$, the function $g_{(\ox)}$ as in Theorem \ref{Thm:Unstable}, the (local) unstable manifolds $W^{u}(\ox)$ defined in   Section \ref{SEC:4.1}, and we denote by $\nu_{(\ox)}$ the induced volume on $W^{u}(\ox)$.
	   	\item For $\om\in \Om$, $n\in \N$ and $y\in \cB$, we use $f^{-n}_{\om}(y)$ to denote the unique point satisfied that   $f^{n}_{\te^{-n}\om}(f^{-n}_{\om}(y))=y$, if such a point exists.
	   	\item For $(\ox)\in \Gamma_{\ell}$, $\cS(\ox)$ is the unstable stack of local unstable manifolds as in \eqref{eq:stack}.
	   	      The random unstable stack $\cS$ is defined in Section \ref{SEC:Random US}.
	   \end{itemize}
\section{Preliminaries}\label{SEC:Pre}
\subsection{Measurability and random compactness}
This subsection collects some known results about measurability and random compactness that we needed in this paper.

Suppose that $(\Om,\cF)$ is a measurable space, so that $\Om$ is a set, and $\cF$ is a $\sigma$-algebra of subsets of $\Om$.
A \textit{Polish space} $X$ is a separable topological space whose topology is metrizable by a complete metric.
For a Polish space $X$, denote by $\mathscr{B}(X)$ the Borel $\sigma$-algebra of $X$.
Let $d$ be a complete metric on $X$, and let $\cF \otimes \mathscr{B}(X)$ be the product $\sigma$-algebra of $\Om \times X$.

The most useful criterion for proving that a given map is measurable on $\Om \times X$ is provided by the following lemma:
\begin{lemma}\label{Lem:MPolish}
	Suppose that $(\Om,\cF)$ is a measurable space and $X$ is a separable metric space. Let $Y$ be another metric space. Suppose that
	$g: \Om \times X \rightarrow Y$ satisfies
	\begin{enumerate}
		\item[(a)] $\om \mapsto g(\om,x)$ is measurable for each $x\in X$;
		\item[(b)] $x \mapsto g(\om,x)$ is continuous for each $\om\in \Om$.
	\end{enumerate}
    Then, $g$ is a measurable map on $\Om \times X$.
\end{lemma}
The proof of this lemma is proposed in \cite[Lemma 1.1]{Crauel02}.

Let $\bP$ be a probability measure on $\Om$. A probability space $(\Omega,\cF,\bP)$ is \textit{complete} if any subset of a measurable set with zero measure is measurable (and has zero measure).

The following projection proposition of Castaing and Valadier \cite[Theorem III.23]{Castaing77} is well-known.
\begin{proposition}\label{Prop:projec P}
	Let $(\Omega,\cF,\bP)$ be a complete probability space and $X$ be a Polish space.
	Let $\cF \otimes \mathscr{B}(X)$ be the product $\sigma$-algebra of $\Om \times X$.
	Denote by $\pi_{\Om}$ the canonical projection from $\Om \times X$ onto $\Om$. Then $\pi_{\Om}(E)\in \cF$ for every $E\in \cF \otimes \mathscr{B}(X)$.
\end{proposition}

For a closed subset $Z\subset X$ and for $x\in X$, put $d(x,Z):=\inf_{y\in Z} d(x,y)$.
Recall that a subset $\Lambda \subset \Om \times X$ is \textit{random compact} if
$\Lambda_{\omega}:=\{x\in \cB:(\omega,x)\in \Lambda\}$
is compact for every $\om\in \Om$,
and the mapping $\om \mapsto d(x,\Lambda_{\omega})$ is measurable for every $x\in X$.

The following proposition \cite[Theorem III.30]{Castaing77} (also see \cite[Proposition 2.4]{Crauel02})
gives characterizations of random compact subsets.
\begin{proposition}\label{Prop:random compact}
	Let $(\Omega,\cF,\bP)$ be a probability space and $X$ be a Polish space.
	Suppose $K \subset \Omega \times X$ satisfied that $K_{\om}$ is compact for every $\om\in \Om$.
	Consider the following conditions
	\begin{enumerate}
		\item[(i)]   $\{\om:K_{\om}\cap U\neq \emptyset\}\in \cF$ for any open set $U\subset X$;
		\item[(ii)]  $\om \mapsto d(x,K_{\omega})$ is measurable for every $x\in X$;
		\item[(iii)] $K$ is a measurable subset of $\Omega \times X$, means that $K\in \cF \otimes \mathscr{B}(X)$.
	\end{enumerate}
    Then, (i) and (ii) are equivalent, and either of them implies (iii). Furthermore, (i), (ii), and (iii) are equivalent if $(\Omega,\cF,\bP)$ is a complete probability space.
\end{proposition}

The following proposition \cite[Proposition 2.15]{Crauel02} gives a version of tightness for random compact sets.
\begin{proposition}\label{Prop:Trandom compact}
	Let $(\Omega,\cF,\bP)$ be a probability space and $X$ be a Polish space.
	Suppose $K \subset \Omega\times X$ is random compact.
	Then, for any $\epsilon>0$ there exists a (non-random) compact subset $K_{\epsilon}\subset \cB$ such that
	$\bP\{\omega:K_{\omega}\subset K_{\epsilon}\}>1-\epsilon$.
\end{proposition}

\subsection{Gaps and distance between closed linear subspaces}\label{SEC:Gaps}
In this subsection, we recall the definitions of gap and distance which are taken from Kato \cite[Chapter 4.2]{Kato95} (see also in \cite[Section 3]{Young17}).

Let ($\cB,|\cdot|$) be a separable Banach space, and let $\cG(\cB)$ be the Grassmannian of all closed subspaces of the Banach space $\cB$.
We consider the metric topology on $\cG(\cB)$ defined by the Hausdorff distance $d_H$ between unit spheres:
for non-trivial subspaces $E,F\in \cG(\cB)$
\begin{align*}
	d_{H}(E,F)=\max\{\sup_{v\in S_E} d(v,S_F),\sup_{u\in S_F} d(u,S_E)\},
\end{align*}
where $S_E=\{v\in E:|v|=1\}$ and $d(v,S_F)=\inf_{u\in S_F} |v-u|$.
For every $k\in\N$, we denote by $\cG_{k}(\cB)$ the collection of $k$-dimensional subspaces of $\cB$, and denote by $\cG^{k}(\cB)$ the collection of $k$-codimensional closed subspaces of $\cB$.
\begin{lemma}[\cite{Quas12} Lemma B.11]\label{Lem:SPGk}
	Suppose that ($\cB,|\cdot|$) is a separable Banach space, then ($\cG_{k}(\cB),d_H$) is a separable metric space.
\end{lemma}

For convenience computations, it is often to work with the \textit{gap} $G(E,F)$ between non-trivial subspaces $E,F\in \cG(\cB)$, defined by
\begin{align*}
G(E,F):=\sup_{v\in S_{E}} d(v,F)
\end{align*}
where $d(v,F)=\inf_{u\in F}|v-u|$.
Owing to $G(\cdot,\cdot)$ is not symmetric, for non-trivial subspaces $E,F\in \cG(\cB)$ we let
\begin{align*}
\hat{G}(E,F)=\max\{G(E,F),G(F,E)\}.
\end{align*}
The relations between $G(E,F)$ and $G(F,E)$ are the following:
\begin{lemma}[\cite{Blumenthal16} Lemma 2.6]\label{Lem:gap}
	Let $k>0$, suppose $E,F\in \cG_{k}(\cB)$ or that $E,F\in \cG^{k}(\cB)$. Assume $G(E,F)<1/k$, then
	$$G(F,E)\leq \dfrac{kG(E,F)}{1-kG(E,F)}.$$
\end{lemma}
On Hilbert space, $\hat{G}$ is a metric and coincides with the operator norm of the difference between orthogonal projections.
On Banach space,  $\hat{G}$ is not a metric since it does not satisfies the triangle inequality in general \cite[Chapter 4.2]{Kato95}.
However, the gap $\hat{G}$ and the distance $d_{H}$ are related by the following proposition (see \cite[Chapter 4]{Kato95} or \cite[Lemma 2.4]{Young16}):
\begin{equation}\label{eq:gap}
\hat{G}(E,F) \leq d_{H}(E,F) \leq  2\hat{G}(E,F).
\end{equation}

Consider a splitting $\cB=E\oplus F$ for $E,F\in \cG(\cB)$, denote by $\pi_{E//F}$ and $\pi_{F//E}$ the projections onto $E$ and $F$ via the splitting respectively.
Then $\pi_{E//F}$ and $\pi_{F//E}$ are bounded linear operators by the closed graph theorem.
\begin{lemma}\label{Lem:ProDh}
	Let $\cB=E\oplus F$ for $E,F\in \cG(\cB)$.
	If a sequence of closed subspaces $E_n\in \cG(\cB),n\in \N$ satisfied that $d_{H}(E_n,E)\rightarrow 0$ as $n\rightarrow \infty$.
	Then, $|\pi_{E//F}|_{E_n}|\rightarrow 1$ and $|\pi_{F//E}|_{E_n}|\rightarrow 0$ as $n\rightarrow \infty$.
\end{lemma}
\begin{proof}
	We first prove that $|(\pi_{F//E})|_{E_n}|\rightarrow 0$ as $n\rightarrow \infty$.
	Fix a unit vector $v\in E_n$.
	By the definition of $d_H$, for every $c>1$ arbitrarily close to 1, there exists a unit vector $v'\in E$ such that $|v-v'|\leq cd_H(E,E_n)$.
	Then,
	$$|\pi_{F//E} v|=|\pi_{F//E} (v-v')|\leq |\pi_{F//E}|\cdot |v-v'|\leq c|\pi|_{F//E}|\cdot d_H(E,E_n).$$
	Therefore, we get
	$$|\pi_{F//E}|_{E_n}|\leq  |\pi|_{F//E}|\cdot d_H(E,E_n).$$
    So, one has $|\pi|_{F//E}|_{E_n}|\rightarrow 0$ as $n\rightarrow \infty$.

	Note that $Id|_{E_n}=\pi_{E//F}|_{E_n}+\pi_{F//E}|_{E_n}$. This implies that
	$$1-|\pi_{F//E}|_{E_n}|\leq |\pi_{E//F}|_{E_n}| \leq 1+|\pi_{F//E}|_{E_n}|.$$
	Since $|\pi_{F//E}|_{E_n}|\rightarrow 0$, we have $|(\pi_{E//F})|_{E_n}|\rightarrow 1$ as $n\rightarrow \infty$.
	This completes the proof.
\end{proof}

\begin{lemma}\label{Lem:CdhA}
   	Let $k>0$ and $\cB=E\oplus F$ for $E\in \cG_k(\cB), F\in \cG^{k}(\cB)$.
   	Assume $T$ is an injective linear operator from $E$ into $F$, and $T_n:E\rightarrow F, n\geq 0$ is a sequence of injective linear operators.
   	Denote by $E_T=(Id+T)E$ and $E_n=(Id+T_n)E$.
   	If $d_{H}(E_T,E_n)\rightarrow 0$ as $n\rightarrow \infty$, then $|T_n-T|\rightarrow 0$ as $n\rightarrow \infty$.
\end{lemma}
\begin{proof}
	For each $n\in \N$, denote $L_n:E_T \rightarrow F$ by $L_n:= (T_n-T)\circ \pi_{E//F}$.
	Then, $E_n=(Id+L_n)E_T$ for every $n\in \N$.
	Note that $\cB=E_T \oplus F$.
	Fix $\epsilon>0$ arbitrarily small.
    For every unit vector $v \in E_T$, let $u=v+L_n v\in E_n$.
    By the definition of $d_{H}$, there exists a unit vector $u'\in E_T$ such that $|u'-(u/|u|)| \leq d_H(E_T,E_n)+\epsilon$.
    Then,
	$$|L_n v|= |u| \cdot |\pi_{F//E_T}(u'-(u/|u|))|\leq (1+|L_n|) \cdot |\pi_{F//E_T}| \cdot (d_H(E_T,E_n)+\epsilon).$$
	Therefore, we have
	$$ \dfrac{|L_n|}{|\pi_{F//E_T}|\cdot(1+|L_n|)}\leq d_H(E_T,E_n).$$
	Since $d_{H}(E_T,E_n)\rightarrow 0$, one has $|L_n|\rightarrow 0$ as $n\rightarrow \infty$.

	By $T$ is injective, the minimum norm $m(\pi_{E//F}|_{E_{T}}):=\min \{|\pi_{E//F} v|:v\in E_T, |v|=1\}$ is strictly larger than $0$.
	Then
    $$|T_n-T|\leq |L_n|\cdot (m(\pi_{E//F}|_{E_{T}}))^{-1}.$$
	By $|L_n|\rightarrow 0$, one has $|T_n-T|\rightarrow 0$ also.
\end{proof}

\subsection{Induced volumes}\label{SEC:Induced volumes}
We explain what is meant by induced volumes in Definition \ref{Def:SRB}.
Those definitions and properties are taken from \cite[Section 3]{Young17}.

We first introduce the notion of induced volumes on finite-dimensional subspaces.
Let $E\subset \cB$ be a finite-dimensional subspace, we denote by $m_E$ the unique Haar measure on $E$ for which
$$m_E\{u\in E:|u|\leq 1\}=w_k$$
where $k=\dim E$ and $w_k$ is the volume of the Euclidean unit ball in $\R^{k}$.
Note that $m_E$ depends on the choice of the norm $|\cdot|$ on $\cB$.
We referred to the measures of the form $m_E$ as a \textit{induced volume} on $E$ from $(\cB,|\cdot|)$.

The notion of \textit{determinant} is defined by induced volumes naturally.
For two Banach spaces $(\cB_1,|\cdot|_1)$, $(\cB_2,|\cdot|_2)$ and a bounded linear operator $A:\cB_1\rightarrow \cB_2$.
Let $E\subset \cB_1$ be a finite-dimensional subspace of $\cB_1$, then
\begin{equation*}
	\det(A|_{E}):=
	\begin{cases}
		\dfrac{m_{A(E)}(A(B_E))}{m_E(B_E)} &\dim A(E)=E,\\
		0                         &\text{otherwise},
	\end{cases}
\end{equation*}
where $B_E=\{v\in E: |v|_1\leq 1\}$.
Note that $\det(A|_{E})$ depends on the choices of the norms $|\cdot|_1$, $|\cdot|_2$ on $\cB_1$ and $\cB_2$ respectively.
We write some basic properties about determinants.
\begin{proposition}[\cite{Young17} Section 3]\label{Prop:Det}
	Let $E,F,G$ be subspaces of $\cB$ of the same finite $k$-dimension, and $A,B:\cB\rightarrow\cB$ be bounded linear maps for which $AE\subset F$, $BF\subset G$.
	Then:
	\begin{enumerate}
		\item[(1)] $m_F(A(S))=\det(A|E)\cdot m_E(S)$ for every Borel set $S\subset E$;
		\item[(2)] $\det(BA|E)=\det(B|F)\cdot\det(A|E)$;
		\item[(3)] there exists a unit basis $\{v_i\}^{k}_{i=1}$ of $E$ (independent of $A$) such that
		           $$\det(A|E)\leq k^{\frac{k}{2}} \prod_{i=1}^{k}|Av_i|;$$
		\item[(4)] assume $A:E\rightarrow F$ is invertible, and there are splitting $E=E_1\oplus E_2, F=F_1\oplus F_2$ for which $AE_1=F_1$, $AE_2=F_2$,
		           then there exists a constant $C_k$ (depends only on $k$) such that
		           $$\dfrac{1}{(|\pi_{F_1//F_2}|^q C_k)} \leq \dfrac{\det(A|E)}{\det(A|E_1)\det(A|E_2)}\leq |\pi_{E_1//E_2}|^q C_k,$$
		           where $\pi_{E_1//E_2}$ is the projection from $E_1\oplus E_2$ onto $E_1$ and $q=\dim E_1$.
	\end{enumerate}
\end{proposition}

For the determinant to make sense, we must guarantee some regularity of the dependence on subspaces and bounded linear operators.
\begin{proposition}[\cite{Young17} Proposition 3.15]\label{Prop:CofDet}
	Let $(\cB_1,|\cdot|_1)$, $(\cB_2,|\cdot|_2)$ be Banach spaces.
	For any $k\geq 1$ and any $M>1$ there exist $L,\delta$ such that,
	for every two injective bounded linear operators $A_1,A_2:\cB_1 \rightarrow \cB_2$ and two $k$-dimensional subspaces $E_1,E_2$ of $\cB_1$ satisfied that
	\begin{align*}
		|A_j|&, |(A_j|_{E_j})^{-1}|\leq M \quad \text{for} \ j=1,2, \\
		|A_1-A_2|&, d_H(E_1,E_2)\leq \delta,
	\end{align*}
	then, we have
	$$|\log \dfrac{\det(A_1|_{E_1})}{\det(A_2|_{E_2})}|\leq L(|A_1-A_2|+d_H(E_1,E_2)).$$
\end{proposition}
\begin{remark}\label{Rem:SOTC}
	Fixed a finite-dimensional subspace $E$, the same argument in \cite[proof of Proposition 3.5]{Young17} shows that if there is a sequence of injective bounded linear operators $T_n, n\in\N$ satisfied that $T_n$ converges to $T$ on strong operator topology $SOT(\cB)$, then we have $\det(T_n|_E) \rightarrow \det(T|_E)$ as $n\rightarrow\infty$.
\end{remark}

Once the regularity of determinants is proved, it is natural to define the induced volumes on embedded (or injectively immersed) finite-dimensional submanifolds.
Let $U\subset \R^{k}$ be an open set, $\phi:U\rightarrow \cB$ be a $C^{1}$ Fr\'{e}chet embedding and $W:=\phi(U)$.
For each Borel set $V\subset W$, we define the induced volume $\nu_{W}$ by
$$\nu_{W}(V)=\int_{\phi^{-1}(V)}\det(D\phi(y))dy$$
where $\det(D\phi(y))$ defined by the linear map $D\phi(y)$ from $\R^{k}$ with Euclidean volume onto the tangent space $T_{\phi(y)} W$ with the induced volume $m_{T_{\phi(y)} W}$.
One can easily show that $\nu_{W}$ does not depend on the choices of $(U,\phi)$ by the change of variables formula for Lebesgue measures, and so it is well defined.
This idea of volumes is easily extended to injectively immersed finite dimensional submanifolds, such as unstable manifolds.
Moreover, assume that $W,{W}'$ are two embedded $k$-dimensional submanifolds and $f:\cB\rightarrow \cB$ is a $C^1$ map that maps $W$ diffeomorphically onto ${W}'$, then for each $y \in {W}'$ we have
$$\dfrac{df_{\ast}(\nu_{W})}{d\nu_{{W}'}}(y)=\dfrac{1}{\det(Df(f^{-1}(y))|T_{f^{-1}(y)}W)}$$
where $f_{\ast}(\nu_{W})$ is the pushforward of the measure $\nu_{W}$ by $f$.
\section{Add on multiplicative ergodic theorem}\label{SEC:3}
	Let $T: X\rightarrow X$ be an invertible measure preserving transformation of a probability space $(X,\mu)$.
	For a given measurable function $\psi:X\rightarrow (0,\infty)$, we call $\psi$ is $T$-\textit{temperate} (with respect to $\mu$), if for $\mu$-almost every $x\in X$
	$$\lim_{n\rightarrow \pm \infty} \frac{1}{n} \log\psi(T^n(x))=0.$$

	The following lemmas are well-known, see Lemmas 8 and 9 in \cite{Walters93} for the first result, and Lemma 5.4 in \cite{Young17} for the second result.

	\begin{lemma} \label{Lem:Tem}
		Let $T: X\to X $ be an invertible measure preserving transformation of a probability space $(X,\mu)$, and let $\psi:X\rightarrow \R$ be a measurable function. Assume that either $(\psi\circ T-\psi)^{+}$ or $(\psi\circ T-\psi)^{-}$ is integrable, then $\psi\circ T-\psi$ is integrable, and
		$$\lim_{n \rightarrow \pm \infty} \frac{1}{n} \psi\circ T^n(x)=0 \quad \mu\text{-}a.e. \ x\in X.$$
	\end{lemma}
	\begin{lemma}\label{Lem:slowly}
		Let $T: X \to X$ be an invertible measure preserving transformation of a probability space $(X,\mu)$, and let $\psi:X \rightarrow \R$ be a $T$-temperate function.
		Then, for every $\varepsilon>0$, there exists a  measurable function ${\psi}':X \rightarrow [1,\infty)$ such that
		${\psi}'(x) \geq \psi(x)$ and
		${\psi}'(T^{\pm}x) \leq e^{\varepsilon}{\psi}'(x)$ for $\mu$-almost every  $x\in X$.
	\end{lemma}

	\textit{In the rest of this paper, we assume that $(\Phi,\mu)$ satisfies (H1)-(H3). For simplicity the discussion, we assuming $\mu$ is $\Phi$-ergodic from here through Section \ref{SEC:NON-ergodic}.}

	\subsection{Oseledets splitting}
	Fix $v\in \cB$, the discussion in the remark of basic assumptions implies that $\om \mapsto Df_{\om}(x) v$ is measurable for each $x\in \cB$.
	Since $x \mapsto Df_{\om}(x) v$ is continuous for each $\om\in \Om$, it follows from Lemma \ref{Lem:MPolish} that $(\ox)\mapsto Df_{\om}(x) v$ is measurable.
	Hence, we have $(\ox) \mapsto Df_{\om}(x)$ is strongly measurable.

	Since $\mu$ is $\Phi$-ergodic, the function $l_{\alpha}$ which we defined in (H3) is constant for $\mu$-almost everywhere.
	Let $0>\lambda_{\alpha}>l_{\alpha}$.
	We have the following theorem which gives the existence of Lyapunov exponents of $(\Phi,\mu)$.
	\begin{theorem}[Z. Lian and K. Lu \cite{Lian10}]\label{MET}
		Let $(\Phi,\mu)$ and $\lambda_{\alpha}$ be as in above.
		Then there exists a $\Phi$-invariant subset $\Gamma$ of $\Omega\times \cB$ and at most finitely many real numbers
		$$\lambda_1>\lambda_2>\cdots>\lambda_{r}$$
		with $\lambda_{r}>\lambda_{\alpha}$ such that for every $(\omega,x)\in \Gamma$, there is a unique splitting of the tangent space $\cB_x$
		$$\cB_x=E_1(\omega,x) \bigoplus \cdots E_{r}(\omega,x)  \bigoplus F(\omega,x)$$
		with the following properties:
		\begin{enumerate}
			\item[(a)] for $i=1,\cdots,r$, $\dim E_{i}(\omega,x)=m_i(\ox)$ is finite and
            $Df_{\omega}(x)E_{i}(\omega,x)=E_{i}(\theta(\omega),f_{\omega}(x))$,
			for any $v\in E_{i}(\omega,x) \setminus \{0\}$, we have
			$$\lim_{n\rightarrow \pm \infty} \frac{1}{n}\log |Df_{\omega}^{n}(x)v|=\lambda_{i},$$
			where $Df_{\omega}^{-n}(x)=(Df_{\theta^{-n}\omega}^{n}(x)|_{E_{i}(\theta^{-n}\omega,f_{\omega}^{-n}(x))})^{-1}$ is well defined by injectivity;
			\item[(b)] $F(\omega,x)$ is closed and finite co-dimensional, satisfies that
			$Df_{\omega}(x) F(\omega,x)$ $\subset F(\theta(\omega),f_{\omega}(x))$ and
			$$\limsup_{n\rightarrow \infty} \frac{1}{n}\log |Df_{\omega}^{n}(x)|_{F(\omega,x)}|\leq \lambda_{\alpha};$$
			\item[(c)] the mappings $(\omega,x)\mapsto E_{i}(\omega,x)$, $i=1,\cdots r$, and $(\omega,x)\mapsto F(\omega,x)$ are measurable from $\Om\times \cB$ into $\cG(\cB)$;
			\item[(d)] denote by $\pi_{i}(\omega,x)$, $\pi_{F}(\omega,x)$ the projection of $\cB_x $ onto $E_{i}(\omega,x)$ and $F(\ox)$ via the splitting,
			then for every $\pi=\pi_{i}$, any $i$ and $\pi=\pi_{F}$ we have that $(\ox) \mapsto \pi(\ox)$ is strongly measurable and for $\mu$-almost every $(\ox)$
			$$\lim_{n\rightarrow \pm\infty} \frac{1}{n}\log |\pi(\Phi^{n}(\omega,x))|=0.$$
		\end{enumerate}
	\end{theorem}

	\begin{lemma}\label{Lem:M3}
		For every $E=E_i$, any $i$, and $E=F$ the following maps are measurable:
		\begin{enumerate}
			\item[(i)]
			$(\omega,x)\mapsto |Df_{\omega}(x)|_{E(\omega,x)}|$,
			\item[(ii)]
			$(\omega,x)\mapsto m(Df_{\omega}(x)|_{E(\omega,x)})$ where $m(T|_{V})$ is the minimum norm,
			\item[(iii)]
			$(\omega,x)\mapsto \det(Df_{\omega}(x)|_{E(\omega,x)})$ for $E=E_i$.
		\end{enumerate}
	\end{lemma}
	\begin{proof}
		Let $E=E_i$ and $m=\dim E$.
		Consider the maps $(T,V)\mapsto |T|_{V}|$, $(T,V)\mapsto m(T_{V})$ which defined on $B(\cB)\times \cG_{m}(\cB)$, and the map $(T,V)\mapsto \det(T|V)$ which defined on $B_{inj}(\cB) \times \cG_{m}(\cB)$, where $B_{inj}(\cB)$ is the space of all injective linear bounded operators.

		For fixed $T$, the maps $V\mapsto |T|_{V}|$, $V\mapsto m(T_{V})$ are continuous on $\cG_{m}(\cB)$, and $V\mapsto \det(T|V)$ is also continuous on $\cG_{m}(\cB)$ by Proposition \ref{Prop:CofDet}.
		For fixed $V$, $T\mapsto |T|_{V}|$, $T\mapsto m(T_{V})$ are continuous on $B(\cB)$ with respect to $SOT(\cB)$, and $T \mapsto \det(T|V)$ is also continuous on $B_{inj}(\cB)$ with respect to $SOT(\cB)$ by Remark \ref{Rem:SOTC}.
		Note that $\cG_{m}(\cB)$ is separable (see Lemma \ref{Lem:SPGk}).
		Then, by Lemma \ref{Lem:MPolish} the three maps above depend measurable on $(T,V)$.
		Therefore, it suffices to show $(\omega,x)\mapsto (Df_{\omega}(x),E(\omega,x))$ is measurable.

		By Theorem \ref{MET}, $(\omega,x)\mapsto E(\omega,x)$ is measurable.
		Since $(\omega,x)\mapsto Df_{\omega}(x)$ is measurable with respect to the strong $\sigma$-algebra on $B(\cB)$, we have $(\omega,x)\mapsto (Df_{\omega}(x),E(\omega,x))$ is measurable.

		Let $E=F$, and let $\{v_n\}_{n=1}^{\infty}\subset \cB$ be a countable dense subset of $\cB$.
		Then, one has that $\{\pi_{F}(\ox) v_n\}_{n=1}^{\infty}$ is a countable dense subset of the subspace $F(\ox)$.
		Note that for any $c\geq 0$,
		\begin{align*}
			&\{(\ox):|Df_{\omega}(x)|_{F(\omega,x)}|\leq c\}=\bigcap_{n>0}\{(\ox):|Df_{\omega}(x)\pi_{F}(\ox) v_n|\leq c|\pi_{F}(\ox) v_n |\}\\
			&\{(\ox):m(Df_{\omega}(x)|_{F(\omega,x)})\geq c\}=\bigcap_{n>0}\{(\ox):|Df_{\omega}(x) \pi_{F}(\ox) v_n|\geq c|\pi_{F}(\ox) v_n|\}.
		\end{align*}
		By Theorem \ref{MET} (d), the function $(\ox)\mapsto |\pi_{F}(\ox) v_n |$ is measurable for each $n$.
		Note that the map $(A_1,A_2)\mapsto A_1A_2$ is continuous from $B(\cB)\times B(\cB)$ to $B(\cB)$ on the strong operator topology $SOT(\cB)$.
		By strongly measurability of $(\ox)\mapsto D\fo(\ox)$ and $(\ox)\mapsto \pi_{F}(\ox)$, one has
		$(\omega,x) \mapsto Df_{\omega}(x) \pi_{F}(\ox)$ is also strongly measurable.
		So, the two sets we have written above are measurable.
	\end{proof}

	\begin{proposition}\label{Prop:DetL}
		For every $k\leq r$ and any
		$1\leq i_1\leq \cdots \leq i_k\leq r$
		the map
		$(\omega,x)\mapsto
		\det(Df_{\omega}(x)|\oplus_{j=1}^{k}E_{i_j}(\omega,x))$ is measurable, and for $\mu$-amolst every $(\omega,x)\in \Gamma$ we have
		$$\lim_{n\rightarrow \infty} \frac{1}{n}\log
		\det(Df^{n}_{\omega}(x)|\bigoplus_{j=1}^{k}E_{i_j}(\omega,x))=\sum_{j=1}^{k}m_{i_j}\lambda_{i_j}.$$
	\end{proposition}
	\begin{proof}
		We follow the proof of Proposition 4.7 in \cite{Young17}.

		We first prove for $k=1$, and writing $\lambda,E,m$ instead of $\lambda_{i_1},E_{i_1},m_{i_1}$.
		Let $\psi(\ox):=\log \det(D\fo(x)|E(\ox))$ for $(\ox)
		\in \Gamma$.
		It follows from Lemma \ref{Lem:M3} immediately that $\psi$ is measurable.
		The Proposition \ref{Prop:Det} (3) tells us
		$$\psi(\ox)\leq m\log m^{\frac{1}{2}} |\fo|_{B(A_{\omega},r_0)}|_{C^2}.$$
		So, by assumption (H2) (iii) one has
		$\psi^{+}(\ox)\in L^1({u})$.
		Using the Birkhoff ergodic theorem, there is a constant $\gamma\in [-\infty,\infty)$ such that for $\mu$-almost every $(\ox)$,
		$$ \gamma:=\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1} \psi(\Phi^{i}(\ox))=\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{i=0}^{n-1} \psi(\Phi^{-i}(\ox)).$$
		It suffices to show that $\gamma=m\lambda$.
		On the one hand, by Proposition \ref{Prop:Det} (3) we choose a unit basis $\{v_i\}_{i=1}^{m}$ for $E(\ox)$ such that
		$$\det(Df^{n}_{\omega}(x)|E(\omega,x))\leq m^{m/2} \prod_{i=1}^{m}|Df^{n}_{\omega}(x)v_i|.$$
		By Theorem \ref{MET}, we have that $\gamma\leq m\lambda$.
		On the other hand, note that
		\begin{align*}
			\sum_{i=0}^{n-1} \psi(\Phi^{-i}(\ox))&=\det(Df^{n}_{\te^{-n}\omega}(\fo^{-n}x)|E(\Phi^{-n}(\ox)))\\
			&=(\det(Df^{-n}_{\omega}(x)|E(\ox)))^{-1}.
		\end{align*}
		We also choose a unit basis $\{u_i\}_{i=1}^{m}$ for $E(\ox)$ such that
		$$\det(Df^{-n}_{\omega}(x)|E(\omega,x))\leq m^{m/2} \prod_{i=1}^{m}|Df^{-n}_{\omega}(x)u_i|.$$
		Also by Theorem \ref{MET}, we have that $\gamma\geq m\lambda$.
		This proves the case when $k=1$.

		Proceeding to the general case, it suffices to prove the case $k=2$. By Proposition \ref{Prop:Det}, we have
		\begin{align*}
			\dfrac{1}{C_{m_{i_1}+m_{i_2}}\cdot|\pi_{i_1}(\Phi^{n}(\ox))|^{m_{i_1}}}&\leq \dfrac{\det(Df^{n}_{\omega}(x)|E_{i_1}(\omega,x)\oplus E_{i_2}(\omega,x))}{\det(Df^{n}_{\omega}(x)|E_{i_1}(\omega,x)
				)\det(Df^{n}_{\omega}(x)|E_{i_2}(\omega,x))}\\
			&\leq C_{m_{i_1}+m_{i_2}}|\pi_{i_1}(\ox)|^{m_{i_1}}.
		\end{align*}
		Recall that Theorem \ref{MET} tells us $|\pi_{i_1}|$ is $\Phi$-temperate.
		So, together with the case $k=1$, the case for $k=2$ is also valid.
	\end{proof}
    \begin{remark}
    	Note that the fact
    	$\int \log \det(D\fo(x)|E(\ox)) d\mu<\infty$
    	is proved in the proof of the above proposition.
    \end{remark}

	\subsection{Lyapunov norms}\label{SEC:Lyapunov norms}
	Recall that $\lambda_{\alpha}<0$, it is sufficient to make a distinction between unstable, center, and stable subspaces, defined by
	\begin{align*}
		&E^{u}(\omega,x)=\bigoplus\{E_{i}(\omega,x):\lambda_i>0\}, \
		E^{c}(\omega,x)=\bigoplus\{E_{i}(\omega,x):\lambda_i=0\}\\
		&\text{and} \ E^{s}(\omega,x)=\bigoplus\{E_{i}(\omega,x):\lambda_i<0\}
		\bigoplus F(\omega,x).
	\end{align*}
	Let $\pi^{\tau}_{(\omega,x)}$ be the projection onto  $E^{\tau}(\omega,x)$ according to the splitting
	$\cB_x=E^{u}(\omega,x)\oplus E^{c}(\omega,x)\oplus E^{s}(\omega,x)$
	for $\tau=u,c,s$ respectively, and let $\pi^{cs}_{(\ox)}=\pi^{c}_{(\ox)}+\pi^{s}_{(\ox)}$.

	We now introduce a new norm $\Lnorm{\cdot}{(\ox)}$ on the tangent space $\cB_x$, in which the expansions and contractions of vectors are reflected in a single time step and called \textit{Lyapunov norms}.
	Let $\lambda^{+}=\min\{\lambda_{i}:\lambda_{i}>0\}$ and $\lambda^{-}=\max\{\lambda_{\alpha},\lambda_{i}:\lambda_{i}<0\}$.
	Denote by $\lambda_0=\min\{\lambda^{+},-\lambda^{-}\}$, and fix $\varepsilon_0\ll \lambda_0$, let $\lambda=\lambda_0-\varepsilon_0$.
	For each $(\omega,x)\in \Gamma$, we let
	\begin{equation*}
		\begin{alignedat}{2}
		|u|'_{(\omega,x)}:=&\sum_{n\geq 0}\dfrac{|Df^{-n}_{\omega}(x) u|}{e^{-n\lambda}}
			\ &\text{for}& \  u\in E^{u}(\omega,x), \\
		|v|'_{(\omega,x)}:=&\sum_{n \in \Z}\dfrac{|Df^{n}_{\omega}(x) v|}{e^{|n|\varepsilon_0}}
			\ &\text{for}& \  v\in E^{c}(\omega,x),\\
		|w|'_{(\omega,x)}:=&\sum_{n\geq 0}\dfrac{|Df^{n}_{\omega}(x) w|}{e^{-n\lambda}}
			\ &\text{for}& \  w\in E^{s}(\omega,x)
		\end{alignedat}
		\end{equation*}
	For each $p\in \cB$, let
	$$|p|'_{(\omega,x)}:=
	\max\{\Lnorm{\pi^{u}_{(\omega,x)}p}{(\ox)},\Lnorm{\pi^{c}_{(\omega,x)}p}{(\ox)},\Lnorm{\pi^{s}_{(\omega,x)}p}{(\ox)}\}.$$
	To estimate the difference between these new norms and the original ones, we let
	\begin{align*}
		&C_{u}(\omega,x)=\sup_{n\geq 0}\dfrac{\sup_{v\in E^{u}(\omega,x),|v|=1}|Df_{\omega}^{-n}(x) v|}{e^{-n(\lambda_0-\frac{\varepsilon_0}{2})}} \\
		&C_{c}(\omega,x)=\sup_{n\in \Z}\dfrac{\sup_{v\in E^{c}(\omega,x),|v|=1}|Df_{\omega}^{n}(x) v|}{e^{\frac{|n|\varepsilon_0}{2}}}\\
		&C_{s}(\omega,x)=\sup_{n\geq 0}\dfrac{\sup_{v\in E^{s}(\omega,x),|v|=1}|Df_{\omega}^{n}(x) v|}{e^{-n(\lambda_0-\frac{\varepsilon_0}{2})}}.
	\end{align*}
	and let
	$$C(\ox)=\max\{C_{u}(\omega,x),C_{c}(\omega,x),C_{s}(\omega,x),|\pi^{u}_{(\omega,x)}|,|\pi^{c}_{(\omega,x)}|,|\pi^{s}_{(\omega,x)}|\}.$$
	By Theorem \ref{MET} and Lemma \ref{Lem:M3}, these functions all are measurable and finite-valued on $\Gamma$. Moreover, one can easily show that for $\tau=u,c,s$, either
	$$(\log C_{\tau}(\omega,x)-\log C_{\tau}(\Phi(\omega,x)))^{+}\leq \log^{+} \sup_{x\in B(A_{\omega},r_0)}|Df_{\omega}(x)|$$
	or
	$$(\log C_{\tau}(\Phi(\omega,x))-\log C_{\tau}(\omega,x))^{+}\leq \log^{+} \sup_{x\in B(A_{\omega},r_0)}|Df_{\omega}(x)|.$$
	Therefore, by condition (H2) (iii) and Lemma \ref{Lem:Tem}, we have $C_{\tau}$ is $\Phi$-temperate for $\tau=u,c,s$.
	Together with Theorem \ref{MET} (d), we have $C$ is $\Phi$-temperate.
	\begin{lemma}[Linear picture]\label{Lem:LinP}
		The following hold for all $(\omega,x)\in \Gamma$:
		\begin{align}
			e^{\lambda}|u|'_{(\omega,x)} \leq &|Df_{\omega}(x) u|'_{\Phi(\omega,x)} \label{eq:41};  \\
			e^{-\varepsilon_0}|v|'_{(\omega,x)}\leq &|Df_{\omega}(x) v|'_{\Phi(\omega,x)} \leq e^{\varepsilon_0} |v|'_{(\omega,x)}  \label{eq:42};\\
			&|Df_{\omega}(x) w|'_{\Phi(\omega,x)} \leq e^{-\lambda}|w|'_{(\omega,x)} \label{eq:43}.
		\end{align}
		Where $u\in E^{u}(x,h)$, $v\in E^{c}(x,h)$ and $w\in E^{s}(x,h)$. Moreover, the relations between the norms are
		\begin{equation}\label{eq:norms}
			\frac{1}{3}|\cdot|\leq |\cdot|'_{(\omega,x)}\leq
			\frac{3}{1-e^{-\frac{\varepsilon_0}{2}}}C(\omega,x)^{2} |\cdot|
		\end{equation}
	\end{lemma}
	The proof is a simple computation and we omitted it here.

	For each $(\omega,x)\in \Gamma$.
	Let $\widetilde{B}_{(\omega,x)}(r):=\{v\in \cB_x:\Lnorm{v}{(\ox)} \leq r\}$. Let the connecting map
	$\widetilde{f}_{(\omega,x)}:=\exp^{-1}_{f_{\omega}(x)} \circ f_{\omega}\circ \exp_{x}$,
	and for each $n\in \Z$ denote by
	$\widetilde{f}^{n}_{(\omega,x)}:=\exp^{-1}_{f^{n}_{\omega}(x)} \circ f^{n}_{\omega}\circ \exp_{x}$.
	Recall the constant $r_0$ as in assumption (H2) (iii).
	\begin{lemma}[Nonlinear picture]\label{Lem:NlinP}
		Define a measurable function
		${\ell}':\Gamma\rightarrow [1,\infty)$
		by
		$${\ell}'(\ox):=\frac{27}{1-e^{-\frac{\varepsilon_0}{2}}} C(\Phi(\ox))^{2}\cdot \max\{1,|(\fo|_{B(A_{\omega},r_0)})|_{C^2}\}.$$
		Then, for every $0<\delta<r_0$ and every $(\omega,x)\in \Gamma$ the following holds for nonlinear map
		$$\widetilde{f}_{(\omega,x)}: (\widetilde{B}_{(\omega,x)}(\delta {\ell}'(\ox)^{-1}),|\cdot|'_{(\omega,x)}) \rightarrow (\cB_{f_{\omega}(x)},|\cdot|'_{\Phi(\omega,x)}).$$
		\begin{enumerate}
			\item[(1)] $\Lip(\widetilde{f}_{(\omega,x)}-Df_{\omega}(x))\leq \delta$;
			\item[(2)]
			the mapping $z\mapsto D\widetilde{f}_{(\omega,x)}(z)$ satisfies $\Lip (D\widetilde{f}_{(\omega,x)})\leq {\ell}'(\ox)$.
		\end{enumerate}
	\end{lemma}
	\begin{proof}
		We denote by $|\cdot|'$ the operator norm of bounded linear operators with respect to the Lyapunov norms.

		For every $y,z\in \wB_{(\ox)}(\delta {\ell}'(\ox)^{-1})$, we have $|y-x|\leq r_0$ and $|z-x|\leq r_0$. Thus,
		\begin{align*}
			|D\wf_{(\ox)}(y)-D\wf_{(\ox)}(z)|'
			&\leq
			\frac{9}{1-e^{-\frac{\varepsilon_0}{2}}} C(\Phi(\ox))^{2} \cdot |(\fo|_{B(A_{\omega},r_0)})|_{C^2} \cdot |z-y|\\
			&\leq \frac{27}{1-e^{-\frac{\varepsilon_0}{2}}} C(\Phi(\ox))^{2}\cdot |(\fo|_{B(A_{\omega},r_0)})|_{C^2} \cdot \Lnorm{z-y}{(\ox)}\\
			&\leq {\ell}'(\ox)\Lnorm{z-y}{(\ox)},
		\end{align*}
		we have (2) holds. The following estimation suffices to show the item (1):
		\begin{align*}
			\Lip(\wf_{(\ox)}-D\wf_{(\ox)}(0))
			&\leq \sup_{y\in \wB_{(\ox)}(\delta {\ell}'(\ox)^{-1})}|D(\wf_{(\ox)}-D\wf_{(\ox)}(0))(y)|'\\
			&\leq \sup_{y\in \wB_{(\ox)}(\delta {\ell}'(\ox)^{-1})}|D\wf_{(\ox)}(y)-D\wf_{(\ox)}(0)|'\\
			&\leq \Lip (D\widetilde{f}_{(\omega,x)})\cdot \Lnorm{y}{(\ox)} \leq \delta.
		\end{align*}
	\end{proof}
    By assumption (H2) (iii) and Lemma \ref{Lem:Tem} we have $\omega\mapsto |(\fo|_{B(A_{\omega},r_0)})|_{C^2}$ is $\theta$-temperate.
	Since $C$ is $\Phi$-temperate, one has the function ${\ell}'$ is also $\Phi$-temperate.
	So, let $\varepsilon_1\ll \varepsilon_0$, by Lemma \ref{Lem:slowly} there exists a measurable function $\ell:\Gamma\rightarrow [1,\infty)$  such that $\ell \geq {\ell}'$ and
	\begin{equation}\label{eq:functionl}
		\ell(\Phi^{\pm}(\ox))\leq e^{\varepsilon_1}\ell(\ox).
	\end{equation}
	By the definition of the function $\ell$, for each $(\ox)\in \Gamma$, we have
	\begin{equation}\label{eq:lox}
		\begin{aligned}
			|(\fo|_{B(A_{\om},r_0)})|_{C^2}\leq \ell(\ox), \
			\frac{1}{3}\Lnorm{\cdot}{(\ox)}&\leq |\cdot|\leq \ell(\ox)\Lnorm{\cdot}{(\ox)}\\ \text{and} \ \max\{|\pi^{u}_{(\ox)}|,|\pi^{cs}_{(\ox)}|\}&\leq \ell(\ox).
		\end{aligned}
	\end{equation}

	Choose $0<\delta_1<r_0$ small enough such that
	\begin{equation}\label{eq:delta1}

		e^{\varepsilon_0}+\delta_1<e^{\frac{1}{2}\lambda}<e^{\lambda}-\delta_1.
	\end{equation}
	\begin{corollary}\label{Cor:NlinP}
		Let $(\omega,x)\in \Gamma$.
		Denote by $\wf_{k}:=\wf_{\te^{k}\om}$, $\Lnorm{\cdot}{k}:=\Lnorm{\cdot}{\Phi^k(\ox)}$, $\ell(k)=\ell(\Phi^{k}(\ox))$ and $\pi^{\tau}_{k}=\pi^{\tau}_{\Phi^k(\ox)}$ for $k\in \Z$ and $\tau=u,cs$.
		For every $0<\delta\leq\delta_1$, if $u,v\in \cB_x$ and $n\in \N$ satisfied that
		$\Lnorm{\wf^{k}_{0}(u)}{k},\Lnorm{\wf^{k}_{0}(v)}{k} \leq \delta \ell(k)^{-1}$ for $k=0,1,\cdots,n-1$,
		then the following holds:
		\begin{enumerate}
			\item[(1)] If $\Lnorm{u-v}{0}=\Lnorm{\pi^{u}_{0}(u-v)}{0}$, then for $k=1,\cdots,n$
			\begin{align*}
				\Lnorm{\wf^{k}_{0}(u)-\wf^{k}_{0}(v)}{k}&=\Lnorm{\pi^{u}_{k}(\wf^{k}_{0}(u)-\wf^{k}_{0}(v))}{k}\\
				&\geq (e^{\lambda}-\delta)^{k}\Lnorm{u-v}{0}.
			\end{align*}
			\item[(2)]
			If $\Lnorm{\wf^{n}_{0}(u)-\wf^{n}_{0}(v)}{n}=\Lnorm{\pi^{cs}_{n}(\wf^{n}_{0}(u)-\wf^{n}_{0}(v))}{n}$, then for $k=0,1,\cdots,n-1$
			\begin{align*}
				\Lnorm{\wf^{k}_{0}(u)-\wf^{k}_{0}(v)}{k}&=  \Lnorm{\pi^{cs}_{k}(\wf^{k}_{0}(u)-\wf^{k}_{0}(v))}{k}\\
				&\leq (e^{\varepsilon_0}+\delta)^{k}\Lnorm{u-v}{0}.
			\end{align*}
		\end{enumerate}
	    Furthermore, if $E^{c}(\omega,x)=\{0\}$, then the right-hand side in item (2) is
	    $$\leq (e^{-\lambda}+\delta)^{k}\Lnorm{u-v}{0}.$$
	\end{corollary}
    \begin{proof}
         For each $u,v\in \cB_x$ with $\Lnorm{u}{0},\Lnorm{v}{0}\leq \delta \ell(0)^{-1}$.
         We first assume that $\Lnorm{u-v}{0}=\Lnorm{\pi^{u}_{0}(u-v)}{0}$.
         By Lemma \ref{Lem:NlinP} we have
         \begin{equation}\label{eq:21}
          	\begin{aligned}
            \Lnorm{\pi^{u}_{1}(\wf_{0} u-\wf_{0} v)}{1}\geq& \Lnorm{\pi^{u}_{1}(D\wf_{0}(0)(u-v))}{1}\\
                                                           &-\Lnorm{\pi^{u}_{1}  ((\wf_{0}-D\wf_{0}(0))u-(\wf_{0}-D\wf_{0}(0))v)}{1}\\
                                                       \geq& \Lnorm{D\wf_{0}(0)(\pi^{u}_{0}(u-v))}{1}-\delta\Lnorm{u-v}{0}\\
                                                       \geq& (e^{\lambda}-\delta)\Lnorm{u-v}{0}.
          	\end{aligned}
         \end{equation}
         Similarly, one can show that
         \begin{equation}\label{eq:22}
         	\Lnorm{\pi^{cs}_{1}(\wf_{0} u-\wf_{0} v)}{1}\leq (e^{\varepsilon_0}+\delta)\Lnorm{u-v}{0}.
         \end{equation}
         By the choice of $\delta_1$, we have
         $\Lnorm{\pi^{u}_{1}(\wf_{0} u-\wf_{0} v)}{1}\geq \Lnorm{\pi^{cs}_{1}(\wf_{0} u-\wf_{0} v)}{1}$. This implies that
         $\Lnorm{\wf_{0} u-\wf_{0} v}{1}=\Lnorm{\pi^{u}_{1}(\wf_{0} u-\wf_{0} v)}{1}$ when $\Lnorm{u-v}{0}=\Lnorm{\pi^{u}_{0}(u-v)}{0}$.

         Now we assume that $u,v$ and $n$ satisfied the conditions in statements.
         For item (1), if $\Lnorm{u-v}{0}=\Lnorm{\pi^{u}_{0}(u-v)}{0}$. By induction, one has
         $\Lnorm{\wf^{k}_{0}(u)-\wf^{k}_{0}(v)}{k}=\Lnorm{\pi^{u}_{k}(\wf^{k}_{0}(u)-\wf^{k}_{0}(v))}{k}$ for every $k=1,\cdots n$.
         In this case, (\ref{eq:21}) implies that
         $$\Lnorm{\wf^{k}_{0}(u)-\wf^{k}_{0}(v)}{k}\geq (e^{\lambda}-\delta)\Lnorm{\wf^{k-1}_{0}(u)-\wf^{k-1}_{0}(v)}{k-1}$$
         for every $k=1,\cdots n$. This proves item (1).

         For item (2), if $\Lnorm{\wf^{n}_{0}(u)-\wf^{n}_{0}(v)}{n}=\Lnorm{\pi^{cs}_{n}(\wf_{n}(u)-\wf_{n}(v))}{n}$,
         then we must have
         $\Lnorm{\wf^{k}_{0}(u)-\wf^{k}_{0}(v)}{k}=\Lnorm{\pi^{cs}_{k}(\wf^{k}_{0}(u)-\wf^{k}_{0}(v))}{k}$
         for every $k=0,\cdots n-1$.
         Together with (\ref{eq:22}), we have
         $$\Lnorm{\wf^{k}_{0}(u)-\wf^{k}_{0}(v)}{k}\leq (e^{\varepsilon_0}+\delta)\Lnorm{\wf^{k-1}_{0}(u)-\wf^{k-1}_{0}(v)}{k-1}$$
         for every $k=1,\cdots n$. This proves item (2).

         Moreover, if $E^{c}(\ox)=\{0\}$, then (\ref{eq:22}) is
         $$\Lnorm{\pi^{s}_{1}(\wf_{0} u-\wf_{0} v)}{1}\leq (e^{-\lambda}+\delta)\Lnorm{u-v}{0}.$$
         Using the same argument above one can show that
         $$\Lnorm{\wf^{k}_{0}(u)-\wf^{k}_{0}(v)}{k}=  \Lnorm{\pi^{s}_{k}(\wf^{k}_{0}(u)-\wf^{k}_{0}(v))}{k}\leq (e^{-\lambda}+\delta)^{k}\Lnorm{u-v}{0}$$
         when $\Lnorm{\wf^{n}_{0}(u)-\wf^{n}_{0}(v)}{n}=\Lnorm{\pi^{s}_{n}(\wf_{n}(u)-\wf_{n}(v))}{n}$. This completes the proof.
    \end{proof}

	\subsection{Continuity of certain spaces on uniformly sets}\label{SEC:CCS}
	For $\ell_0\geq 1$, we let
	$$\Gamma_{\ell_0}:=\{(\ox)\in \Gamma: \ell(\ox)\leq \ell_0\}$$
	and called the sets of the form $\Gamma_{\ell_0}$ as \textit{uniformly sets}. Since $\Omega \times \cB$ is a Polish space, one can assume that $\Gamma_{\ell_0}$ is compact for each $\ell_0$.

	In finite-dimensional case or Hilbert case, there are more general results about the continuity of Oseledets subspaces, which could reference \cite{Simion16} and \cite{Froyland18}.
	\begin{proposition}\label{Prop:CCS}
		For each $\om\in \Gamma$.
		The maps $x \mapsto E^{cs}(\ox)$ and $x \mapsto E^{u}(\ox)$ continuous on $(\Gamma_{\ell_0})_{\omega}$ with respect to the $d_H$-metric on $\cG(B)$.
		More specifically, for any $\epsilon>0$, there exists $\delta(\ell_0,\varepsilon)>0$ and ${\delta}'(\ell_0,\varepsilon,\omega)>0$
		such that for every $(\ox),(\oy)\in \Gamma_{\ell_0}$,
		if $|x-y|\leq \delta$ then  $d_H(E^{cs}(\ox),E^{cs}(\oy))<\epsilon$,
		if $|x-y|\leq {\delta}'$ then $d_H(E^{u}(\ox),E^{u}(\oy))<\epsilon.$
	\end{proposition}
	\begin{proof}
		For every $n\in \N$ and every $(\ox), (\omega,y)\in \Gamma_{\ell}$.
		We first prove that,
		$$|Df^n_{\omega}(x)-Df^n_{\omega}(y)|\leq C(n,\ell)\cdot|x-y|.$$
		Indeed, we have
		\begin{align*}
			|Df^n_{\omega}(x)-Df^n_{\omega}(y)|
			&\leq (\prod^{n-1}_{m=0}|(f_{\te^m \omega}|_{B(A_{\te^m \om,r_0})})|_{C^2})\cdot \sum_{j=0}^{n-1}|f^{j}_{\om}x-f^{j}_{\om}y|\\
			&\leq n\ell^{2n} e^{(n^2+n)\varepsilon_1 }\cdot |x-y|=C(n,\ell)\cdot|x-y|.
		\end{align*}

		Assume that $(\ox),(\oy)\in \Gamma_{\ell_0}$.
		Fix a unit vector $v\in E^{cs}(\oy)$.
		Let $v=v^s+v^u$ be the decomposition with respect to the splitting $E^{cs}(\ox)\oplus E^{u}(\ox)$.
		For each $n\in \N$.
		On the one hand, by Lemma \ref{Lem:LinP} and (\ref{eq:lox}) we have
		\begin{align*}
			|D\fo^{n}(x) v|&\geq |D\fo^{n}(x) v^u|-|D\fo^{n}(x) v^{cs}| \\
			&\geq (3\ell_0)^{-1}e^{n(\lambda-\varepsilon_1)}|v^u|-3\ell_0 e^{n\varepsilon_0}\cdot|\pi^{cs}_{(\ox)}|
		\end{align*}
		On the other hand,
		\begin{align*}
			|D\fo^{n}(x) v|&\leq
			|D\fo^{n}(y)v|+|(D\fo^{n}(x)-D\fo^{n}(y))v|\\         &\leq 3\ell_0e^{n\varepsilon_0}+C(n,\ell_0)|x-y|.
		\end{align*}
		Recall that $\max\{|\pi^{u}_{(\omega,z)}|,|\pi^{cs}_{(\omega,z)}|\}\leq \ell(\omega,z)$ for every $(\omega,z)\in \Gamma$.
		Therefore, we have that
		$$|v^u|\leq (3\ell_0)^2(1+\ell_0)e^{-n(\lambda-\varepsilon_0-\varepsilon_1)}+3\ell_0 e^{-n\lambda}C(n,\ell_0)|x-y|.$$
		For each $\epsilon>0$, take ${n}':={n}'(\ell_0,\epsilon)$ large enough so that $(3\ell_0)^2(1+\ell_0)e^{-{n}'(\lambda-\varepsilon_0-\varepsilon_1)}<\frac{\epsilon}{4}$,
		and take $\delta:=\delta(\ell_0,\epsilon)$ small enough such that $3\ell_0 e^{-{n}'\lambda}C({n}',\ell_0)\delta< \frac{\epsilon}{4}$.
		Then, by definition of the gap $G(\cdot,\cdot)$ we have
		$G(E^{cs}(\oy),E^{cs}(\ox))\leq \frac{\epsilon}{2}$ when $|x-y|\leq \delta$.
		Symmetrically, replacing $(x,y)$ by $(y,x)$ one has $G(E^{cs}(\ox),E^{cs}(\omega,y))<\frac{\epsilon}{2}$.
		So, we get that $d_{H}(E^{cs}(\ox),E^{cs}(\omega,y))<\epsilon$ when $|x-y|\leq \delta$.

		Note that we cannot iterate backwards to estimate $d_H(E^{u}(\ox),E^{u}(\oy))$.
		Fix a unit vector $v\in E^{u}(\omega,y)$, let $v=v^u+v^s$ be the decomposition with respect to the splitting $E^{u}(\ox)\oplus E^{cs}(\ox)$.
		Denote by $x_{-n}$ and $y_{-n}$ the points $f^{-n}_{\omega}(x)$ and $f^{-n}_{\omega}(y)$ respectively, and let $v_{-n}$ be the unique vector in $E^{u}(\te^{-n}\omega,y_{-n})$ such that $Df^n_{\te^{-n}\omega}(y_{-n}) v_{-n}=v$.
		We decompose $v_{-n}$ by
		$$v_{-n}=v^u_{-n}+v^s_{-n}\in E^{u}(\te^{-n} \omega,x_{-n})\oplus E^{cs}(\te^{-n}\omega,x_{-n}).$$
		By (\ref{eq:lox}) and Lemma \ref{Lem:LinP}, we have $|v_{-n}|\leq 3\ell_0e^{-n\lambda}$.
		We bounded $v^s$ by
		$$|v^s|\leq |v^s-Df^n_{\te^{-n}\omega}(x_{-n}) v^s_{-n}|+|Df^n_{\te^{-n}\omega}(x_{-n}) v^s_{-n}|.$$
		The first term is upper-bounded by the following
		$$\leq |\pi^{cs}_{(\omega,x)}|\cdot |v-Df^n_{\te^{-n}\omega}(x_{-n}) v_{-n}|\leq 3\ell^{2}_0 e^{-n\lambda} C(n,\ell_0e^{n\varepsilon_1})|x_{-n}-y_{-n}|.$$
		The second term has the following estimation:
		\begin{align*}
			|Df^n_{\te^{-n}\omega}(x_{-n}) v^s_{-n}|&\leq 3\ell_0e^{n(\varepsilon_0+\varepsilon_1)} |v^s_{-n}|\\
			&\leq3\ell_0 e^{n(\varepsilon_0+\varepsilon_1)}|\pi^{cs}_{\Phi^{-n}(\ox)}|\cdot|v_{-n}|\\
			&\leq9\ell_0^{3}e^{-n(\lambda-\varepsilon_0-2\varepsilon_1)}.
		\end{align*}
		For each $\epsilon>0$ take $k:=k(\epsilon,\ell_0)$ large enough such that the second term is upper-bounded by $\epsilon/4$.
		Therefore, by definition of $G(\cdot,\cdot)$ one has
		$$G(E^u(\oy),E^u(\ox))\leq \frac{\epsilon}{4}+ 3\ell^{2}_0 e^{-k\lambda}C(k,\ell_0e^{k\varepsilon_1})\cdot |x_{-k}-y_{-k}|.$$
		Choose ${\delta}'':={\delta}''(\epsilon,\ell_0)$ small enough, such that when $|x_{-k}-y_{-k}|<{\delta}''$ we have $G(E^u(\oy),E^u(\ox))< \frac{\epsilon}{2}$.
		Symmetrically, replacing $(x,y)$ by $(y,x)$ we have $G(E^u(\ox),E^u(\omega,y))<\frac{\epsilon}{2}$, so we get $d_{H}(E^u(\ox),E^u(\omega,y))< \epsilon$ when $|x_{-k}-y_{-k}|<{\delta}''$.

		To make $|x_{-k}-y_{-k}|<{\delta}''$, we use the fact that $f^{-k}_{\omega}$ continuous on the compact subset $A_{\omega}$.
		Then, choose a real number ${\delta}'(\epsilon,\ell_0,\omega)$ small enough, such that for every $x,y\in A_{\om}$ if
		$|x-y|\leq {\delta}'$ then $ |x_{-k}-y_{-k}|<{\delta}''$.
		This completes the proof.
	\end{proof}
\section{Nonuniform hyperbolic theory}\label{SEC:4}
	\subsection{Unstable manifolds}\label{SEC:4.1}
	For each $(\omega,x)\in \Gamma$.
	Let $\widetilde{B}^{\tau}_{(\omega,x)}(r):=\{v\in E^{\tau}(\omega,x):{|v|}'_{(\omega,x)}\leq r\}$ for $\tau=u,cs$ and $r>0$.
	The definition of the Lyapunov norms implies that $\widetilde{B}_{(\ox)}(r)=\widetilde{B}^{u}_{(\ox)}(r)+\widetilde{B}^{cs}_{(\ox)}(r)$.

	\begin{theorem}[Local unstable manifolds] \label{Thm:Unstable}
		There exists ${\delta}'_1>0$ such that, ${\delta}'_1<\delta_1$ (where $\delta_1$ as in Corollary \ref{Cor:NlinP})
		and for every $\delta<{\delta}'_1$ there exists a unique family of continuous maps
		$$\{g_{(\omega,x)}:\widetilde{B}^{u}_{(\omega,x)}(\delta \ell(\omega,x)^{-1})\rightarrow \widetilde{B}^{cs}_{(\omega,x)}(\delta \ell(\omega,x)^{-1})  \}_{(\omega,x)\in\Gamma}$$
		satisfied that
		$$g_{(\omega,x)}(0)=0 \ \text{and} \ \widetilde{f}_{(\omega,x)}\graph(g_{(\omega,x)})\supset \graph(g_{\Phi(\omega,x)}).$$
		With respect to the $|\cdot|_{x}$ norms on $\widetilde{B}_{x}(\delta \ell(\ox)^{-1})$, the family $\{g_{(\omega,x)}\}_{(\omega,x)\in \Gamma}$ has the following additional properties:
		\begin{enumerate}
			\item[(1)] $g_{(\omega,x)}$ is $C^{1+\Lip}$ Fr\'{e}chet differentiable, with $Dg_{(\omega,x)}(0)=0$;
			\item[(2)] $\Lip g_{(\omega,x)}\leq 1/10$ and $\Lip(Dg_{(\omega,x)})\leq C_0 \ell(\omega,x)$ where $C_0> 0$ is independent of $(\omega,x)$;
			\item[(3)] if $u_i\in \graph(g_{(\omega,x)})$ and $\widetilde{f}_{(\omega,x)}u_i\in \graph(g_{\Phi(\omega,x)})$ for $i=1,2$, then
			\begin{align*}
				\Lnorm{ \wf_{(\omega,x)}(u_1)-\wf_{(\omega,x)}(u_2) }{\Phi(\omega,x)}
				\geq (e^\lambda-\delta) \Lnorm{u_1-u_2}{(\omega,x)}
			\end{align*}
		\end{enumerate}
	\end{theorem}
	\begin{remark}\label{Rem:mofg}
		Since the mapping $(\ox)\mapsto E^{u}(\ox)$ is measurable, we can find a measurable basis $\{e_i{(\ox)}\}_{i=1}^{\dim E^{u}}$ spanning $E^{u}(\ox)$ (see \cite[Corollary 7.3]{Lian10}).
		Moreover, for each $v=(v_i)_{i=1}^{\dim E^{u}}\in \R^{\dim E^{u}}$ be fixed, define
		$$v_{\ox}=:\sum_{i=1}^{\dim E^{u}} v_i e_i{(\ox)}.$$
		Then, by \cite[Remark 9.15]{Lian10}, the mapping $(\ox)\mapsto g_{(\omega,x)}(v_{\ox})$ is measurable on the measurable subset
		$\{(\ox):\Lnorm{v_{\ox}}{(\ox)}\leq \delta \ell(\ox)^{-1}\}$.
	\end{remark}
    These results are well-known for finite-dimensional systems.
	For infinite dimensional random dynamical systems, stable and unstable manifolds theorem be proved in \cite{Varzaneh21} on fields of Banach spaces, in \cite[Chapter 9]{Lian10} on Banach spaces, and in \cite{Li13} on Hilbert spaces, also be mentioned in \cite{Young17} and \cite{Young16} for certain maps on a Banach space.

	For each $(\omega,x)\in \Gamma$. Let $\wW_{\delta}^{u}(\omega,x)=\graph(g_{(\omega,x)})$ and define the \textit{local unstable manifold} at $(\ox)$ by
	$$W_{\delta}^{u}(\ox)=\exp_x \wW_{\delta}^{u}(\ox).$$
	The \textit{global unstable manifold} at $(\ox)$ defined to be
	$$W^{u}(\ox)=\bigcup_{n\geq 0}f^{n}_{\theta^{-n}(\omega)} W_{\delta}^{u}(\Phi^{-n}(\ox)).$$

	\begin{remark}\label{Rem:ivinm}
		Let $(\ox)\in \Gamma$ and $0<\delta<\delta_1'$, one may not have $W_{\delta}^{u}(\ox)\subset A_{\om}$. However, by Theorem \ref{Thm:Unstable} one has $W_{\delta}^{u}(\ox)\subset f^n_{\te^{-n}\om} W_{\delta}^{u}(\Phi^{-n}(\ox))$ for each $n>0$. Hence, for every $y\in W_{\delta}^{u}(\ox)$, we denote by $f^{-n}_{\om}(y) $
		the unique point in $W_{\delta}^{u}(\Phi^{-n}(\ox))$ such that $f^{n}_{\te^{-n}(\om)} (f^{-n}_{\om}(y))=y$.
	\end{remark}

	The key point to prove the unstable manifolds theorem is the result of graph transforms.
	Take $(\ox)\in \Gamma$ and $\delta \leq {\delta}'_1$.
    All the following statements should be understood to the ``$\Lip$" with respect to the $\Lnorm{\cdot}{(\ox)}$ norm.
	Let
	$$\cW(\ox):=\{g:\widetilde{B}^{u}_{(\ox)}(\delta \ell(\ox)^{-1})\rightarrow E^{cs}(\ox) \ \text{with} \ g(0)=0, \ \Lip(g) \leq \frac{1}{10}\}.$$
	For $g \in \cW(\ox)$, we denote by $\cT_{(\ox)}(g)$ the graph transform of $g$ if it is defined;
	means that,
	$\cT_{(\ox)}(g): \widetilde{B}^{u}_{\Phi(\ox)}(\delta \ell(\Phi(\ox))^{-1}) \rightarrow E^{cs}(\Phi(\ox))$ is the map satisfied
	$$\wf_{(\omega,x)}(\graph(g)) \supset \graph(\cT_{(\ox)}(g)).$$
	\begin{lemma}[\cite{Young17} Lemma 6.2]\label{Lem:Gtran}
		Let $(\omega,x)\in \Gamma$,
		\begin{enumerate}
			\item[(1)] for every $g \in \cW(\ox)$, $\cT_{(\ox)}(g)$ is defined and belong to $\cW(\Phi(\ox))$.
			\item[(2)] there exists a constant $c \in (0,1)$ independent of $(\ox)$ such that for all $g_1, g_2 \in \cW(\ox)$
			$$|||\cT_{(\ox)}(g_1)-\cT_{(\ox)}(g_2) |||_{\Phi(\ox)}\leq c|||g_1-g_2|||_{(\ox)} $$
			where
			$$|||g|||_{(\ox)}=\sup_{v\in \widetilde{B}^{u}_{x}(\delta \ell(\ox)^{-1})\setminus \{0\}} \dfrac{\Lnorm{g(v)}{(\ox)}}{\Lnorm{v}{(\ox)}}.$$
		\end{enumerate}
	\end{lemma}
	Since in infinite dimensions one cannot iterate backward,
	the ``backward graph transformation'' is defined only in the case that $E^{c}(\ox)=0$, which is different from the case for finite dimensions. We now give the formal definition below.

	Take $\delta >0$ sufficiently small. Let $(\omega,x)\in \Gamma$ and assume that $E^{c}(\ox)=\{0\}$, define
	\begin{align*}
		\cW^{s}(\ox):=\{h:\widetilde{B}^{s}_{(\ox)}(\delta \ell(\ox)^{-1}) &\rightarrow E^{u}(\ox) \ \text{with}\\ \Lnorm{h(0)}{(\ox)}
		&\leq \frac{\delta}{2} \ell(\ox)^{-1}, \ \Lip(h) \leq \frac{1}{10}\}.
	\end{align*}
    If there exists an ${h}'\in\cW^{s}(\Phi^{-1}(\ox))$ such that
	$$\wf_{\Phi^{-1}(\omega,x)}(\graph {h}')\subset \graph h,$$
	then we call ${h}'$ is a backwards graph transformation of $h$ by $ \wf_{\Phi^{-1}(\omega,x)}$, written as $\cT^{s}_{(\ox)}(h):={h}'$.
	\begin{lemma}[\cite{Young17} Lemma 6.4]\label{Lem:GtranS}
		Let $(\omega,x)\in \Gamma$. Assume that $E^{c}(\ox)=0$ and $h\in \cW^{s}(\ox)$. Then
		\begin{enumerate}
			\item[(1)] $\cT^{s}_{(\ox)}(h)$ exists, and unique.
			\item[(2)] for $z_1,z_2\in \graph (\cT^{s}_{(\ox)}(h))$ ,
			$$\Lnorm{\wf_{\Phi^{-1}(\omega,x)}z_1-\wf_{\Phi^{-1}(\omega,x)}z_2}{(\ox)}\leq (e^{-\lambda}+\delta)\Lnorm{z_1-z_2}{\Phi^{-1}(\ox)}.$$
		\end{enumerate}
		(1) and (2) continue to hold for graph transforms by maps which are $C^{1}$ sufficiently close to $\wf_{\Phi^{-1}(\omega,x)}$.
	\end{lemma}

	Note that we do not assume $E^c=\{0\}$ in the following lemma, which is different from \cite[Lemma 6.3]{Young17}.
	\begin{lemma} \label{Lem:CUnstable}
		For each $(\ox)\in \Gamma$ and for every $0<\delta\leq {\delta}'_1$.
		\begin{enumerate}
			\item[(a)]	The following set equivalent to $W_{\delta}^{u}(\ox)$:
			\begin{equation}\label{eq:CofW}
				\begin{aligned}
					\exp_{x}\{v\in \wB_{(\ox)}(\delta \ell(\ox)^{-1}): \forall n \in &\N, \exists v_n \in \wB_{\Phi^{-n}(\ox)}(\delta_1 \ell(\Phi^{-n}(\ox))^{-1}) \\ \text{such that} \
					\widetilde{f}^{n}_{\te^{-n}\omega}v_n=v \ &\text{and} \ \limsup_{n\rightarrow \infty} \frac{1}{n}\log |v_n|\leq -\frac{\lambda}{2}\}.
				\end{aligned}
			\end{equation}
			Additionally, if $E^{c}(\ox)=\{0\}$ then the description that
			$$\limsup_{n\rightarrow \infty} \frac{1}{n}\log |v_n|\leq -\frac{\lambda}{2}$$ in (\ref{eq:CofW}) can be omit.
			\item[(b)] For every $y\in W_{\delta}^{u}(\ox)$ the tangent space $T_{y} W_{\delta}^{u}(\ox)$ characterized by
			\begin{equation}\label{Eq:CofT}
				\{v\in \cB_{y}: \ \forall n \in \N, Df^{-n}_{\omega}(y)v \ \text{exists and} \
				\limsup_{n\rightarrow \infty} \frac{1}{n}\log |Df^{-n}_{\omega}(y)v|\leq -\lambda\}.
			\end{equation}
		\end{enumerate}
	\end{lemma}
	\begin{proof}
		To simplify the notations, write $\wf_{-n}:=\wf_{\Phi^{-n}(\ox)}$, $g_{-n}:=g_{\Phi^{-n}(\ox)}$, $\Lnorm{\cdot}{-n}:=\Lnorm{\cdot}{\Phi^{-n}(\ox)}$ and $\pi^{\tau}_{-n}:=\pi^{\tau}_{\Phi^{-n}(\ox)}$ for $n\geq 0$ and $\tau=u,cs$.

		For item (a), $W^{u}_{\delta}(\ox)$ is a subset of the set \eqref{eq:CofW} follows from Theorem \ref{Thm:Unstable}.
		It suffices to prove the reverse relation holds.

		If there exists $v$ belong to the set \eqref{eq:CofW} but $v\notin \wW^{u}_{\delta}(\ox)$,
		then there exists $u\in \wW^{u}_{\delta}(\ox)$ such that $\pi^{u}_{0}u=\pi^{u}_{0}v$ and $\pi^{cs}_{0}u\neq \pi^{cs}_{0}v$.
		Therefore, for every $n\in \N$ by Corollary \ref{Cor:NlinP} we have
		\begin{equation}\label{eq:L42}
			\Lnorm{u_n-v_n}{-n}=\Lnorm{\pi^{cs}_{-n}(u_n-v_n)}{-n} \geq (e^{\varepsilon_0}+\delta)^{-n}\Lnorm{u-v}{0},
		\end{equation}
		where $u_n\in \wW^{u}_{\delta}(\Phi^{-n}(\ox))$ is the unique point satisfied that $\wf^{n}_{-n}(u_n)=u$.
		Hence, together with $(\ref{eq:norms})$ we have
		$$\liminf_{n\rightarrow\infty}\frac{1}{n} \log |u_n-v_n|\geq -\log(e^{\varepsilon_0}+\delta),$$
		which is a contradiction with
		$$\limsup_{n\rightarrow\infty}\frac{1}{n}\log|u_n|\leq -\frac{\lambda}{2} \ \text{and} \ \limsup_{n\rightarrow\infty}\frac{1}{n}\log|v_n|\leq -\frac{\lambda}{2},$$
		So, we must have $v\in \wW^{u}_{\delta}(\ox)$.

		Moreover, if $E^{c}(\ox)=0$, then by Corollary \ref{Cor:NlinP} for each $n\in \N$ the inequality (\ref{eq:L42}) is
		$$\Lnorm{u_n-v_n}{\Phi^{-n}(\ox)}=\Lnorm{\pi^{s}_{-n}(u_n-v_n)}{-n}\geq (e^{-\lambda}+\delta)^{-n}\Lnorm{u-v}{(\ox)},$$
		which is a contradiction with
		$$\Lnorm{u_n-v_n}{\Phi^{-n}(\ox)} \leq  2\delta_1 \ell(\Phi^{-n}(\ox))^{-1}\leq 2\delta_1e^{n\varepsilon_1} \ell(\ox)^{-1}\quad \forall n\geq 0.$$
		This proves item (a).

		For item (b), fixed $y\in W_{\delta}^{u}(\ox)$.
		Let $y^{x}:=y-x$ and denote by $y^{x}_{-n}\in \wW^{u}_{\delta}(\Phi^{-n}(\ox))$ the unique point satisfies  $\wf^{n}_{-n}y^{x}_{-n}=y^{x}$ for each $n\geq 0$.
		We will use the $|\cdot|'$ to denote the operator norm of bounded linear operators with respect to the Lyapunov norms.

		By Theorem \ref{Thm:Unstable}, for every $n\geq 0$, we have
		$$
		T_{y^{x}_{-n}} \wW_{\delta}^{u}(\Phi^{-n}(\ox))=(Id+Dg_{\Phi^{-n}(\ox)}(y^{x}_{-n}))E^{u}(\Phi^{-n}(\ox)).
		$$
		By item (a), one can choose $n_0\in \N$ such that for every $n\geq n_0$ we have $|y^{x}_{-n}|\leq e^{-\frac{n\lambda}{3}}$.
		Then, for every $n\geq n_0$, by (\ref{eq:lox}) and Theorem \ref{Thm:Unstable}, one has
		\begin{align*}
			|Dg_{-n}(y^{x}_{-n})|'&\leq  C_0 e^{2n\varepsilon_0} \ell(\ox)^{2} |y^{x}_{-n}|\\
			                                 &\leq C_0e^{-n(\frac{\lambda}{3}-2\varepsilon_0)}\ell(\ox)^{2},
		\end{align*}
		and by (\ref{eq:lox}) and Lemma \ref{Lem:NlinP}, one has
		\begin{equation}\label{eq:L43}
			\begin{aligned}
			 |D\wf_{-n}(y^{x}_{-n})-D\wf_{-n}(0)|'&\leq  e^{2n\varepsilon_0} \ell(\ox)^{2} |y^{x}_{-n}|\\
			 &\leq  e^{-n(\frac{\lambda}{3}-2\varepsilon_0)} \ell(\ox)^{2}.
			\end{aligned}
		\end{equation}
        By (\ref{eq:lox}) and the choice of the number ${\delta}'_1$,
        \begin{align*}
        	|D\wf_{-n}(y^{x}_{-n})|'&\leq3\ell(\Phi^{-n+1}(\ox)) \cdot |(f_{\te^{-n} \om}|_{B(A_{\te^{-n} \om},r_0)})|_{C^2}\\
        	                       &\leq 3e^{(n-1)\varepsilon_0}\ell(\ox)\ell(\Phi^{-n}(\ox)) \\
        	                       &\leq 3e^{2n\varepsilon_0}\ell(\ox)^{2}.
        \end{align*}
        Take $c>0$ arbitrarily small.
        Enlarging $n_0$ if necessary such that for every $n\geq n_0$,
        $$\max\{3C_0\ell(\ox)^{2}, C_0, 1\}\cdot e^{-n(\frac{\lambda}{3}-4\varepsilon_0)} \ell(\ox)^{2}\leq \max\{1,\dfrac{c}{2}e^{\lambda}\}.$$
		For each $n\geq n_0$ and each $v=u+Dg_{-n}(y^{x}_{-n})u \in T_{y^{x}_{-n}} \wW_{\delta}^{u}(\Phi^{-n}(\ox))$, where $u\in E^{u}(\Phi^{-n}(\ox))$.
		Since $\Lnorm{Dg_{-n}(y^{x}_{-n})}{}<1$, by the definition of the Lyapunov norms we have $\Lnorm{v}{-n}=\Lnorm{u}{-n}$.
		Then,
		\begin{align*}
			\Lnorm{D\wf_{-n}(y^{x}_{-n})v}{-n+1}\geq&  \Lnorm{D\wf_{-n}(0)u}{-n+1}-\Lnorm{(D\wf_{-n}(y^{x}_{-n})-D\wf_{-n}(0))u}{-n+1}\\-&\Lnorm{D\wf_{-n}(y^x_{-n})Dg_{-n}(y^x_{-n})u}{-n+1}\\
			\geq&e^{\lambda}\Lnorm{u}{-n}-ce^{\lambda}\Lnorm{u}{-n}=(1-c)e^{\lambda}\Lnorm{v}{-n}.
		\end{align*}
	    Therefore, for every $v_0\in T_yW^{u}_{\delta}(\ox)$ and every $n\geq n_0$ we have
	    $$\Lnorm{Df^{-n-1}_{\om}(y)v_0}{-n-1}\leq (1-c)^{-1}e^{-\lambda} \Lnorm{Df^{-n}_{\om}(y)v_0}{-n}.$$
		This together with (\ref{eq:norms}), one has
		$$\limsup_{k\rightarrow \infty}\frac{1}{k}\log|Df^{-k}_{\omega}(y)v_0|\leq -\lambda-\log(1-c).$$
		By $c$ is arbitrariness, this prove that $T_yW^{u}_{\delta}(\ox)$ is a subset of (\ref{Eq:CofT}).

		To prove the reverse holds. We denote $E(\omega,y)$ by
		$$\{v\in \cB_{y}: \ \forall n \in \N, Df^{-n}_{\omega}(y)v \ \text{exists and} \
		\limsup_{n\rightarrow \infty} \frac{1}{n}\log |Df^{-n}_{\omega}(y)v|\leq -\lambda\}.$$
		It is easy to show that $E(\omega,y)$ is a subspace of $\cB_y$ and $Df_{\om}(y)E(\omega,y)=E(\Phi(\omega,y))$.
		Assume that there exists $v \in E(\oy)$ but $v\notin E^{u}(\oy)$. Denote by $v_{-n}:=Df^{-n}_{\omega}(y) v$.
	    By \eqref{eq:L43}, for every $n\geq n_0$ we have
		$$|D\wf_{-n}(y^{x}_{-n})-D\wf_{-n}(0)|'\leq e^{-n(\frac{\lambda}{3}-2\varepsilon_0)} \ell(\ox)^{2}=C(n).$$
        For every $n\geq n_0$, every $u,w \in \cB$ with $\Lnorm{u-w}{-n}=\Lnorm{\pi^{u}_{-n} (u-w)}{-n}$, we have
		\begin{align*}
			&\Lnorm{\pi^{u}_{-n+1}(D\wf_{-n}(y^{x}_{-n})(u-w))}{-n+1}\\
			\geq &\Lnorm{\pi^{u}_{-n+1}(D\wf_{-n}(0)(u-w))}{-n+1}
			-\Lnorm{(D\wf_{-n}(y^{x}_{-n})-D\wf_{-n}(0))(u-w)}{-n+1}\\
			\geq &e^{\lambda}\Lnorm{\pi^{u}_{-n} (u-w)}{-n}-C(n)\Lnorm{(u-w)}{-n}\\
			\geq &(e^{\lambda}-C(n))\Lnorm{(u-w)}{-n},
		\end{align*}
		and similarly, we also have
		\begin{align}\label{eq:LSS}
			\Lnorm{\pi^{cs}_{-n+1}(D\wf_{-n}(y^{x}_{-n})(u-w))}{-n+1}\leq (e^{\varepsilon_0}+C(n)) \Lnorm{(u-w)}{-n}.
		\end{align}
		Take $n_1>n_0$ large enough, such that for every $n\geq n_1$ one has
		$$e^{\lambda}-C(n)> e^{2\varepsilon_0}> e^{\varepsilon_0}+C(n).$$
		For each $n\geq n_1$, if $u,w \in \cB$ with $\Lnorm{u-w}{-n}=\Lnorm{\pi^{u}_{-n} (u-w)}{-n}$ then
		$$\Lnorm{D\wf_{-n}(y^{x}_{-n})(u-w)}{-n+1} =\Lnorm{\pi^{u}_{-n+1}(D\wf_{-n}(y^{x}_{-n})(u-w))}{-n+1}.$$
		Let $u_{-n_1}\in T_{y^{x}_{-n_0}}\wW^{u}(\Phi^{-n_1}(\ox))$ with $\pi^{u}_{-n_1}u_{-n_1}=\pi^{u}_{-n_1}v_{-n_1}$. Note that $v\notin T_{y}W^{u}(\ox)$ implies that $v_{-n_1}\neq u_{-n_1}$.
		Denote by $u_{-n}:=D\wf^{n_1-n}_{-n_1}(y^x_{-n_1})u_{-n_1}$,
		we have $$\Lnorm{u_{-n_1}-v_{-n_1}}{-n_1}=\Lnorm{\pi^{cs}_{-n_1}(u_{-n_1}-v_{-n_1})}{-n_1}>0,$$
		then for every $n>n_1$ one has
		$$\Lnorm{u_{-n}-v_{-n}}{-n}=\Lnorm{\pi^{cs}_{-n}(u_{-n}-v_{-n})}{-n}.$$
		Using inequality (\ref{eq:LSS}) we have that for every $n>n_1$
		$$\Lnorm{u_{-n}-v_{-n}}{-n}\geq e^{-2(n-n_1)\varepsilon_0}\Lnorm{u_{-n_1}-v_{-n_1}}{-n_1}.$$
		Therefore, together with (\ref{eq:norms}) one has
		$$\liminf_{n\rightarrow \infty} \frac{1}{n}\log|u_{-n}-v_{-n}|\geq -2\varepsilon_0.$$
		This is a contradiction with
		$$\limsup_{n\rightarrow\infty}\frac{1}{n}\log|u_{-n}|\leq -\lambda \ \text{and} \ \limsup_{n\rightarrow\infty}\frac{1}{n}\log|v_{-n}|\leq -\lambda.$$
		Hence, one has $v\in E^{u}(\oy)$. This completes the proof.
	\end{proof}

	The following corollary is direct.
	\begin{corollary}\label{Cor:CofT}
		For each $(\ox)\in\Gamma$ and every $(\oy)\in\Gamma$ with $y\in W^{u}(\ox)$. Then, we have $T_y W^{u}(\ox)=E^{u}(\oy)$.
	\end{corollary}

	\subsection{Distortion properties}\label{SEC:Distortion}
	Note that the local unstable manifold $W_{\delta}^{u}(\ox)$ is embedded finite dimensional submanifolds and, the global one is injectively immersed finite dimensional submanifolds for $(\ox)\in \Gamma$.
	Using the notion of induced volumes we introduced in Section \ref{SEC:Induced volumes}, for each $(\ox)\in \Gamma$ we denote by $\nu_{(\ox)}$ the induced volume $\nu_{W^{u}(\ox)}$.

	We also use $E^{u}(\omega,y)$ to denote the tangent space to $W^{u}(\ox)$ at $y\in W^{u}(\ox)$.
	This meaning is clear.
	If the Lyapunov exponents for $(\omega,y)$ exist, then by Corollary \ref{Cor:CofT} the unstable subspace of $(\oy)$ equivalent to the tangent space $T_y W^{u}(\ox)$.
	Note that $f_{\omega}$ maps $W^{u}(\ox)$ diffeomorphically onto $W^{u}(\Phi(\ox))$, we have the relations between the measure $\nu_{\Phi(\ox)}$ and the measure $(f_{\omega})_{\ast}\nu_{(\ox)}$ on $W^{u}(\Phi(\ox))$ are (recall $f^{-1}_{\te\omega}(y)$ is the unique point such that $f_{\om}(f^{-1}_{\te\omega}(y))=y$)
	\begin{equation*}
		\dfrac{d((f_{\omega})_{\ast}\nu_{(\ox)})}{d\nu_{\Phi(\ox)}}(z)=\dfrac{1}{\det(Df_{\omega}(f^{-1}_{\te\omega}(z))|_{E^u(\omega,f^{-1}_{\omega}(z))})} \ \text{for} \ z\in W^{u}(\Phi(\ox)).
	\end{equation*}
	The distortion property of the determinant which we have written below is important for us.
	Let $W^{u}_{\loc}(\ox):=W^{u}_{{\delta}'_1}(\ox)$, where ${\delta}'_1$ be chosen in Theorem \ref{Thm:Unstable}.
	Recall the definition of the uniformly set $\Gamma_{\ell}:=\{(\ox): \ell(\ox)\leq \ell\}$.
	\begin{proposition}\label{Prop:dis}
		For every $\ell\geq 1$, there is a constant $D_{\ell}$ such that for every $(\ox)\in \Gamma_{\ell}$ the following holds:
		\begin{enumerate}
			\item[(a)] For all $y_1,y_2\in W^{u}_{\loc}(\ox)$ and all $n\geq 1$, we have
			\begin{equation}
				 \bigg| \log \dfrac{\det(Df^{n}_{\te^{-n}\om}(f^{-n}_{\om}y_1)|E^u(\te^{-n}\omega,f^{-n}_{\om}y_1))}{\det(Df^{n}_{\te^{-n}\om}(f^{-n}_{\omega}y_2)|E^u(\te^{-n}\om,f^{-n}_{\om}y_2))} \bigg| \leq D_{\ell} |y_1-y_2|.
			\end{equation}
			\item[(b)] For fixed ${x}'\in  W^{u}_{\loc}(\ox)$, the sequence of functions $y \mapsto \log \Delta_{N}({x}',y)$, where
			\begin{equation}\label{eq:Delta n}
				\Delta_{N}({x}',y):= \prod_{k=1}^{N} \dfrac{\det(Df_{\te^{-k}\om}(f^{-k}_{\om}{x}')|E^u(\te^{-k}\om,f^{-k}_{\omega}{x}'))}{\det(Df_{\te^{-k}\om}(f^{-k}_{\om}y)|E^u(\te^{-k}\om,f^{-k}_{\om}y))} \ N=1,2,\cdots
			\end{equation}
			defined for $y\in W^{u}_{\loc}(\ox)$ converges uniformly as $N\rightarrow \infty$ to a Lipschitz function with constant $\leq D_{\ell}$ in the $|\cdot|$ norm.
		\end{enumerate}
	\end{proposition}
	For certain maps on Banach spaces, the distortion property is given by \cite[Proposition 6.8]{Young17}, and the same method can be carried out in the random case.
	We give the detailed proofs as follows.

	Fixed $(\ox)\in \Gamma$, for ease notations, we denote by $\wf_{-k}:=\wf_{\Phi^{-k}(\ox)}$, $\Lnorm{\cdot}{-k}:=\Lnorm{\cdot}{\Phi^{-k}(\ox)}$,
	$g_{-k}:=g_{\Phi^{-k}(\ox)}$, $\ell(-k):=\ell(\Phi^{-k}(\ox))$ for $k\in \N$, and denote by $|\cdot|'$ the operator norm of bounded linear operators with respect to the Lyapunov norms.
	For $k\in \Z$ and a finite-dimensional subspace $E\subset \cB_{f^{-k}_{\om}(x)}$, we denote by $m'_{E}$ the induced volume on $E$ from the Lyapunov norm $\Lnorm{\cdot}{-k}$,
	and we use the notion ``$\det'$" to denote the determinant with respect to these volumes.

	\begin{lemma}[\cite{Young17}, Proposition 6.9]\label{Lem:Distortion}
		For any $\ell>0$, there is a constant $D'_{\ell}$ with the following property.
		Fixed $(\ox)\in \Gamma$.
		Then for any $z^1,z^2\in \graph g_{0}$ with $\Lnorm{z^1-z^2}{0}\leq \delta'_1 (D'_{\ell(0)})^{-1}$ and any $n\in \N$ we have that
		\begin{align*}
			\bigg| \log  \dfrac{\det'(D\wf^n_{-n}(z^1_{-n})|E^1_{-n})}{\det'(D\wf^n_{-n}(z^2_{-n})|E^2_{-n})} \bigg|\leq D'_{\ell(0)}\Lnorm{z^1-z^2}{0},
		\end{align*}
		where $z^i_{-n}$ is the unique point in $\graph g_{-n}$ with $D\wf^n_{-n}(z^i_{-n})=z^i$, and $E^i_{-n}$ is the tangent space to $\graph g_{-n}$ at $z^i_{-n}$.
	\end{lemma}
	\begin{proof}
		Fixed $n\in \N$.
		By Proposition \ref{Prop:Det}, for arbitrary $z^1,z^2\in \graph g_{0}$ we have
		\begin{equation}\label{eq:4.51}
			 \dfrac{\det'(D\wf^n_{-n}(z^1_{-n})|E^1_{-n})}{\det'(D\wf^n_{-n}(z^2_{-n})|E^2_{-n})}
			=\prod_{k=1}^{n} \dfrac{\det'(D\wf_{-k}(z^1_{-k})|E^1_{-k})}{\det'(D\wf_{-k}(z^2_{-k})|E^2_{-k})}.
		\end{equation}

		We will use the following refinement version of Proposition \ref{Prop:CofDet}, which emphasizes the dependence of the constants and can be referred to \cite[Remark 3.16]{Young17}.
		\begin{lemma}[\cite{Young17} Lemma 6.10]\label{Lem:CofDet}
			For any $m\in \N$, there is a constant $C_m>1$ with the following property.
			Let $(\cB_1,|\cdot|_1)$, $(\cB_2,|\cdot|_2)$ be Banach spaces and fixed $M>1$.
			Assume $A_1,A_2:\cB\rightarrow \cB$ are two injective bounded linear operators and $E_1,E_2$ are $m$-dimensional subspaces of $\cB_1$ satisfied that
			\begin{align*}
				|A_j|&, |(A_j|_{E_j})^{-1}|\leq M \quad \text{for} \ j=1,2, \\
				|A_1-A_2|&, d_H(E_1,E_2)\leq \dfrac{1}{C_m M^{10m}},
			\end{align*}
			then, we have
			$$|\log \dfrac{\det(A_1|_{E_1})}{\det(A_2|_{E_2})}|\leq C_m M^{10m}(|A_1-A_2|+d_H(E_1,E_2)).$$
		\end{lemma}

		Let $m_u:=\dim E^{u}(\ox)$.
		For each fixed $1\leq k\leq n$, we choose $M_k$ so that $D\wf_{-k}(z^i_{-k})$, $E^i_{-k}$, $i=1,2$ and $C_{m_u}$ satisfy the hypotheses of Lemma \ref{Lem:CofDet}. We need the following:
		\begin{align}
			&\Lnorm{D\wf_{-k}(z^i_{-k})}{},
			 \Lnorm{(D\wf_{-k}(z^i_{-k})|_{E^i_{-k}})^{-1}}{}\leq M_k \ \text{for} \ i=1,2, \label{eq:111} \\
			&\Lnorm{D\wf_{-k}(z^1_{-k})-D\wf_{-k}(z^2_{-k})}{}
			, d'_H(E^1_{-k},E^2_{-k})\leq C_{m_u} M^{10m_u},\label{eq:222}
		\end{align}
		where $d'_H$ refers to the Hausdorff distance in the adapted norm $\Lnorm{\cdot}{-k}$.

		First we choose $M_k$ so that $\eqref{eq:111}$ holds.
		Note that $\Lnorm{(D\wf_{-k}(z^i_{-k})|_{E^i_{-k}})^{-1}}{}\leq 1$ for $i=1,2$.
		By the choice of the function $\ell(\ox)$, for $i=1,2$ we have
		$$\Lnorm{D\wf_{-k}(z^i_{-k})}{}\leq 3\ell(-k)\cdot |(f_{\te^{-k}\om}|_{B(A_{\te^{-k}\om},r_0)})|_{C^2}\leq 3e^{2k\varepsilon_1}\ell(0)^{2}.$$
		So, on setting $M_k:=3e^{2k\varepsilon_1}\ell(0)^{2}$, $\eqref{eq:111}$ satisfied.

		Next, we estimate the two terms on $\eqref{eq:222}$.
		By Lemma \ref{Lem:NlinP} and Theorem $\ref{Thm:Unstable}$ (3) we have
		\begin{align*}
			\Lnorm{D\wf_{-k}(z^1_{-k})-D\wf_{-k}(z^2_{-k})}{}&\leq \ell(k) \cdot\Lnorm{z^1_{-k}-z^2_{-k}}{-k}\\
			&\leq  (\dfrac{e^{\varepsilon_1}}{e^{\lambda}-\delta_1})^k\cdot \ell(0) \cdot \Lnorm{z^1-z^2}{0}.
		\end{align*}
		By a simple computation and Theorem $\ref{Thm:Unstable}$ (2),
		\begin{align*}
			d'_H(E^1_{-k},E^2_{-k})&\leq 2|Dg_{-k}(u^1_{-k})-Dg_{-k}(u^2_{-k})|'\\
			&\leq  2C_0\ell(-k)\cdot\Lnorm{u^1_{-k}-u^2_{-k}}{-k}\\
			&=     2C_0\ell(-k)\cdot\Lnorm{z^1_{-k}-z^2_{-k}}{-k}\\
			&\leq  (\dfrac{e^{\varepsilon_1}}{e^{\lambda}-\delta_1})^k\cdot 2C_0\ell(0) \cdot \Lnorm{z^1-z^2}{0},
		\end{align*}
		where $z^i_{-k}=u^i_{-k}+g_{-k}(u^i_{-k})$ for some $u^i_{-k}\in E^u(\Phi^{-k}(\ox))$ and $i=1,2$.
		To fulfill the inequality $\eqref{eq:222}$, we take $\Lnorm{z^1-z^2}{0}$ small enough so that
		$$\max\{2C_0,1\}(\dfrac{e^{\varepsilon_1}}{e^{\lambda}-\delta_1})^k\cdot \ell(0) \cdot \Lnorm{z^1-z^2}{0}\leq( C_{m_u} M^{10m_u}_{k})^{-1}.$$
		That is,
		$$\Lnorm{z^1-z^2}{0}\leq (3^{10m_u}C_{m_u}\ell(0)^{20m_u+1} \max\{2C_0,1\}(\dfrac{e^{(20m_u+1)\varepsilon_1}}{e^{\lambda}-\delta'_1})^k)^{-1}.$$
		By the choice of $\varepsilon_1\ll \lambda$ one may assume $e^{(20m_u+1)\varepsilon_1}<e^{\lambda}-\delta'_1$.
		So, we take $D'_{\ell(0)}$ large enough for which
		$$D'_{\ell(0)}\geq \delta'_1 3^{10m_u}C_{m_u}\ell(0)^{20m_u+1} \max\{2C_0,1\},$$
		one has \eqref{eq:222} holds for every $z^1,z^2\in \graph g_{0}$ with $\Lnorm{z^1-z^2}{0}\leq \delta'_1 (D'_{\ell})^{-1}$.

		Therefore, by Lemma \ref{Lem:CofDet} for every $z^1,z^2\in \graph g_{0}$ with $\Lnorm{z^1-z^2}{0}\leq \delta'_1 (D'_{\ell(0)})^{-1}$ and every $1\leq k\leq n$ we have
		\begin{equation}\label{eq:45.2}
			\begin{aligned}
			\bigg| \log	\dfrac{\det'(D\wf_{-k}(z^1_{-k})|E^1_{-k})}{\det'(D\wf_{-k}(z^2_{-k})|E^2_{-k})}\bigg|\leq
			K' \ell(0)^{20m_u+1} \Lnorm{z^1-z^2}{0}(\frac{e^{(20m_u+1)\varepsilon_1}}{e^{\lambda}-\delta'_1})^k,
			\end{aligned}
		\end{equation}
	    where $K':=C_{m_u}3^{10m_u}(2C_0+1)$.
	    Together with \eqref{eq:4.51}, one has
	    \begin{align*}
	    	\bigg| \log  \dfrac{\det'(D\wf^n_{-n}(z^1_{-n})|E^1_{-n})}{\det'(D\wf^n_{-n}(z^2_{-n})|E^2_{-n})} \bigg| &\leq K' \ell(0)^{20m_u+1} \Lnorm{z^1-z^2}{0} \sum_{k=1}^{n}(\frac{e^{(20m_u+1)\varepsilon_1}}{e^{\lambda}-\delta'_1})^k\\
	    	&\leq K''\ell(0)^{20m_u+1}\Lnorm{z^1-z^2}{0},
	    \end{align*}
	    where $K''$ is independent of $n$ and $(\ox)$.
	    By increasing $D'_{\ell(0)}$ once more so that $D'_{\ell(0)}\geq K'\ell(0)^{20m_u+1}$, the statement of Lemma \ref{Lem:Distortion} follows.
	\end{proof}
    We now give the proof of Proposition \ref{Prop:dis}.
    \begin{proof}[proof of Proposition \ref{Prop:dis}]
    	Fixed $(\ox)\in \Gamma_{\ell}$. For $y^1,y^2\in W^{u}_{\loc}(\ox)$.
    	We denote by $z^i:=\exp_{x}^{-1}y_i$ and $y^{i}_{-k}:=\fo^{-k}y_i$ for $k\in \N$ and $i=1,2$, and we will use the same notation as in \ref{Lem:Distortion}, such as $z^i_{-n}$ and $E^{i}_{-n}$.
    	Since $\ell(\ox)\leq \ell$, by the choice of $D'_{\ell(0)}$ one may assume $D'_{\ell}\geq D'_{\ell(0)}$ so that Lemma \ref{Lem:CofDet} also holds for $D'_{\ell}$.

    	For item (a), we first assume that $y^1,y^2\in W^{u}_{\loc}(\ox)$ with $\Lnorm{z^1-z^2}{0}\leq \delta'_1 (D'_{\ell})^{-1}$.
    	For fixed $n\geq 1$, we have
    	$$ \dfrac{\det(D\wf^n_{-n}(z^1_{-n})|E^1_{-n})}{\det(D\wf^n_{-n}(z^2_{-n})|E^2_{-n})}=
    	\underbrace{ \dfrac{dm_{E^{1}_{0}}/dm'_{E^{1}_{0}} }{dm_{E^{2}_{0}}/dm'_{E^{2}_{0}}}}_{I}
    	\underbrace{ \dfrac{dm_{E^{1}_{-n}}/dm'_{E^{1}_{-n}} }{dm_{E^{1}_{-n}}/dm'_{E^{1}_{-n}}}}_{II}
    	\underbrace{\dfrac{\det'(D\wf^n_{-n}(z^1_{-n})|E^1_{-n})}{\det'(D\wf^n_{-n}(z^2_{-n})|E^2_{-n})}}_{III}.$$

    	For $i=1,2$, let $z^i=u^i_{0}+g_{-k}(u^i_{0})$ for some $u^i_{0}\in E^u(\ox)$.
    	For term $ I $, observe that if $L:E^u(\ox)\rightarrow E^{cs}(\ox)$ is a linear map with $\Lnorm{L}{0}\leq 1$, then the definition of $\Lnorm{\cdot}{0}$ implies that $Id+L$ is an isometry on $E^u(\ox)$ with respect to the norm $\Lnorm{\cdot}{0}$.
    	By the definition of induced volumes, one has
    	$$\dfrac{dm_{(Id+L)E^{u}(\ox)}}{dm'_{(Id+L)E^{u}(\ox)}}=
    	  \dfrac{m_{(Id+L)E^{u}(\ox)}((Id+L)B)}{m'_{(Id+L)E^{u}(\ox)}((Id+L)B)}=
    	  \dfrac{m_{(Id+L)E^{u}(\ox)}((Id+L)B)}{m'_{E^{u}(\ox)}(B)},$$
    	where $B$ is the unit closed ball of $E^{u}(\ox)$.
    	Then, since $\Lip g_0\leq 1$, the term $ I $ is
    	$$I=\dfrac{\det((Id+Dg_0(u^1_{0}))|E^{u}(\ox))}{\det((Id+Dg_0(u^2_{0}))|E^{u}(\ox))},$$
    	where the determinants involved are in the natural norm $|\cdot|$.

    	We now estimate $I$ by Lemma \ref{Lem:CofDet}.
    	On the one hand
    	\begin{align*}
    		|Id+Dg_0(u^1_{0})|&\leq 3\ell \Lnorm{Id+Dg_0(u^1_{0})}{}=3\ell,\\
    		|(Id+Dg_0(u^1_{0})|_{E^{u}(\ox)})^{-1}|&\leq 3\ell
    		\Lnorm{(Id+Dg_0(u^1_{0})|_{E^{u}(\ox)})^{-1}}{}\leq 3\ell.
    	\end{align*}
        So, we take $M=3\ell$.
        On the other hand, by Theorem \ref{Thm:Unstable} (ii)
       \begin{align*}
       	|Dg_0(u^1_{0})-Dg_0(u^2_{0})|&\leq 3\ell\Lnorm{Dg_0(u^1_{0})-Dg_0(u^2_{0})}{}\\
       	&\leq 3C_0\ell^2\Lnorm{u^1_{0}-u^2_{0}}{0}\\
       	&=3C_0\ell^2\Lnorm{z^1-z^2}{0}.
       \end{align*}
       So, it suffices to enlarge $D'_{\ell}$ if necessary for which $D'_{\ell}\geq (3\ell)^{10m_u+1}\ell\delta'_1C_0C_{m_u}$.
       Then, by Lemma \ref{Lem:CofDet} we have
       $$|\log I|\leq C_{m_u}C_03^{10m_u+1}\ell^{10m_u+2}\Lnorm{z^1-z^2}{0}=K'''\ell^{10m_u+2}\Lnorm{z^1-z^2}{0}.$$

       Using the same argument, term $II$ has the following estimate
       \begin{equation}\label{eq:45.3}
       	\begin{aligned}
       	|\log II|&\leq K'''\ell(-n)^{10m_u+2}\Lnorm{z^1_{-n}-z^2_{-n}}{-n}\\
       	&\leq K'''\ell(0)^{10m_u+2} (\dfrac{e^{(10m_u+2)\varepsilon_1}}{e^{\lambda}-\delta'_1})^n \Lnorm{z^1-z^2}{0}\\
       	&\leq K'''\ell^{10m_u+2}\Lnorm{z^1-z^2}{0}.
       	\end{aligned}
       \end{equation}

       By Lemma \ref{Lem:Distortion}, we have
       $$|\log III|\leq D'_{\ell}\Lnorm{z^1-z^2}{0}.$$
       Therefore, take $D_{\ell}\geq \ell(2K'''\ell^{10m_u+2}+D'_{\ell})$.
       This completes the proof of item (a) for $y^1,y^2\in W^{u}_{\loc}(\ox)$ with $\Lnorm{z^1-z^2}{0}\leq \delta'_1 (D'_{\ell})^{-1}$.

       For $y^1,y^2\in W^{u}_{\loc}(\ox)$ but $\Lnorm{z^1-z^2}{0}>\delta'_1 (D'_{\ell})^{-1}$, we only need to choose a finite sequence of points $\{u'_i\}^{N}_{i=0}$ on the line segment with the endpoints $u^1_{0}$ and $u^2_{0}$, so that $u'_0=u^1_{0}$,
       $u'_N=u^2_{0}$ and $\Lnorm{u'_{i}-u'_{i-1}}{0}\leq \delta'_1 (D'_{\ell})^{-1}$ for any $1\leq i\leq N$.
       Take $\hat{z}^{i}=u'_{i}+g_0u'_{i}$, then
       $\Lnorm{\hat{z}^{i}-\hat{z}^{i-1}}{0}=\Lnorm{u'_{i}-u'_{i-1}}{0}\leq \delta'_1 (D'_{\ell})^{-1}$ for any $1\leq i\leq N$. Using the argument above one has
       \begin{align*}
       	\bigg| \log \dfrac{\det(D\wf^n_{-n}(z^1_{-n})|E^1_{-n})}{\det(D\wf^n_{-n}(z^2_{-n})|E^2_{-n})}\bigg|
       	&=\bigg|\log\prod_{i=1}^{N} \dfrac{\det(D\wf^n_{-n}(\hat{z}^{i}_{-n})|\hat{E}^{i}_{-n})}{\det(D\wf^n_{-n}(\hat{z}^{i-1}_{-n})|\hat{E}^{i-1}_{-n})}\bigg|\\
       	&\leq D_{\ell}(\sum_{i=1}^{N}|\hat{z}^{i}-\hat{z}^{i-1}|)
       	=D_{\ell}|z^1-z^2|,
       \end{align*}
     where $\hat{z}^i_{-n}$ is the unique point in $\graph g_{-n}$ with
     $D\wf^n_{-n}(\hat{z}^i_{-n})=\hat{z}^i$, and $\hat{E}^i_{-n}$ is the tangent space to $\graph g_{-n}$ at $\hat{z}^i_{-n}$ for $0\leq i\leq N$.
     This completes the proof of item (a).

     For item (b), it suffices to show that the sequence of functions $y \mapsto \log \Delta_{k}({x}',y), k\in \N$ is a Cauchy sequence. Note that for $M<N$,
     $$\bigg| \log \dfrac{\Delta_{N}({x}',y)}{\Delta_{M}({x}',y)}\bigg|= \bigg| \log \prod_{k=M+1}^{N} \dfrac{\det(Df_{\te^{-k}\om}(f^{-k}_{\om}{x}')|E^u(\te^{-k}\om,f^{-k}_{\omega}{x}'))}{\det(Df_{\te^{-k}\om}(f^{-k}_{\om}y)|E^u(\te^{-k}\om,f^{-k}_{\om}y))}\bigg|.$$
     Using the estimations in \eqref{eq:45.2} and \eqref{eq:45.3} one can show the right term above is upper-bounded by
     \begin{align*}
     	&\leq (2K'''\ell(\ox)^{10m_u+2}+K'\ell(\ox)^{20m_u+1})\cdot 3\sum_{i=M}^{N}(\dfrac{e^{(20m_u+1)\varepsilon_1}}{e^{\lambda}-\delta'_1})^i \Lnorm{x'-y}{0}\\& \leq(2K'''\ell(\ox)^{10m_u+1}+K'\ell(\ox)^{20m_u})\cdot 6\sum_{i=M}^{N}(\dfrac{e^{(20m_u+1)\varepsilon_1}}{e^{\lambda}-\delta'_1})^i \delta'_1
     \end{align*}
     for every $y\in W^{u}_{\loc}(\ox)$. This completes the proof of item (b).
    \end{proof}

	\subsection{Unstable stacks of local unstable manifolds}\label{SEC:4.2}
	Now, we discuss the regularity of local unstable manifolds.
	To avoid using more than one norm which depends on the point, we let $B^{\tau}_{(\ox)}(r)=\{v\in E^{\tau}(\ox):|v|\leq r\}$ for $\tau=u,cs$.

	Recall the definitions of the uniformly set $\Gamma_{\ell}:=\{(\ox): \ell(\ox)\leq \ell\}$ and the $\om$-section $(\Gamma_{\ell})_{\om}:=\{x\in\cB:(\ox)\in \Gamma_{\ell}\}$.
	\begin{lemma}\label{Unstacks}
		Let $\ell_0$ be fixed so that $\mu(\Gamma_{\ell_0})>0$, and fixed $(\om, x_{\om})\in \Gamma_{\ell_0}$.
		For every $r>0$ and every $x \in (\Gamma_{\ell_0})_{\om}$ we let
		$$U_{\omega}(x,r):=(\Gamma_{\ell_0})_{\omega} \cap \{z : |z-x| \leq r\},$$
		and let $g_{(\ox)}$ be as in Theorem \ref{Thm:Unstable}.
		Then, for each $0<\delta<{\delta}'_1$ sufficiently small, there exists $r_1(\ell_0,\delta,\omega)>0$ such that
		\item[(1)]
		for $y\in U_{\omega}(x_{\om},r_1)$, there exists a continuous function
		$g_{(\om,x_{\om})}^{y}: B^{u}_{(\om,x_{\om})}(2\delta \ell_0^{-3}) \rightarrow E^{cs}(\om,x_{\om})$
		such that
		\begin{align*}
		\exp_{y}\graph(g_{(\oy)}|_{B^{u}_{(\oy)}(\delta \ell_{0}^{-3})})
		&\subset
		\exp_{x_{\om}}\graph(g_{(\om,x_{\om})}^{y})\\
		&\subset
		\exp_{y}\graph(g_{(\oy)}|_{B^{u}_{(\oy)}(4\delta \ell_{0}^{-3})}).
		\end{align*}
		\item[(2)]
		the mapping $\Theta_{\omega}:U_{\omega}(x,r_0)\rightarrow C(B_{(\omega, x_{\om})}^{u}(2\delta \ell_0^{-3}),E^{cs}(\om,x_{\om}))$ defined by $\Theta_{\omega}(y)=g_{(\om,x_{\om})}^{y}$ is continuous with respect to the $C^0$ norm $||\cdot||$.
	\end{lemma}
	\begin{proof}
		With a minor modification of the proof of \cite[Lemma 6.5]{Young17}, we give the proof of this lemma in the following.

		Let $(\ox),(\oy)\in \Gamma_{\ell_0}$.
		For ease notations, we denote by $g_y:=g_{(\oy)}$, $E^{\tau}_y:=E^{\tau}(\oy)$, $\pi^{\tau}_{y}:=\pi^{\tau}_{(\oy)}$ and
		$W^{u}_{x,\delta,y}:=\exp^{-1}_x \exp_y \graph(g_y|_{\wB^{u}_{(\oy)}(\delta \ell^{-2}_{0})})$ for $\tau=u,cs$ and $\delta>0$.

		We first show that, $\pi^{u}_{x}$ is invertible on $W^{u}_{x,4\delta,y}$ when $|x-y|$ and $\delta$ are small enough.
		This is guaranteed if for all $u_1, u_2\in W^{u}_{x,4\delta,y}$, one has $|\pi^{cs}_{x}(u_1-u_2)|<|u_1-u_2|$.
		By Theorem \ref{Thm:Unstable}, for every $u_1, u_2\in W^{u}_{x,4\delta,y}$ we have
		$$|\pi^{cs}_{x}(u_1-u_2)|\leq (|\pi^{cs}_x|_{E^{u}_y}|+|\pi^{cs}_x|_{E^{cs}_y}|\cdot \Lip(g_y))\cdot|u_1-u_2|,$$
		where $\Lip(g_y)$ is the Lipschitz constant of $g_y$ with respect to the norm $|\cdot|$.
		We show that $\Lip(g_y)\ll 1$ when $\delta$ small enough as follows: for every $u_1,u_2\in W^{u}_{x,4\delta,y}$,
		\begin{align*}
			|g_yu_1-g_yu_2|&\leq 3 \Lnorm{g_yu_1-g_yu_2}{(\oy)}\\
			               &\leq 3{\Lip}'_{y}(g_y) \cdot \Lnorm{u_1-u_2}{(\oy)} \\
			               &\leq 3\ell_{0} {\Lip}'_{y}(g_y) \cdot |u_1-u_2|\\
			               &\leq 3\ell_{0} (\sup_{v\in \wB^{u}_{(\oy)}(4\delta\ell_0^{-2})} \Lnorm{Dg_y (v)}{}) \cdot |u_1-u_2|\\
			               &\leq 12\delta \ell_{0}^{-1} {\Lip}'_{y}(Dg_y) \cdot |u_1-u_2|,
		\end{align*}
		where ``${\Lip}'_{y}$" is the Lipschitz constant with respect to the norm $\Lnorm{\cdot}{(\oy)}$.
		By Theorem \ref{Thm:Unstable}, we have ${\Lip}'_{y}(Dg_y)\leq C_0\ell_0$.
		Therefore, we have $\Lip(g_y)\leq 12C_0 \delta$ which we may assume $\ll 1$.

		By Lemma \ref{Lem:ProDh}, we have $|\pi^{cs}_x|_{E^{u}_y}|\rightarrow 0, |\pi^{cs}_x|_{E^{cs}_y}| \rightarrow 1$ whenever
		$d_H(E^u_y,E^u_x)\rightarrow 0$ and $d_H(E^{cs}_y,E^{cs}_x)\rightarrow 0$.
		Thus, with Proposition \ref{Prop:CCS}, take ${r}'_1(\ell_0, \om)$ small enough such that for every $x, y\in U_{\om}(x_{\om},{r}'_1)$, $d_H(E^u_y,E^u_x)$ and $d_H(E^{cs}_y,E^{cs}_x)$ are sufficiently small so that
		$$|\pi^{cs}_x|_{E^{u}_y}|+|\pi^{cs}_x|_{E^{cs}_y}|\cdot \Lip(g_y)<1.$$

		We define the function $g^{y}_{(\ox)}$ by $g^{y}_{(\ox)}(u):=\pi^{cs}_x((\pi^{u}_x)^{-1}(u))$ for $u\in \pi^{u}_x(W^{u}_{x,4\delta,y})$, where $(\pi^{u}_x)^{-1}(u)$ is the unique point in $W^{u}_{x,4\delta,y}$ such that $\pi^{u}_x((\pi^{u}_x)^{-1}(u))=u$.

		Take $r_1(\ell_0,\delta,\om)<{r}'_1$ small enough, so that for every $x, y\in U_{\om}(x_{\om},r_1)$ one has
		\begin{align*}
			 \pi^u_x (\exp^{-1}_x\circ \exp_y\graph(g_y|_{B^{u}_{(\oy)}(\delta \ell_{0}^{-3})}))
			&\subset B^{u}_{(\ox)}(3\delta \ell_0^{-3})\\
			&\subset \pi^u_x(\exp^{-1}_x\circ \exp_y\graph(g_y|_{B^{u}_{(\oy)}(4\delta \ell_{0}^{-3})})).
		\end{align*}
	    Then, by definition $g^{y}_{(\ox)}$	is well-defined and continuous on $B^{u}_{(\ox)}(3\delta \ell_0^{-3})$.

		In particularly, for each $y\in U_{\om}(x_{\om},r_1)$ we consider the function $g^{y}_{(\om,x_{\om})}$ which defined on $B^{u}_{(\om,x_{\om})}(2\delta \ell_0^{-3})$, then by definition one has $g^{y}_{(\om,x_{\om})}$ satisfying item (1).

		For item (2), for each $x\in U_{\omega}(x_{\om},r_1)$, let $y^n\in U_{\omega}(x_{\om},r_1), n\in \N$ be a sequence with $y^n \rightarrow x$ as $n\rightarrow \infty$.
		Shirking $r_1(\ell_0,\delta,\om)$ if necessary, such that for every $x, y\in U_{\om}(x_{\om},r_1)$ one has
		$$\exp_{x_{\omega}} \graph(g^{y}_{(\om,x_{\om})}|_{B^{u}_{(\om,x_{\om})}(2\delta \ell_0^{-3})})\subset
		  \exp_{x} \graph( g^{y}_{(\ox)}|_{B^{u}_{(\ox)}(3\delta \ell_0^{-3})}).$$
		Therefore, to prove $||\Theta_{\om}(y^n)-\Theta_{\om}(x)||\rightarrow 0$ as $n\rightarrow \infty$.
		It suffices to prove $||g^{y^n}_{(\ox)}-g_{(\ox)}||\rightarrow 0 $ on $B^{u}_{(\ox)}(3\delta \ell_0^{-3})$ as $n\rightarrow \infty$.
		Let $\gamma>0$ be fixed, we prove that $||g^{y^n}_{(\ox)}-g_{(\ox)}||<\gamma$ for $n$ large enough.

		For $k\in \N$, denote by $x_{-k}=f^{-k}_{\omega}x$ and $y^n_{-k}=f^{-k}_{\omega}y^n$.
	    For each $n,k\in \N$,
		let
		$$\textbf{0}_{y^n_{-k}}: \wB^{u}_{\Phi^{-k}(\omega,y^n)}({\delta}'_1 \ell(\Phi^{-k}(\omega,y^n))^{-1})\rightarrow E^{cs}(\Phi^{-k}(\omega,y^n))$$
		be the function that is identically equal to $0$, and let
		$$\phi_{y^n_{-k}}:=\cT_{\Phi^{-1}(\omega,y^n)}\circ\cdots\circ \cT_{\Phi^{-k}(\omega,y^n)} \textbf{0}_{y^n_{-k}},$$
		where $\cT$ is the graph transformation as in Lemma \ref{Lem:Gtran}.
		Likewise, define $\phi_{x_{-k}}$.

		Note that for fixed $k\in \N$, we have $y_{-k}^n, x_{-k}\in (\Gamma_{\ell_0e^{k\varepsilon_1}})_{\te^{-k}\omega}$ for every $n\in \N$.
	    By Proposition \ref{Prop:CCS}, for fixed $k$ one has $E^u(\te^{-k}\omega,y_{-k}^n)\rightarrow E^u(\te^{-k}\omega,x_{-k})$ as $n\rightarrow \infty$.
	    This implies that
	    $$\exp_{y^n_{-k}} B^{u}_{\Phi^{-k}(\om,y^n)}(3\delta \ell_0^{-3}) \rightarrow \exp_{x_{-k}} B^{u}_{\Phi^{-k}(\ox)}(3\delta \ell_0^{-3}) \ \text{as} \ n\rightarrow \infty,$$
	    and so, their $f_{\te^{-k} \om}^{k}$-images as well.
	    Then, one can define the map $\phi^{y^n_{-k}}_{(\ox)}$ on $B^{u}_{(\ox)}(3\delta \ell_0^{-3})$ as in item (1) when $n$ large enough and
	    $||\phi_{x_{-k}}-\phi^{y^n_{-k}}_{(\ox)}||\rightarrow 0$ on $B^{u}_{(\ox)}(3\delta \ell_0^{-3})$ as $n\rightarrow \infty$.

	    Recall the uniform equivalence of the $|\cdot|$ and $\Lnorm{\cdot}{(\oy)}$ on $\Gamma_{\ell_0}$.
		By Lemma \ref{Lem:Gtran}, there exists a $k_0$ large enough such that for all $n$ large enough we have
		$$||\phi^{y^n_{-k_0}}_{(\ox)}-g_{(\ox)}^{y^n}||<\gamma/3 \ \text{and}\
		  ||\phi_{x_{-k_0}}-g_{(\ox)}^{x}||<\gamma/3.$$
		To finish, we estimate $||g_{(\ox)}^{y^n}-g_{(\ox)}^{x}||$ by
		\begin{align*}
			||g_{(\ox)}^{y^n}-g_{(\ox)}^{x}||\leq ||g_{(\ox)}^{y^n}-\phi^{y^n_{-k_0}}_{(\ox)}||+||\phi^{y^n_{-k_0}}_{(\ox)}-\phi_{x_{-k_0}}||+||g_{(\ox)}^{x}-\phi_{x_{-k_0}}||.
		\end{align*}
		Since the first and third terms above are $<\gamma/3$, we choose $n$ large enough such that the middle term is $< \gamma/3$.
		This completes the proof.
	\end{proof}
	We referred to sets of the form
	\begin{equation}\label{eq:stack}
		\cS(\om,x_{\om})=\bigcup_{y\in \bar{U}_{\omega}} \exp_{x_{\om}}(\graph \Theta_{\omega}(y)),
	\end{equation}
	where $x_{\om},U_{\omega}(x_{\om},r_0)$ and $\Theta$ as in above and $\bar{U}_{\omega}\subset U_{\omega}(x_{\om},r_0)$ is a compact subset,
	as a \textit{unstable stack of local unstable manifolds} through the point $(\om,x_{\om})$.
	By Lemma \ref{Unstacks} (2), the set $\cS(\om,x_{\om})$ is a compact subset of $\cB$.
	\begin{lemma}\label{UStaL}
		For fixed $(\om,x_{\om})\in \Gamma_{\ell_0}$. Let
		$\Theta_{\omega}$ be defined in Lemma \ref{Unstacks}.
		For $y_i \in U_{\omega}(x_{\om},r_1), i=1,2$ with
		$$\exp_{x_{\om}}(\graph \Theta_{\omega}(y_1))\neq \exp_{x_{\om}}(\graph \Theta_{\omega}(y_2)).$$
		Then, we have
		$$\exp_{x_{\om}}(\graph \Theta_{\omega}(y_1)) \cap \exp_{x_{\om}}(\graph \Theta_{\omega}(y_2))=\varnothing.$$
	\end{lemma}
	\begin{proof}
		Denote by $G_1:=\exp_{x_{\om}}(\graph \Theta_{\omega}(y_1))$ and $G_2:=\exp_{x_{\om}}(\graph \Theta_{\omega}(y_2))$.
		If there exists $z \in G_1\cap G_2$, then $z$ contained in some local unstable manifolds of $(\omega,y_1)$ and $(\omega,y_2)$.
		For each ${z}'\in G_1$, by Lemma \ref{Lem:CUnstable} one has
		\begin{align*}
			\limsup_{n\rightarrow\infty}\frac{1}{n}
			\log|\fo^{-n}({z}')- \fo^{-n}(y_2)|
			&\leq \limsup_{n\rightarrow\infty} \frac{1}{n}\log(|\fo^{-n}({z}')- \fo^{-n}(y_1)|\\
		   +&|\fo^{-n}(z)- \fo^{-n}(y_1)|+|\fo^{-n}(z)- \fo^{-n}(y_2)|) \leq -\frac{\lambda}{2}.
		\end{align*}
		Therefore, also by Lemma \ref{Lem:CUnstable} one can show that ${z}'$ is contained in some local unstable manifold of $(\omega,y_2)$.
		This implies that ${z}'\in G_2$ and so, $G_1 \subset G_2$.
		Symmetrically, we also have $G_1 \supset G_2$.
		This completes the proof.
	\end{proof}
	\begin{lemma}\label{GUStacks}
		Let $\cS(\omega,x_{\om})$ be defined above, and let $\Sigma_{(\omega,x_{\om})}:=\exp_{x_{\om}}^{-1}\cS(\omega,x_{\om}) \cap E^{cs}(\omega,x_{\om})$.
		Then $\cS(\omega,x_{\om})$ is homeomorphic to
		$\Sigma_{(\omega,x_{\om})}\times B^{u}_{(\omega,x_{\om})}(2\delta \ell^{-3}_0)$
		under the mapping $\Psi(\sigma,u):=\exp_{x_{\omega}}(u+g^{\omega}_{\sigma}(u))$,
		where $g^{\omega}_{\sigma}=\Theta_{\omega}(y)$ is the unique leaf of $\cS_{\omega}$ such that $\Theta_{\omega}(y)(0)=\sigma$.
	\end{lemma}
	\begin{proof}
		By Lemma \ref{UStaL}, the mapping $g^{\omega}_{\sigma}$ is well defined.
		Since $\Psi$ is a bijection from two compact sets, we only need to show that $\Psi$ is continuous.
		This is guaranteed by the fact that $\sigma\mapsto  g^{\omega}_{\sigma}$ is continuous, which in turn is a direct consequence of Lemma \ref{Unstacks}.
	\end{proof}

	\subsection{Random unstable stacks}\label{SEC:Random US}
	We now construct the \textit{random unstable stacks}, which is a random compact subset of $\Omega\times \cB$ with positive $\mu$-measure, and for each $w$ fixed, the $\omega$-section $\cS_{\omega}$ has a measurable partition which is consisted of local unstable manifolds.

	Choose $\ell_0$ large enough for which $\mu(\Gamma_{\ell_0})>0$ and fixed $\delta>0$ sufficiently small.
	By Proposition $\ref{Prop:Trandom compact}$, for each $\epsilon>0$ small enough there exists a compact subset $A_{\epsilon}\subset \cB$ such that
	$\bP\{\omega:A_{\omega}\subset A_{\epsilon}\}>1-\epsilon$.
	This implies that $\mu(\Omega\times A_{\epsilon})>1-\epsilon$ and so one can assume that
	$$\mu(\Gamma_{\ell_0}\cap (\Omega\times A_{\epsilon}))>0.$$
	Let $r_2\ll \delta$, by compactness one can choose a point $x_{\ast}$ as a reference point satisfies that
	$$\mu(\Gamma_{\ell_0}\cap (\Omega\times (B(x_{\ast},r_2)\cap A_{\epsilon}))>0,$$
	where $B(x_{\ast},r_2):=\{y:|y-x_{\ast}|\leq r_2\}$.
	Denote by $A_{\ast}:= B(x_{\ast},r_2)\cap A_{\epsilon}$ which is compact, and denote by $W^{u}_{\delta,(\oy)}:=\exp_{y}\graph(g_{(\omega,y)}|_{B^{u}_{(\oy)}(\delta \ell_{0}^{-3})})$.
	Let
	$$S_{\omega}:=\bigcup \{W^{u}_{\delta,(\oy)}\cap B(x_{\ast},2r_2): y\in A_{\ast} \ \text{such that} \ (\omega,y)\in \Gamma_{\ell_0} \},$$
	By chosen $r_2$ small enough (depend only on $\delta, \ell_{0}$), for each $y_i \in A_{\ast}$ with $(\omega,y_i)\in \Gamma_{\ell_0}$, $i=1,2$,
	we can assume that: $W^{u}_{\delta,(\omega,y_i)}\cap B(x_{\ast},2r_2)$ is connected, and either
	$$W^{u}_{\delta,(\omega,y_1)}\cap B(x_{\ast},2r_2)=W^{u}_{\delta,(\omega,y_2)}\cap B(x_{\ast},2r_2),$$
	or the two terms in the above equation are disjoint.

	By Lemma \ref{Unstacks}, for fixed $\omega$, the subset $\cS_{\omega}$ is compact, and the mapping $y \mapsto W^{u}_{\delta,(\omega,x_y)}\cap B(x_{\ast},2r_2)$ is continuous on $\cS_{\om}$ with respect to the Hausdorff distance between compact subsets of $\cB$, where $x_y\in A_{\ast}\cap (\Gamma_{\ell_{0}})_{\om}$ is the unique point such that $y\in W^{u}_{\delta,(\omega,x_y)}\cap B(x_{\ast},2r_2)$.
	Indeed, we can choose a small number $r_{1}(\omega)$, such that for each $x\in (\Gamma_{\ell_0})_{\om}$, the unstable stacks
	$$\cS(\ox):=\bigcup_{y\in \bar{U}_{\omega}} \exp_{x}(\graph \Theta_{\omega}(y))$$ be defined, where $\bar{U}_{\omega}=A_{\ast}\cap U_{\omega}(x,r_1)$ is a compact subset.
	By the choice of $r_2\ll \delta$ one can assume that
	$$W^{u}_{\delta,(\ox)}\cap B(x_{\ast},2r_2)=\exp_{x}(\graph \Theta_{\omega}(y))\cap B(x_{\ast},2r_2).$$
	Then, by compactness, there is a finitely subset $\{x_{\omega,i}\}^{k(\om)}_{i=1}\subset (\Gamma_{\ell_0})_{\om} \cap A_{\ast}$
	such that
	\begin{equation}\label{eq:cSom}
		\cS_{\omega}= \bigcup^{k(\om)}_{i=1} \cS(\om, x_{\omega,i})\cap B(x_{\ast},2r_2).
	\end{equation}
    Since for each $i=1,\cdots,k(\om)$, one has $\cS(\om, x_{\omega,i})$ is compact, and so $\cS_{\omega}$ is also compact.
	By Lemma \ref{GUStacks}, it is easy to deduce that
	$y \mapsto W^{u}_{\delta,(\om,x_y)}\cap B(x_{\ast},2r_2)$ is continuous on $\cS(\om, x_{\omega,i})$ for each $i=1,\cdots,k(\om)$, where $x_y\in \bar{U}_{\om}(x_{\omega,i},r_1)$ is the unique point such that $y\in \exp_{x_{\omega,i}}\graph(\Theta_{\omega}(x_y))$.
	So, the mapping is also continuous on $\cS_{\om}$.

	We define the subset $\cS$ of $\Om \times \cB$ by
	$$\cS:=\bigcup \{\{\omega\}\times S_{\omega}: \exists y\in A_{\ast} \ \text{such that} \ (\oy)\in\Gamma_{\ell_0} \}.$$
	\begin{lemma}\label{Lem:MMM}
		The set $\cS$ is random compact, and so is measurable with $\mu(\cS)>0$.
	\end{lemma}
	\begin{proof}
		By Proposition \ref{Prop:random compact}, it suffices to show that, for every open set $U$ of $\cB$ the set $\{\omega:S_{\om}\cap U\neq \varnothing\}$ is measurable.
		By the construction of $\cS$, we have
		$$
		\{\omega:S_{\om}\cap U\neq \varnothing\}=\pi_{\Omega}
		\{(\ox)\in \Gamma_{\ell_0}\cap (\Omega\times A_{\ast}):W^{u}_{\delta,(\ox)}\cap B(x_{\ast},2r_2)\cap U\neq \varnothing \},
		$$
		where $\pi_{\Omega}$ be the canonical projection from $\Omega\times \cB$ onto $\Omega$.
		So, by Proposition \ref{Prop:projec P} we only need to show that
		$$\{(\ox)\in \Gamma_{\ell_0}\cap (\Omega\times A_{\ast}): W^{u}_{\delta,(\ox)}\cap B(x_{\ast},2r_2)\cap U\neq \varnothing
		\}$$
		is a measurable subset of $\Omega\times \cB$.

		Let $m:= \dim E^{u}$, and let $\{v_k\}_{k>0}$ be a countable dense subset of $\R^{m}$.
		Choose a measurable basis $\{e_i{(\ox)}\}_{i=1}^{m}$ spanning $E^{u}(\ox)$, for each $v_k=(v_k^i)\in \R^{m}$, let $v_k(\ox):= \sum_{i}v_k^i e_i{(\ox)}$.
		Then, the mapping $(\ox)\mapsto v_k(\ox)$ is measurable for every $k$.
		By density, restricted to the subset $\Gamma_{\ell_0}\cap (\Omega\times A_{\ast})$ we have
		\begin{align*}
			\{(\omega,x): W^{u}_{\delta,(\omega,x)}\cap B(x_{\ast},2r_2)\cap U\neq \varnothing\}
			=\bigcup_{k>0}(\{(\ox): |v_k(\ox)|\leq \delta \ell^{-3}_0\}\bigcap&\\
			\{(\ox): v_k(\ox)+g_{(\ox)}v_k(\ox)+x\in B(x_{\ast},2r_2)\cap U\}&).
		\end{align*}
		By Remark \ref{Rem:mofg}, the map $(\ox) \mapsto g_{(\omega,x)}v_k(\om,x)$ is measurable on the measurable subset
		$\{(\ox): |v_k(\ox)|\leq \delta \ell^{-3}_0\}$ for every $k$.
		Therefore, one has
		$$\{(\omega,x)\in \Gamma_{\ell_0}\cap (\Omega\times A_{\ast}): W^{u}_{\delta,(\omega,x)}\cap B(x_{\ast},2r_2)\cap U\neq \varnothing
		\}$$
		is measurable.

		The construction of $\cS$ implies that $\Gamma_{\ell_0}\cap (\Omega\times A_{\ast})$ is a subset of $\cS$, so one has $\mu(\cS)>0$.
	\end{proof}

	For each $\omega$, let $\xi_{\omega}$ be a partition of $\cS_{\omega}$ set by
	$$\xi_{\omega}:=\{W^{u}_{\delta,(\oy)}\cap B(x_{\ast},2r_2): y\in A_{\ast} \ \text{such that} \ (\omega,y)\in \Gamma_{\ell_0} \}.$$
	For each $W \in \xi_{\omega}$, we call $W$ is a \textit{unstable leave} of $\cS_{\omega}$.
	Since $S_{\omega}$ is contained in the finite union of unstable stacks of local unstable manifolds.
	By Lemma \ref{GUStacks}, one can easily deduced that $\eta_{\omega}$ is a measurable partition of $\cS_{\omega}$.
	We call the set $\cS$ as a random unstable stack with the center $x_{\ast}$.

	\subsection{Properties of maps on unstable stacks}\label{SEC:4.4}
	Let the random unstable stack $\cS$ be defined above.

	\begin{lemma}\label{Lem:inverse}
		For all $n\in \N$, the map $f^{-n}_{\omega}$ is defined and continuous on $S_{\omega}$.
	\end{lemma}
	\begin{proof}
		For each $n\in \N$, the map $f^{-n}_{\omega}$ is defined on $S_{\omega}$ follows from Theorem \ref{Thm:Unstable}.
		We now prove it is continuous.
		Let $r_1(\om)$ be a small number satisfying Lemma \ref{Unstacks}, so that for each $x\in (\Gamma_{\ell_0})_{\om}$ the unstable stacks
		$$\cS(\ox):=\bigcup_{y\in \bar{U}} \exp_{x_{\omega}}(\graph \Theta_{\omega}(y))$$
		is defined, where $\bar{U}:=A_{\ast}\cap U_{\omega}(x,r_1)$.
		Since there is a finitely subset $\{x_{\omega,i}\}^{k(\om)}_{i=1}\subset (\Gamma_{\ell_0})_{\om} \cap A_{\ast}$ such that
		$$\cS_{\omega}\subset \bigcup^{k(\om)}_{i=1} (\cS(\om, x_{\omega,i})\cap B(x_{\ast},2r_2)).$$
		It suffices to show $f^{-n}_{\omega}$ is continuous on $\cS(\ox)$ for every $x\in (\Gamma_{\ell_0})_{\om} \cap A_{\ast}$.

		We claim that, by shrinking $r_1>0$, there exists $n_0\in \N$, for each $n\geq n_0$ there exists a compact subset $V_{n,x}\subset (\Gamma_{\ell_0e^{n\varepsilon_1}})_{\te^{-n}\omega} $ such that one can define a compact unstable stack $\cS_{n,x}$ of local unstable manifolds with $f^{n}_{\te^{-n}\omega}(\cS_{n,x})\supset \cS(\ox)$.

		Assuming the claim holds. For $n\geq n_0$, by $f^{n}_{\te^{-n}\omega}$ is injective, one has $f^{-n}_{\omega}$ is continuous on $\cS(\ox)$. For $n<n_0$, only need to write $f^{n}_{\omega}=f^{n_0-n}_{\te^{-{n_0}}\omega} f^{-n_0}_{\omega}$.

		It remains to show that the claim holds.
		Note that $f^{-n}_{\omega} \bar{U}\subset (\Gamma_{\ell_0e^{n\varepsilon_1}})_{\te^{-n}\omega}$,
		so $z\mapsto E^{u}(\te^{-n}\omega,z)$ and $z\mapsto E^{cs}(\te^{-n}\omega,z)$ are continuous on $f^{-n}_{\omega} \bar{U}$.
		Shrinking $r_1>0$, such that for every $z\in f^{-n}_{\omega} \bar{U}$, one has $d_H(E^{cs}(\te^{-n}\omega,z), E^{cs}(\te^{-n}\omega,\fo^{-n}x))$ and
		$d_H(E^{u}(\te^{-n}\omega,z), E^{u}(\te^{-n}\omega,\fo^{-n}x))$ small enough,

		so that the map $g^{\fo^{-n}z}_{\Phi^{-n}(\ox)}$ as in Lemma \ref{Unstacks} is defined on $D_{n,x}$,
		where $D_{n,x}:=B^{u}_{\Phi^{-n}(\ox)}(2\delta(e^{n\varepsilon_1}\ell_0)^{-3})$.
		Let $V_{n,x}:=f^{-n}_{\omega} \bar{U}$, and let
		$$\cS_{n,x}=\bigcup_{z\in V_{n,x}} \exp_{f^{-n}_{\omega}x} \graph(g^{\fo^{-n}z}_{\Phi^{-n}(\ox)}|_{D_{n,x}}).$$
		Shrinking $r_1>0$ if necessary, one may assume that for every $z\in V_{n,x}$
		$$\exp_{z} \graph (g_{(\te^{-n}\om,z)}|_{\wB^{u}_{(\te^{-n}\om,z)}(\frac{1}{6}\delta(e^{n\varepsilon_1}\ell_0)^{-3})})
		  \subset \exp_{f^{-n}_{\omega}x} \graph(g^{\fo^{-n}z}_{\Phi^{-n}(\ox)}|_{D_{n,x}}).$$
		By Theorem \ref{Thm:Unstable} (iii) and Lemma \ref{Unstacks} (1), we can choose $n_0$ large enough (depends only on $\ell_0$), such that for every $n\geq n_0$
		\begin{align*}
			& \wf^{n}_{(\te^{-n}\omega,z)}\graph (g_{(\te^{-n}\om,z)}|_{\wB^{u}_{(\te^{-n}\om,z)}(\frac{1}{6}\delta(e^{n\varepsilon_1}\ell_0)^{-3})})\\
	\supset & \graph (g_{(\om,f^{n}_{\te^{-n}\om}(z))}|_{\wB^{u}_{(\om,f^{n}_{\te^{-n}\om}(z))}(4\delta \ell_{0}^{-2})})\\
	\supset & \graph \Te_{\om}(f^{n}_{\te^{-n}\om}(z)).
		\end{align*}
		Therefore, we have $f^{n}_{\te^{-n}\omega}\cS_{n,x}\subset \cS(\ox)$ for each $n\geq n_0$.
	\end{proof}

	Let $\xi_{\omega}$ be the measurable partition of $\cS_{\omega}$ into unstable leaves.
	Let $\xi:=\{\{\omega\}\times W:W\in \xi_{\omega}\}$ be a partition of $\cS$, since $\Omega$ is Polish space one has $\xi$ is also measurable.
	For each $(\ox)\in \cS$, denote by $\xi(\ox)$ the element of $\xi$ which contained $(\ox)$ and denote by $\xi(\ox)_{\om}$ the $\om$-section of the subset $\xi(\ox)$ of $\Om\times \cB$.
	Then by construction one has $\xi_{\omega}(x)=\xi(\ox)_{\om}:=\{y:(\oy)\in \xi(\ox)\}$.

	Recall the induced volume $\nu_{(\ox)}$ which we defined in Section \ref{SEC:4.1}.
	\begin{lemma}\label{Lem:MofV}
		For any measurable set $E \subset \Om\times \cB$, the map $ (\ox) \mapsto \nu_{(\ox)}((E \cap \xi(\ox))_{\om})$ is measurable on $\Gamma_{\ell_0}\cap (\Om\times A_{\ast})$.
	\end{lemma}
	\begin{proof}
		Let $m:=\dim E^{u}$, choose a measurable basis $\{e_i{(\ox)}\}_{i=1}^{m}$ spanning $E^{u}(\ox)$.
		For each $t=(t_1,\cdots,t_m)\in \R^{m}$, let $I_{(\ox)}(t)=\sum_{i=1}^{m}t_ie_i{(\ox)}$.
		Then, for
		$(\ox)\in \Gamma_{\ell_0}\cap (\Omega\times A_{\ast})$
		we have
		$$
		\nu_{(\ox)}((E \cap \xi(\ox))_{\om}):=\int_{E_{(\ox)}}\det(Id+Dg_{(\ox)}(I_{(\ox)}(t)))dt.
		$$
		Where $E_{(\ox)}:=\{t\in \R^{m}: I_{(\ox)}(t)+g_{(\ox)}(I_{(\ox)}(t))+x\in E_{\om}\cap B(x_{\ast},2r_2)\}
		      \cap \{t\in \R^{m}:|I_{(\ox)}(t)|\leq \delta \ell^{-3}_0\}$
		and the ``$\det$" with respect to the Euclidean volume on $\R^{m}$ and the induced volume $m_{T_{I_{(\ox)}(t)} \wW^{u}_{\delta}(\ox)}$.

		Since $\Omega\times \cB$ is a Polish space, without loss of generality, we can assume $E$ is a compact set.
	    For fixed $t\in \R^{m}$ with $|I_{(\ox)}(t)|\leq \delta \ell^{-3}_0$.
	    By Remark \ref{Rem:mofg}, we have $(\ox)\mapsto \chi_{E(\ox)}(t)$ is a measurable function on $\Gamma_{\ell_0}\cap (\Omega\times A_{\ast})$,
		and the similar discussion in Lemma \ref{Lem:M3} shows that $(\ox)\mapsto \det(Id+Dg_{(\ox)}(I_{(\ox)}(t)))$
		is also measurable on $\Gamma_{\ell_0}\cap (\Omega\times A_{\ast})$.
		Therefore, for every $M\geq 0$, the set
		$$
		E_M:=\{(\ox)\in \Gamma_{\ell_0}\cap (\Omega\times A_{\ast}):\nu_{(\ox)}((E \cap \xi(\ox))_{\om})\leq M\}
		$$
		is a measurable subset of $\Gamma_{\ell_0}\cap (\Omega\times A_{\ast})$.
	\end{proof}

	In the remainder of this section, we assume that $E^{c}(\ox)=0$ for every $(\ox)\in \Gamma$.

	\begin{lemma}\label{Lem:CofU2}
		Assume $E^{c}=\{0\}$. For each $q\in \N$, the map $x\mapsto E^{u}(\omega,x)$ is continuous on $f^{-q}_{\omega} S_{\omega}$
	\end{lemma}
	\begin{proof}
	    We give the proof for $q=0$, and the case for $q\geq 1$ can be proven similarly.

		Let $(\ox)\in \cS$ be fixed, the main difference with the argument in Proposition \ref{Prop:CCS} is that no intrinsically defined the stable subspace $E^{cs}(\ox)$.
		As in \cite[proof of Lemma 7.6]{Young17}, we use the backward graph transform to overcome it.

		For $(\ox)\in \cS$, there exists a point $(\om,x_0)\in \Gamma_{\ell_0}$ with the property that $x\in W^{u}_{\delta,(\om,x_0)}$.
		We write $\wf_{-n}:=\wf_{\Phi^{-n}(\om,x_0)}$, $\Lnorm{\cdot}{-n}:=\Lnorm{\cdot}{\Phi^{-n}(\om,x_0)}$ and
		${x}'_{-n}:=\fo^{-n}x-\fo^{-n}x_{0}$ for each $n\geq 0$.

		Let $F_0=E^{s}(\om,x_0)$, then $\cB=E^{u}(\ox)\oplus F_0$.
		Identifying $\cB_x$ with $\cB_{x_0}$. For $\delta$ small enough, by Lemma \ref{Lem:NlinP}, we may assume that $D\wf_{-n}({x}'_{-n})$ is sufficiently close to $D\wf_{-n}(0)$ so that the backward graph transform (Lemma \ref{Lem:GtranS}) can be applied to the linear map $D\wf_{-n}({x}'_{-n})$ for every $n\geq 0$.
		Then, by Lemma \ref{Lem:GtranS} we get a sequence of subspace $F_n\subset \cB,n\in \N$, such that $D\wf_{-n}({x}'_{-n})(F_n)\subset F_{n-1}$ and for each $v_{-n}\in F_n$, one has
		$$\Lnorm{D\wf_{-n}({x}'_{-n})v_{-n}}{-n+1}\leq (e^{-\lambda}+\delta) \Lnorm{v_{-n}}{-n}.$$
		Since $F_n$ is a graph of some linear map from $E^{s}(\om,x_0)$ to $E^{u}(\om,x_0)$, one has $\cB= E^{u}(\Phi^{-n}(\ox))\oplus F_{n}$.

		For every $y_k\in \cS_{\om}$ with $y_k\rightarrow x$ as $k\rightarrow \infty$. Let $v^k\in E^{u}(\om,y_k)$ and decompose $v^k=v^{k,u}+v^{k,s}$ with respect to the splitting $\cB=E^{u}(\ox)\oplus F_0$.
		Using the same estimation in Proposition \ref{Prop:CCS} one has $v^{k,s}\rightarrow 0$ as $k\rightarrow \infty$.
		Hence, we have $G(E^{u}(\om,y_k),E^{u}(\ox))\rightarrow 0$ as $k\rightarrow \infty$.
		By Lemma \ref{Lem:gap}, this is enough to show that $x\mapsto E^{u}(\omega,x)$ is continuous on $\cS_{\om}$.
	\end{proof}
	Let $J^{u}(\ox)=\det(Df_{\omega}(x)|E^u(\ox))$ for $(\ox)\in \cS$.
	By Proposition \ref{Prop:dis} (b) the following function is well defined on $\xi_{\omega}(x)$ and Lipschitz-continuous:
	$$z\mapsto \Delta^{\omega}(x,z):= \prod_{n=1}^{\infty} \dfrac{J^{u}(\Phi^{-n}(\ox))}{J^{u}(\Phi^{-n}(\omega,z))}, \quad z\in \xi_{\omega}(x).$$

	Define the function $q(\omega,z):\cS \rightarrow \R$ by
	$$q(\omega,z):=\dfrac{\Delta^{\omega}(x,z)}{\int_{\xi_{\omega}(x)} \Delta^{\omega}(x,y) d\nu_{(\ox)}(y)}, \quad z\in \xi_{\omega}(x).$$
	Since for every ${x}'\in \cB$ satisfied that $\xi_{\omega}(x)=\xi_{\omega}({x}')$, one has
	$$\Delta^{\omega}({x}',z)=\Delta^{\omega}(x,z) \cdot \Delta^{\omega}({x}',x),$$
	so the function $q$ is not dependent on the choice of $x\in \xi_{\omega}(z)$ and is well defined as a function of $(\om,z)$.
	Moreover, Lemma \ref{Lem:MofV} and Lemma \ref{Lem:M3} imply that for each fixed $z$, the map $\om\mapsto q(\om,z)$ is measurable on the measurable set
	$\{\om:(\om,z)\in \cS \}$.
	\begin{lemma}\label{Lem:CofFq}
		Assume $E^{c}=\{0\}$. For fixed $\omega$, the function $q(\omega,z):\cS_{\omega}\rightarrow \R$ is continuous.
	\end{lemma}
	\begin{proof}
		We follow the proof of Lemma 7.9 in \cite{Young17}.

		Let $\omega$ be fixed.
		For each $n$, by Lemma \ref{Lem:inverse} we have $\fo^{-n}|_{\cS_{\omega}}$ is continuous,
		by Lemma \ref{Lem:CofU2} we have
		$x\mapsto E^{u}(\ox)$ is continuous on $\fo^{-n}{\cS_{\omega}}$.
		Then, one has $x\mapsto \det(Df_{\te^{-n}\omega}(\fo^{-n}x)|E^u(\Phi^{-n}\ox))$ is continuous on $\cS_{\om}$ by Proposition \ref{Prop:CofDet}.

		Let $r_1(\om)$ be a small number satisfying Lemma \ref{Unstacks}.
		For each $(\ox)\in \Gamma_{\ell_0}\cap (\Om\times A_{\ast})$.
		Define the unstable stack $\cS(\ox)$ by
		$$
		\cS(\ox):=\bigcup_{y\in \bar{U}} \exp_{x_{\omega}}(\graph \Theta_{\omega}(y))
		$$
		where $\bar{U}=A_{\ast}\cap U_{\omega}(x,r_1)$.
		Since $\cS_{\om}$ is contained in the finite union of unstable stacks.
		It suffices to show that $q(\om,\cdot)$ is continuous on $\cS(\ox)\cap B(x_{\ast},2r_2)$.

		By Lemma \ref{GUStacks}, there exists a homeomorphism
		$\Psi_{\ox}: \Sigma_{(\ox)}\times B^{u}_{(\ox)}(2\delta \ell^{-3}_0)\rightarrow \cS(\ox)$,
		where $\Sigma_{(\ox)}$ as in Lemma \ref{GUStacks}.
		For every $z\in \cS(\ox)\cap B(x_{\ast},2r_2)$, denote by $(\sigma_z,u_z):=\Psi^{-1}_{\ox}(z)$ and let $\sigma(z):=\Psi_{\ox}(\sigma_z,0)$.
		Define
		$$z\mapsto \bar{\Delta}_n(z):=\Delta^{\om}_{n}(\sigma(z),z),$$
		where $\Delta^{\om}_{n}(\sigma(z),z)$ as in $(\ref{eq:Delta n})$.
		Since $\sigma(z)$ depend continuously on $z$, one has $z\mapsto \bar{\Delta}_n(z)$ is continuous.
		By Proposition \ref{Prop:dis},
		the sequence $\bar{\Delta}_n$ converges uniformly to $\bar{\Delta}$ which is continuous on $\cS(\ox)$.

		It remains to show that $z\mapsto \int \bar{\Delta} dv_{(\om,z)}$ is continuous on $\cS(\ox)\cap B(x_{\ast},2r_2)$.
		Denote by $E_z:=\pi^{u}_{(\ox)}(\exp^{-1}_{x}\xi_{\omega}(z))$, since $z\mapsto \xi_{\omega}(z)$ is continuous,
		we have $z\mapsto E_z$ is also continuous with respect to the Hausdorff topology.

		By Lemma \ref{GUStacks} and the change of variables formula we have
		$$
		\int_{\xi_{\omega}(\sigma(z))} \bar{\Delta}(z) dv_{(\om,z)}=
		\int_{B^{u}_{(\ox)}(2\delta \ell^{-3}_0)}
		\bar{\Delta}(z) \det(Id+Dg_{\sigma_z}(u))\chi_{E_z}(u)dm(u)$$
		where $m$ is the induced volume on $E^{x}$ and the ``$\det$" is with respect to $m$ and $m_{E^u(\om,z)}$.
	    By Lemma \ref{Lem:CofU2} we have that
	    $(\sigma,u) \mapsto E^u(\om,\Psi_{\ox}(\sigma,u))$
	    is continuous on
	    $\Sigma_{\ox} \times B^{u}_{\ox}(2\delta \ell^{-3}_0)$.
	    So, by Lemma \ref{Lem:CdhA} the map $(\sigma,u) \mapsto Id+Dg_{\sigma}(u)$ is continuous.
	    From this and from properties of the ``det"-function (see Proposition \ref{Prop:Det}),
	    we deduce that the mapping $z \mapsto \det(Id+g_{\sigma_z}(u))$ is continuous for each $u$.
	    Therefore, one has $z\mapsto \int \bar{\Delta} dv_{(\om,z)}$ is continuous. This completes the proof.
	\end{proof}

\section{Proof of main Theorem}\label{SEC:5}
	In this section, we use the same method in \cite{Young85} (also in \cite{Young17} and \cite{Li13}) to give a detailed proof of Theorem \ref{Thm:A} and Theorem \ref{Thm:B} when $\mu$ is ergodic.

	We recall some notations about partitions at the beginning.
	Let $T: X\rightarrow X$ be a measure preserving transformation of a probability space $(X,\mu)$.
	For a measurable partition $\eta$ on $X$, we let $\eta(x)$ denote the atom of $\eta$ containing $x$ for each $x\in X$.
	For two measurable partitions $\eta_1$ and $\eta_2$ , we say $\eta_2$ is a refinement of $\eta_1$ if $\eta_1(x)\supset \eta_2(x)$ for $\mu$-almost every $x$, and written by $\eta_1\leq \eta_2$.
	Let $ \eta_1 \vee \eta_2=\{A\cap B:A\in \eta_1, B\in\eta_2\}$ and let $T^{-1} \eta_1=\{T^{-1}(A):A\in\eta_1\}$. We say $\eta$ is $T$-decreasing if $\eta \leq T^{-1} \eta$.

	We fixed $\ell_0$ large enough such that $\mu(\Gamma_{\ell_0})>0$, and let the random stack $\cS$ as defined in Section \ref{SEC:Random US}.

	\subsection{Partitions subordinate to the unstable foliation}
	Let $\xi_{\omega}$ be the measurable partition of $\cS_{\omega}$ into unstable leaves, and let $\xi:=\{\{\omega\}\times W_{\omega}:W_{\omega}\in \xi_{\omega}\}$ be the measurable partition of $\cS$.
	By definition, for each $(\ox)\in \cS$, one has $\xi(\ox)\subset \{\om\}\times \cB$.
	We denote by $\xi(\ox)_{\om}$ the $\om$-section of the subset $\xi(\ox)$ of $\Om\times \cB$.

	Denote by $\{\mu_{\xi(\ox)}\}_{(\ox)\in \cS}$ the canonical disintegration of $\mu|_{\cS}$,
	which satisfied that,
	$$\mu|_{\cS}=\int_{\cS} \mu_{\xi(\ox)} d\mu(\ox).$$
	The family of measures $\{\mu_{\xi(\ox)}\}$ on $\cB$ (one can identify $\{\omega\}\times\cB$ with $\cB$) exists by Rokhlin's disintegration theorem, and any two families like that are equal on a full $\mu|_{\cS}$-measure subset of $\cS$.

	We give a formal definition of SRB measures for random dynamical systems on Banach spaces as follows:
	\begin{definition}\label{DEF:SRBm}
		We say $\mu$ is a random SRB measure if (i) it has a positive exponent for $\mu$-almost every $(\omega,x)$, and (ii) for every random stack $\cS$ with $\mu(\cS)>0$, $\mu_{\xi(\ox)}$ is absolutely continuous with respect to $\nu_{(\ox)}$ for $\mu$-almost every $(\ox)$ in $\cS$.
	\end{definition}

	\begin{definition}\label{DEF:PSubord}
		We say that a measurable partition $\eta$ is \textit{subordinate to the unstable foliation} $W^{u}$ if for $\mu$-almost every $(\ox)$ we have
		\begin{enumerate}
			\item[(i)]  $\eta(\ox)\subset \{\omega\}\times W^{u}(\ox)$,
			\item[(ii)] $\eta(\ox)_{\omega}$ contains a neighborhood of $x$ in $W^{u}(\ox)$, and
			\item[(iii)]$\eta(\ox)_{\omega} \subset f^{N}_{\te^{-n}\omega} W^{u}_{\loc}(\Phi^{-n}(\ox))$ for some $N\in \N$ depending on $(\ox)$.
	    \end{enumerate}
	\end{definition}

	\begin{lemma}
		The following are equivalent:
		\begin{enumerate}
			\item[(1)] $\mu$ is a random SRB measure;
			\item[(2)] there exists a partition $\eta$ subordinate to $ W^{u}(\ox) $ with $\mu_{\eta(\ox)}\ll \nu_{(\ox)}$ for $\mu$-almost every $(\ox)$;
			\item[(3)] for every partition $\eta$ subordinate to $ W^{u}(\ox) $, one has $\mu_{\eta(\ox)}\ll \nu_{(\ox)}$ for $\mu$-almost every $(\ox)$.
		\end{enumerate}
	\end{lemma}
	\begin{proof}
		We first prove (2) implies (3): Let $\eta_1$ be the partition satisfies (2), for each $\eta_2$ subordinate to
		$W^{u}(\ox)$, we denote by $\eta=\eta_1 \vee \eta_2$ and let $\{\mu_{W}\}_{W\in \eta}$ be the canonical disintegration of $\mu$ with respect to $\eta$. Note that $\mu_{\eta_1(\ox)}\ll \nu_{(\ox)}$ implies that $\mu_{\eta(\oy)} \ll \nu_{(\ox)}$ for every $\eta(\oy) \subset \eta_1(\ox)$. Therefore, by the uniqueness of the canonical disintegration, one has $\mu_{\eta_2(\ox)}\ll \nu_{(\ox)}$ for $\mu$-almost every $(\ox)$.

		The statement (1) implies (2) and, (3) implies (1) follow immediately from the proposition below.
	\end{proof}

	\begin{proposition}\label{Prop:Exist P}
		Assuming that $(\Phi,\mu)$ has a positive Lyapunov exponent, then there is a random stack $\cS$ with positive $\mu$-measure, and a measurable partition $ \eta$ on $\widetilde{\cS}:=\cup_{n\geq0}\Phi^n \cS$ such that
		\begin{enumerate}
			\item[(a)] $\eta$ subordinate to $ W^{u}(\ox) $,
			\item[(b)] $\eta$ is $\Phi$-decreasing,
			\item[(c)] for any measurable set $E \subset \Omega\times \cB$, the function $(\ox) \mapsto \nu_{(\ox)}(E_{\omega} \cap \eta(\ox)_{\omega})$ is finite-valued and measurable on a full $\mu$-measure set, and
			\item[(d)] $\vee_{n\geq 0}\Phi^{-n}\eta$ is partition of $\widetilde{\cS}$ into points modulo  $\mu$-measure $0$.
		\end{enumerate}
	\end{proposition}
	\begin{proof}
		Using the notations in Section \ref{SEC:Random US}, we first construct a random stack.
		Let $r\in (1/2,1)$ and let
		$$\cS^{r}_{\omega}:=\bigcup \{W^{u}_{\delta,(\oy)}\cap \bar{B}(x_{\ast},2rr_2): y\in A_{\ast} \ \text{such that} \ (\omega,y)\in \Gamma_{\ell_0} \},$$
		and define $\cS^{r}:=\bigcup \{\{\omega\}\times S^{r}_{\omega}: \exists y\in A_{\ast} \ \text{such that} \ (\oy)\in\Gamma_{\ell_0} \}$.
		Let $\widetilde{\cS}:=\widetilde{\cS}^r=\cup_{n\geq0}\Phi^n \cS^r$, since $\mu$ is $\Phi$-ergodic one has $\mu(\widetilde{\cS})=1$.

		Let $\xi:=\{\{\omega\}\times W_{\omega}:W_{\omega}\in \xi_{\omega}\}$ be a measurable partition of $\cS^r$, where $\xi_{\omega}$ is the measurable partition of $\cS^r_{\omega}$ into unstable leaves.
		For each $n\geq 0$, let $\xi^n=\{\Phi^n(W):W\in \xi\} \cup \{\widetilde{\cS}-\Phi^n(\cS^r)\}$ be a measurable partition of $\widetilde{\cS}$, and let $\eta=\vee_{n=0}^{\infty}\xi^n$.
		Note that $\eta$ is a refinement of the partition $\eta_0:=\{\{\omega\}\times \cB\}_{\omega\in \Omega}$.
		For every $(\omega,x)\in \widetilde{\cS}$, a point $(\omega,y)\in \eta(\ox)$ if and only if
		\begin{equation}\label{eq:eta}
			\left\{
			\begin{alignedat}{2}
				&f^{-n}_{\omega} y \in \xi_{\te^{-n}\omega}(f^{-n}_{\omega} x)
				\quad &\text{if}& \ \Phi^{-n}(\ox) \in \cS^r; \\
				&\Phi^{-n}(\oy) \notin \cS^r   &\text{if}& \ \Phi^{-n}(\ox) \notin \cS^r.
			\end{alignedat}
			\right.
		\end{equation}
		Therefore, one has that $\eta$ is $\Phi$-decreasing. This proves item (b).

		For each $(\omega,x)\in \widetilde{\cS}$, there exists $N(\ox)\in \N$ such that $\Phi^{-N}(\ox) \in \cS^r$.
		Then, by (\ref{eq:eta}) we have $\eta(\ox)_{\omega}\in f^{N}_{\te^{-n}(\omega)} W^{u}_{\loc}(\Phi^{-N}(\ox))$.
		To show $\eta$ subordinate to $W^{u}$, we only requirement that $\eta(\ox)_{\omega}$ contains a neighborhood of $x$ in $W^{u}_{\loc}(\ox)$ for $\mu$-almost every $(\ox)$ left.

		We now choose $r$ carefully to guarantee that.
		Let
		$$ \partial \cS^{r}_{\omega}:=\bigcup \{W^{u}_{\delta,(\oy)}\cap
		\{z:|z-x_{\ast}|=2rr_2\}: y\in A_{\ast} \ \text{such that} \ (\omega,y)\in \Gamma_{\ell_0} \},$$
		and let
		$$\partial \cS^r:=\bigcup \{\{\omega\}\times \partial \cS^{r}_{\omega}: \exists y\in A_{\ast} \ \text{with} \ (\oy)\in\Gamma_{\ell_0}\}.$$
		For fixed $(\ox)\in \widetilde{\cS}\cap \Gamma$.
		Since $\{\omega\}\times \partial (\eta(\ox)_{\omega})\subset \cup_{n\geq0} \Phi^{n} (\partial (\cS^r))$, we have that for $0<\varepsilon<{\delta}'_{1}$ (${\delta}'_{1}$ be as in Theorem \ref{Thm:Unstable}), if $f^{-n}_{\omega}W^u_{\varepsilon}(\ox)\cap (\partial (\cS^r))_{\te^{-n}\omega}=\varnothing$ then $W^u_{\varepsilon}(\ox)\subset \eta(\ox)_{\omega}$.
		To guarantee that, we needed that for every $y\in W^u_{\varepsilon}(\ox)$ and every $n\in \N$,
		$$|f^{-n}_{\omega}y-f^{-n}_{\omega}x|<
		|(2rr_2-|f^{-n}_{\omega}x-x_{\ast}|)|.$$
		For every $y\in W^u_{\varepsilon}(\ox)$, by Theorem \ref{Thm:Unstable} (iii) we have
		$$3|f^{-n}_{\omega}y-f^{-n}_{\omega}x|\leq \Lnorm{f^{-n}_{\omega}y-f^{-n}_{\omega}x}{\Phi^{-n}(\ox)}
		                                      \leq \Lnorm{x-y}{(\ox)}e^{\frac{-n\lambda}{2}}.$$
		Therefore, we need to take $\varepsilon(\ox)$ small enough such that
		$$\varepsilon<
		\min_{n\in \N}\{{\delta}'_1, \frac{1}{3}\ell(\ox)e^{\frac{n\lambda}{2}} |(2rr_2-|f^{-n}_{\omega}x-x_{\ast}|)|\}.$$
		Take $r\in (1/2,1)$, then the following subset
		\begin{equation}\label{eq:51}
			\{r:\sum_{n=0}^{\infty}\mu(\{(\ox): |(|x-x_{\ast}|-2rr_2)|<e^{\frac{-n\lambda}{2}}\})<\infty \}
		\end{equation}
		have full Lebesgue measure on the interval $(1/2,1)$ by standard estimations (see \cite{Ledrappier82, Young85} for details). Since there are most countable choice of $r$ such that $\mu(\partial \cS^r)>0$, one can choose $r\in (1/2,1)$ such that $r$ belongs to the set (\ref{eq:51}), and $\mu(\partial \cS^r)=0$.
		Moreover, by $\mu$ is $\Phi$ invariant, we also have
		$$\sum_{n=0}^{\infty}\mu(\{(\ox): |(|f^{-n}_{\omega}x-x_{\ast}|-2rr_2)|<e^{\frac{-n\lambda}{2}}\})<\infty.$$
		By a Borel-Cantelli type argument, for $\mu$-almost every $(\ox)$ we have
		$$ |(|f^{-n}_{\omega}x-x_{\ast}|-2rr_2)|\geq e^{\frac{-n\lambda}{2}}$$
		except for a finite number of integers $n\in \N$.
		So, one can choose $\varepsilon(\ox)>0$ such that $W^u_{\varepsilon}(\ox)\subset \eta(\ox)_{\om}$.
		Item (a) be proved.

		It follows from Lemma \ref{Lem:MofV} the sequence of functions
		$$
		(\ox) \mapsto \nu_{(\ox)}(E_{\omega}\cap (\vee_{i=0}^{n-1}\xi^{i}(\ox))_{\omega}) \quad \text{for} \ n\in \N$$
		are measurable on $\cup^{n-1}_{i=0 }\Phi^{i}(\Gamma_{\ell_0}\cap(\Om\times A_{\ast}))$, and decreasing for $n$. Therefore, we have
		$$
		(\ox) \mapsto \nu_{(\ox)}(E_{\omega}\cap \eta(\ox)_{\omega})=\lim_{n\rightarrow \infty}\nu_{(\ox)}(E_{\omega}\cap (\vee_{i=0}^{n}\eta^{i}(\ox))_{\omega})
		$$
		is measurable on a full $\mu$-measure set $\cup^{\infty}_{n=0}\Phi^{i}(\Gamma_{\ell_0}\cap(\Om\times A_{\ast}))$. Item (c) be proved.

		Item (d) follows from (\ref{eq:eta}) and Lemma \ref{Lem:CUnstable} directly.
	\end{proof}
	\subsection{SRB measures implies entropy formula}
	\begin{proof}[proof of Theorem A (ergodic case)]
		Since Ruelle's inequality \cite{Ruelle82} (see also \cite{Li12} for the random case on Banach spaces) tells us that
		$$h({\mu}'|\bP)\leq \int \sum_{i} m_i\lambda^{+}_{i}d{\mu}'$$
		for any $\Phi$-invariant probability measure ${\mu}'$ projection to $\bP$.
		Then, it suffices to show that if $\mu$ is a random SRB measure the reverse inequality holds.
		We assume that $\mu$ is ergodic here.
		Let $\eta$ be constructed in proposition \ref{Prop:Exist P}.
		Recall that $\eta_0:=\{\{\omega\}\times \cB\}_{\omega\in \Omega}$.
		Since $\eta$ is $\Phi$-decreasing and $\eta_0\leq \eta$, we have
		\begin{align*}
			h(\mu|\bP,\eta)&=\lim_{n\rightarrow \infty} \frac{1}{n}H_{\mu}(\vee_{i=0}^{n}\Phi^{-1}\eta| \eta_0)\\
			               &=\frac{1}{n}H_{\mu}(\Phi^{-n}\eta|\eta)=H_{\mu}(\Phi^{-1}\eta|\eta).
		\end{align*}
		It suffices to show that
		$$H_{\mu}(\Phi^{-1}\eta|\eta)=\int \log J^{u}(\ox)d\mu(\ox)=\sum m_i\lambda_{i}^{+},$$
		where $J^{u}(\ox)=\det(Df_{\omega}(x)|E^{u}(\ox))$.
		Note that $\log J^{u}\in L^1(\mu)$, the second equality follows immediately from the Birkhoff ergodic theorem and Proposition \ref{Prop:DetL}.

		By Proposition \ref{Prop:Exist P} (c), we let $\nu$ be the $\sigma$-finite measure defined by
		$$\nu(K)=\int v_{(\ox)}((\eta(\ox) \cap K)_{\omega}) d\mu(\ox).$$
		By assumption, we have $\mu \ll \nu$. Let $\rho=\frac{d\mu}{d\nu}$, then for $\mu$-almost every $(\ox)$ one has
		$$\rho(\omega,y)=\frac{d\mu_{\eta(\ox)}}{d\nu_{(\ox)}}(y) \quad \text{for} \ \nu_{(\ox)}\text{-almost every}\ y\in \eta(\ox)_{\omega}.$$
		Recall that the information function $I(\Phi^{-1}\eta|\eta)$ is
		$$I(\Phi^{-1}\eta|\eta)(\ox)=-\log\mu_{\eta(\ox)}((\Phi^{-1}\eta)(\ox)_{\omega}).$$
		Let $L(\ox)=\rho(\ox)(J^{u}(\ox))^{-1} (\rho(\Phi(\ox)))^{-1}$. For every measurable set $E\subset \cB$.
		On the one hand, since $\eta$ is $\Phi$-decreasing, one has
		\begin{align*}
			\mu_{(\Phi^{-1}\eta)(\ox)}(E)&=\dfrac{\mu_{\eta(\ox)}((\Phi^{-1}\eta)(\ox)_{\omega}\cap E) }{\mu_{\eta(\ox)}((\Phi^{-1}\eta)(\ox)_{\om})}\\
			&=e^{I(\Phi^{-1}\eta|\eta)(\ox)}\int_{((\Phi^{-1}\eta)(\ox))_{\omega}\cap E} \rho(\oy) d\nu_{(\ox)}(y).
		\end{align*}
		On the other hand, by $\mu$ is $\Phi$-invariant, we have for $\mu$-almost every $(\ox)$
		\begin{align*}
			\mu_{(\Phi^{-1}\eta)(\ox)}(E)&=\mu_{\eta(\Phi (\ox))}(f_{\omega}E)\\
			&=\int_{f_{\omega}(((\Phi^{-1}\eta)(\ox))_{\omega}\cap E)} \rho(\te(\om),z) d\nu_{\Phi(\ox)}(z)\\
			&=\int_{f_{\omega}(((\Phi^{-1}\eta)(\ox))_{\omega}\cap E)} \rho(\te(\om),z) J^{u}(\om,f^{-1}_{\te\omega}(z)) d((f_{\omega})_{\ast}\nu_{(\ox)})(z)\\
			&=\int_{((\Phi^{-1}\eta)(\ox))_{\omega}\cap E} \rho(\Phi(\oy)) J^{u}(\oy) d\nu_{(\ox)}(y),
		\end{align*}
		where we use the change of variables formula for induced measures on finite-dimensional manifolds in the third equality
		(recall $f^{-1}_{\te\omega}(y)$ is the unique point such that $f_{\om}(f^{-1}_{\te\omega}(y))=y$).
		Consequently, by $E$ is arbitrary and $ I(\Phi^{-1}\eta|\eta)(\oy)= I(\Phi^{-1}\eta|\eta)(\ox)$ for any
		$y\in ((\Phi^{-1}\eta)(\ox))_{\omega}$, we get that for $\mu$-almost every $(\ox)$
		$$I(\Phi^{-1}\eta|\eta)(\oy)=-\log L(\oy) \quad \text{for} \ \nu_{(\ox)}\text{-almost every}\ y\in ((\Phi^{-1}\eta)(\ox))_{\omega}.$$
		Therefore, for $\mu$-almost every $(\ox)$ one has
		$$I(\Phi^{-1}\eta|\eta)(\ox)=\log J^{u}(\ox)+\log \dfrac{\rho(\Phi(\ox))}{\rho(\ox)}.$$
		Since $\log J^{u}\in L^1(\mu)$ and $I(\Phi^{-1}\eta|\eta)(\ox)\geq 0$, one has $\log^{-} \frac{\rho\circ \Phi}{\rho}\in L^1(\mu)$. Therefore, by Lemma \ref{Lem:Tem} one has $\int \log \frac{\rho\circ \Phi}{\rho}d\mu=0$.
		So, we have
		$$H_{\mu}(\Phi^{-1}\eta|\eta)=
		\int \log J^{u}(\ox)d\mu(\ox).$$
		This completes the proof.
	\end{proof}
	\subsection{Entropy formula implies SRB measures}
	We assume that $(\Phi,\mu)$ satisfies (H1)-(H5) and $\mu$ is ergodic. Let the partition $\eta$ be constructed in Proposition \ref{Prop:Exist P}.
	We first prove the following:
	\begin{lemma}\label{Lem:last}
		Let $(\Phi,\mu)$ with a positive Lyapunov exponent for $\mu$-almost everywhere, and let $\eta$ be as in Proposition \ref{Prop:Exist P}. Assume
		$$h(\mu|\bP,\eta)=\int \log J^{u} d\mu,$$
		then we have $\mu$ is a Random SRB measure.
	\end{lemma}

	\begin{proof}
		Let $\eta,\xi, \cS$ and $\widetilde{\cS}$ be as in Proposition \ref{Prop:Exist P}.

		For $(\om,x)\in \cS$. Recall the Lipschitz-continuous function $z \mapsto \Delta^{\omega}(x,z)$ on $\xi(\ox)_{\om}$ which defined by
		$$z\mapsto \Delta^{\omega}(x,z):= \prod_{n=1}^{\infty} \dfrac{J^{u}(\Phi^{-n}(\ox))}{J^{u}(\Phi^{-n}(\omega,z))}.$$
		And the function $q(\omega,z)$ on $\cS$ defined by
		$$q(\omega,z):=\dfrac{\Delta^{\omega}(x,z)}{\int_{\xi_{\omega}(x)} \Delta^{\omega}(x,y) d\nu_{(\ox)}(y)}, \quad z\in \xi(\ox)_{\om}.$$

		Since $\eta$ is subordinate to $W^{u}(\ox)$,
		for every $(\omega,x)$ there exists $n\geq 0$ such that $f^{-n}_{\te^{-n}\omega}(\eta(\Phi^{-n}(\ox))_{\te^{-n}\omega})$ contained in a leaf of $\cS_{\om}$.
		By assumption (H1) and Theorem \ref{Thm:Unstable}, $f^{n}_{\te^{-n}\omega}$ is a $C^2$ embedding on each unstable leaf.
		Then, we have that for $\mu$-almost every $(\ox)$, the map $z\mapsto \Delta^{\omega}(x,z)$ is Lipschitz-continuous and bounded on $\eta(\ox)_{\omega}$.
		By Proposition \ref{Prop:Exist P} (c), for $\mu$-almost every $(\omega,x)$ we have $\nu_{(\ox)}(\eta(\ox)_{\omega})>0$.
		Therefore, the following function
		$$p(\omega,z):=\dfrac{\Delta^{\omega}(x,z)}{\int_{\eta(\ox)_{\omega}} \Delta^{\omega}(x,y)  d\nu_{(\ox)}(y)}, \ z\in \eta(\ox)_{\omega},$$
		is well defined for $\mu$-almost every $(\omega,x)$.

		Suppose we have that $\mu_{\eta(\ox)}\ll \nu_{(\ox)}$ for $\mu$-almost $(\ox)$.
		Then there exist some function $\rho$ such that $d\mu_{\eta(\ox)}=\rho \nu_{(\ox)}$ for $\nu_{(\ox)}$-almost every
		$y\in \eta(\ox)_{\om}$.
		By the argument in the proof of Theorem A, the function $\rho$ must satisfy $\int_{\eta(\ox)_{\om}}\rho d\nu_{(\ox)}=1$ and $\rho(\Phi(\oy))J^{u}(\oy)\rho(\oy)^{-1}$ must be constant on $(\Phi^{-1}\eta)(\ox)_{\om}$.
		So, one can guess that $\rho=p$.

		The function $p$ is measurable by Lemma \ref{Lem:MofV} and Lemma \ref{Lem:CofFq} with standard arguments.
		So, we can define a probability measure $\upsilon$ on $\widetilde{\cS}$ by
		$$\upsilon(K)=\int(\int_{(K\cap \eta(\ox))_{\omega}} p(\oy) dv_{(\ox)}(y))d\mu(\ox) $$
		for each $K$ measurable.
		By construction, the measure $\upsilon$ and $\mu$ project to the same measure on the quotient space $\widetilde{\cS}/\eta$, and the conditional measure of $\upsilon$ on each element of $\eta$, which we denote by $\upsilon_{\eta(\ox)}$, is absolutely continuous with respect to $\nu_{(\ox)}$ with the densities function $p$.
		It suffices to prove that $\upsilon=\mu$.

		For each $n\in \N$, by the construction we have
		$$\upsilon_{\eta(\ox)} ((\Phi^{-n}\eta)(\ox)_{\omega})=
		\dfrac{\int_{(\Phi^{-n}\eta)(\ox)_{\omega}} \Delta^{\om}(x,y)d\nu_{(\ox)}(y)}{\int_{\eta(\ox)_{\omega}} \Delta^{\om}(x,y)d\nu_{(\ox)}(y)}.$$
		By the change of variables formula (recall $f^{-n}_{\te^n\omega}(y)$ is the unique point such that
		$f^n_{\om}(f^{-n}_{\te^n\omega}(y))=y$),
		\begin{align*}
	     &\int_{(\Phi^{-n}\eta)(\ox)_{\omega}} \Delta^{\om}(x,z)d\nu_{(\ox)}(z)\\
		=&\int_{f^{-n}_{\te^n\om}(\eta(\Phi^{n}(\ox))_{\te^{n}\omega})} \Delta^{\om}(x,z)d\nu_{(\ox)}(z)\\
		=&\int_{\eta(\Phi^{n}(\ox))_{\te^{n}\omega}} \Delta^{\om}(x,f^{-n}_{\te^n\om}(y))d((f^n_{\om})_{\ast}\nu_{(\ox)})(y)\\
		=&\int_{\eta(\Phi^{n}(\ox))_{\te^{n}\omega}} \dfrac{\Delta^{\om}(x,f^{-n}_{\te^n\om}(y))}{\det(Df^n_{\om}(f^{-n}_{\te^n\om}(y))|E^{u}(\om,f^{-n}_{\te^n\om}(y)))}
		d\nu_{\Phi^{n}(\ox)}(y)\\
		=&\dfrac{1}{\det(Df^n_{\om}(x)|E^{u}(\ox))} \int_{\eta(\Phi^{n}(\ox))_{\te^{n}\omega}} \Delta^{\te^n\om}(f^n_\om x,y)d\nu_{\Phi^{n}(\ox)}(y).
		\end{align*}
	    Let
	    $L(\ox):=\int_{\eta(\ox)_{\omega}} \Delta^{\om}(x,y)d\nu_{(\ox)}(y)$, then we have
	    $$\upsilon_{\eta(\ox)} ((\Phi^{-n}\eta)(\ox)_{\omega})=
	    \dfrac{L(\Phi^n(\ox))}{L(\ox)}\cdot
	    \dfrac{1}{\det(Df^n_{\om}(x)|E^{u}(\ox))}.$$
		Using the same argument in the proof of Theorem A, for each $n>0$ we have
		\begin{equation}\label{eq:S7}
			\int -\log \upsilon_{\eta(\ox)} ((\Phi^{-n}\eta)(\ox)_{\omega})d\mu(\ox)=\int \log\det(Df^n_{\omega}(\ox)|E^{u}(\ox))d\mu(\ox).
		\end{equation}
		Note that the main assumption of the lemma tells us
		\begin{equation}\label{eq:S8}
			\begin{aligned}
				 \int -\log \mu_{\eta(\ox)} ((\Phi^{-n}\eta)(\ox)_{\omega})&d\mu(\ox)=H_{\mu}(\Phi^{-n}\eta|\eta)\\
			    =&\int \log\det(Df^n_{\omega}(\ox)|E^{u}(\ox))d\mu(\ox).
			\end{aligned}
		\end{equation}

		Consider the $\widetilde{\cS}/(\Phi^{-n}\eta)$-measurable function
		$$\psi_n(\ox)=\dfrac{\upsilon_{\eta(\ox)} ((\Phi^{-n}\eta)(\ox)_{\omega})}{\mu_{\eta(\ox)} ((\Phi^{-n}\eta)(\ox)_{\omega})},$$
		which is defined for $\mu$-almost everywhere.
		Together with (\ref{eq:S7}) and (\ref{eq:S8}) we have $\int \log \psi_n d\mu=0$.
		Let $\mathbb{B}_n$ be the sub-$\sigma$-algebra of measurable subsets that are unions of elements of $\Phi^{-n}\eta$, and let $\mu_n$ and $\upsilon_n$ be the restriction of $\mu$ and $\upsilon$ on $\mathbb{B}_n$ respectively.
		Decompose $\upsilon_n=\upsilon_{n}^{\mu}+\upsilon^{\perp}_n$ where $\upsilon_{n}^{\mu}\ll \mu_n$ and $ \upsilon^{\perp}_n $ is mutually singular with $\mu_n$ ($\upsilon^{\perp}_n$ can be strictly positive), then $\psi_n=\frac{d\upsilon_{n}^{\mu}}{d\mu_{n}}$.
		Thus, we have $\int \psi_n d\mu\leq 1$.

		By Jensen's inequality we have
		$$0=\int \log \psi_n d\mu \leq \log\int \psi_n d\mu\leq 0$$
		and the equality holds if and only if  $\psi_n$ is constant for $\mu$-almost everywhere.
		So, $\psi_n\equiv1$ for $\mu$-almost every $(\ox)$.
		This implies $\mu$ and $\upsilon$ coincide on $\mathbb{B}_n$. Since $\Phi^{-n}\eta$ partition into points modulo $\mu$-measure 0, one has $ \mathbb{B}_n\nearrow \cF \otimes \mathcal{B}$, and so $\mu=\upsilon$. This completes the proof.
	\end{proof}

	\begin{proof}[proof of theorem B (ergodic case)]
		By Lemma \ref{Lem:last}, it suffices to show that
		$$h(\mu|\bP,\eta)=h(\mu|\bP).$$
		We will use the following lemma which needed that $A$ is visible:
		\begin{lemma}[\cite{Li13} Proposition 4.2]\label{Lem:visivle}
			Let $(\Phi,\mu)$ be a random dynamical system where $\mu$ is $\Phi$-invariant and supported on a random compact set $A$ with $\Phi(A)=A$. For any function $\tau:A\rightarrow (0,1)$ with $\log\tau \in L^1(\mu)$.
			If $A$ is visible, then there exists a measurable partition $\cP$ of $A$ such that
			$h(\mu|\bP,\cP)<\infty$ and $\cP(\ox)_{\omega}\subset B(x,\tau(\ox))$ for $\mu$-almost every $(\ox)$.
		\end{lemma}
		Let $\tau:\Gamma_{\ell_0}\rightarrow \N$ be the first return time to $\Gamma_{\ell_0}$, extend $\tau$ to all of $A$ by setting $\tau|_{A \setminus \Gamma_{\ell_0}}=0$.
		Let $\rho_{\delta}(\ox):=\delta l^{-2}_0e^{2\tau(\ox)\varepsilon_0}$.
		Since $\tau\in L^1(\mu)$ we have $\log \rho_{\delta}\in L^1(\mu)$.
		Using Lemma \ref{Lem:visivle}, there exists a partition $\cP$ of $A$ with finite conditional entropy and $\cP(\ox)_{\omega}\subset B(x,\rho_{\delta}(\ox))$.

		Let $\cP^{+}:=\vee_{n\geq 0}\Phi^{n}\cP$.
		It is straightforward to check that for $\mu$-almost every $(\ox)$, for each $y\in \cP^{+}(\ox)_{\om}$ one has
		$$\Lnorm{\fo^{-n}y-\fo^{-n}x}{\Phi^{-n}(\ox)}\leq \delta \ell(\Phi^{-n}(\ox))^{-1}.$$
		Then, by Lemma \ref{Lem:CUnstable} one has
		$\cP^{+}(\ox)_{\om}\subset W^{u}_{\delta}(\ox)$ (note that we assume $E^{c}=\{0\}$).

		For each $r>0$. Choose a countable partition $\cP_0$ of $A$ with finite conditional entropy such that
		$h(\mu|\bP)<h(\mu|\bP,\cP_0)-r$.
		Let $\cQ:= \cP\vee\{\widetilde{\cS}\cap A,A-\widetilde{\cS}\}\vee \cP_0$. Then, we also have $h(\mu|\bP)<h(\mu|\bP,\cQ)-r$. Let $ \cQ^{+}=\vee_{n\geq 0}\Phi^{n}\cQ$.
		Use the fact that $\cP^{+}(\ox)_{\om}\subset W^{u}_{\delta}(\ox)$ and the construction of $\eta$ we can check that $\eta(\ox)_{\om}\supset \cQ^{+}(\ox)_{\om}$ for $\mu$-almost every $(\ox)$.
		This implies that $\eta \leq \cQ^{+}\vee \eta_0$.
		Then, we have
		\begin{align*}
			h(\mu|\bP,\eta)&=\frac{1}{n} H_{\mu}(\Phi^{-n}\eta|\eta)\\
			&\geq \frac{1}{n} H_{\mu}(\vee_{j=0}^{n} \Phi^{-j}\cQ|\eta)-\frac{1}{n} H_{\mu}(\vee_{j=0}^{n} \Phi^{-j}\cQ|\Phi^{-n}\eta)\\
			&\geq \frac{1}{n} H_{\mu}(\vee_{j=0}^{n} \Phi^{-j}\cQ|\cQ^{+}\vee \eta_0)-
			      \frac{1}{n} H_{\mu}(\vee_{j=0}^{n} \Phi^{-j}\cQ|\Phi^{-n}\eta)\\
			&\geq h(\mu|\bP,\cQ)-\frac{1}{n} H_{\mu}(\vee_{j=0}^{n} \Phi^{-j}\cQ|\Phi^{-n}\eta),
		\end{align*}
		the second term in above can be shown to be $<r$ for large $n$ since $\vee_{n\geq0}\Phi^{-n} \eta$ partition into points $\mu$-modulo $0$. Therefore, we have that
		$$h(\mu|\bP,\eta)\geq   h(\mu|\bP,\cQ)-r>h(\mu|\bP)-2r.$$
		By arbitrarily of $r$,
		we have $h(\mu|\bP,\eta)=h(\mu|\bP)$.
		This completes the proof of Theorem B in the ergodic case.
	\end{proof}
\section{The non-ergodic case}\label{SEC:NON-ergodic}
	The case when $\mu$ is not ergodic can be proved in the same way with the following modifications.

	The non-ergodic case of the Multiplicative Ergodic Theorem is as follows: fixed a measurable function $ \lambda_{\alpha}$ with $0>\lambda_{\alpha}>l_{\alpha}$.
	Then there is a measurable function $r:\Gamma\rightarrow \N$ such that for every $(\ox)\in \Gamma$, there are $r(\ox)$ Lyapunov exponents
	$$\lambda_1(\ox)\geq \cdots \lambda_{r(\ox)}(\ox)(> \lambda_{\alpha}(\ox))$$
	and a unique splitting
	$$\cB=E_1(\ox)\oplus \cdots E_{r(\ox)}(\ox) \oplus F(\ox)$$
	with respect to the properties (a)-(d) in Theorem \ref{MET} hold.
	Moreover, $\lambda_{i}(\ox)$ and $m_i(\ox):=\dim E_{i}(\ox)$ are measurable functions on $\{(\ox):r(\ox)\geq i\}.$
	The limit in Proposition \ref{Prop:DetL} also exists for $\mu$-almost everywhere, but the equality is
	\begin{align*}
		\int \lim_{n\rightarrow \infty} \frac{1}{n}\log
		\det(Df^{n}_{\omega}(x)|\bigoplus_{j=1}^{k}E_{i_j}(\omega,x)) d\mu(\ox)\\
		=\int m_i(\ox)\lambda_{i_j}(\ox)d\mu(\ox).
	\end{align*}

	The unstable, center, and stable subspace in Section \ref{SEC:4.1} are also defined, and let $\lambda^{\pm}(\ox)$ be as before, which are measurable functions with positive (negative) values but not be bounded away from $0$.
	For $m,n\in \N$ and $p,q\in \Z^{+}$, let
	\begin{align*}
		\Gamma(m,n,p,q):=\{(\ox):&\dim E^{u}(\ox)=m, \dim E^{c}(\ox)=n; \\
		&\lambda^{+}(\ox)\geq \frac{1}{p}, \lambda^{-}(\ox)\leq -\frac{1}{q}\}.
	\end{align*}
	Then, each $\Gamma(m,n,p,q)$ is either empty, or it is $\Phi$-invariant, and
	$$\Gamma=\bigcup_{m,n,p,q}\Gamma(m,n,p,q).$$
	Therefore, the results in Section \ref{SEC:3} and Section \ref{SEC:4} are also holding on $\Gamma(m,n,p,q)$.
	Note that $\Gamma(m,n,p,q)$ is not disjoint, we let
	$$\bar{\Gamma}(m,n,p,q)=\Gamma(m,n,p,q)\setminus \Gamma(m,n,p-1,q-1).$$
	Then, $\bar{\Gamma}$ is disjoint, and $\Gamma$ is a countable union of $\bar{\Gamma}(m,n,p,q)$.
	We can see that $\bar{\Gamma}$ is an alternative to ergodic decomposition \cite{Young17}.

	In non-ergodic cases, it may happen that the set $\widetilde{\cS}$ which is defined in Section \ref{SEC:5} is not a full $\mu$-measure set. However, it is no problem that since there is at most a countable union of disjoint sets of the form $\widetilde{\cS}$ will cover a full $\mu$-measure set.

redpajama_set_name: RedPajamaArXiv