\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{mathrsfs}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 67, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/67\hfil Properties of Lyapunov exponents]
{Properties of Lyapunov exponents for quasiperodic cocycles with singularities}

\author[K. Tao \hfil EJDE-2015/67\hfilneg]
{Kai Tao}

\address{Kai Tao \newline
College of Sciences, Hohai University,
1 Xikang Road, Nanjing, Jiangsu 210098, China}
\email{ktao@hhu.edu.cn, tao.nju@gmail.com}

\thanks{Submitted October 10, 2014. Published March 20, 2015.}
\subjclass[2000]{37C55, 37F10}
\keywords{Lyapunov exponent; quasiperodic cocycles; H\"older exponent}

\begin{abstract}
 We consider the quasi-periodic cocycles
 $(\omega,A(x,E)): (x,v)\mapsto (x+\omega, A(x,E)v)$ with $\omega$
 Diophantine. Let $M_2(\mathbb{C})$ be a normed space endowed with the
 matrix norm, whose elements are the $2\times 2$ matrices. Assume
 that $A:\mathbb{T}\times \mathscr{E}\to M_2(\mathbb{C})$ is jointly continuous,
 depends analytically on $x\in\mathbb{T}$ and is H\"{o}lder continuous in
 $E\in\mathscr{E}$, where $\mathscr{E}$ is a compact metric space and
 $\mathbb{T}$ is the torus. We prove that if two Lyapunov
 exponents are distinct at one point $E_0\in\mathscr{E}$, then these two
 Lyapunov exponents are H\"{o}lder continuous at any $E$ in a ball
 central at $E_0$. Moreover, we will give the expressions of the radius
 of this ball and the H\"older exponents of the two Lyapunov exponents.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Denote by $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ the torus equipped
with its Haar measure $\mu$,  and $\mu(\mathbb{T})=1$. Let
$M_d(\mathbb{C})$ be the set of linear operators from $\mathbb{C}^d$
to $\mathbb{C}^d$, i.e. the set of $d\times d$ complex matrices. A
quasi-periodic cocycle is a pair $(\omega, A)$, where $\omega$, the
irrational number, is the frequency and 
$A\in C^0(\mathbb{T},M_d(\mathbb{C}))$ is continuous, defined by a map
$(\omega, A):(x,v)\to (x+\omega, A(x)v)$. The iterates of the
cocycle are given by $(\omega, A)^N=(N\omega, A_N)$, where
\[A_N(x)=\prod_{j=N-1}^0 A(x+j\omega).\]
In this case, the dynamical system
is ergodic and the Oseledets Theorem provides us with a  sequence of
Lyapunov exponents $L_1\leq L_2\leq \dots \leq L_m$, and for almost
every $x\in\mathbb{T}$ there exist an invariant measurable
decomposition $\mathbb{C}^d=\oplus_{j=1}^nE_x^j$, and a non
decreasing surjective map $k:\{1,2,\dots,d\}\to \{1,\dots,n\} $
such that for almost every $x\in\mathbb{T}$, every $1\leq i \leq m$
and every $v\in E_x^{k_i}\backslash \{0\}$ we have
$L_j=\lim_{N\to\infty}\frac{1}{N}\log \|A_N(x) v\|$. Moreover,
$L_i=L_{i+1}$ if and only if $k_i=k_{i+1}$, and the subspace $E_x^j$
has dimension equal to $\sharp k^{-1}(j)$, where $\sharp k^{-1}(j)$
is the number of the elements in the set $\{i|k_i=j\}$.

In the past several years, some researchers focused on the continuity
of Lyapunov exponent for the Schr\"{o}dinger equation:
\[
(S_{x,\omega}\phi)(n)=\phi(n+1)+\phi(n-1)+v(x+n\omega)\phi(n)=E\phi(n),\quad
n\in\mathbb{Z},
\]
with the cocycle
\[
A(x,E)=\begin{pmatrix}
v(x)-E& -1 \\
1& 0
\end{pmatrix}.
\]
 Goldstein and Schlag \cite{GS} proved that if $v(x)$ is
analytic and $\omega$ is a Diophantine, which will mean that
\begin{equation}\label{dio}
\|n\omega\|\geq \frac{c(\omega)}{n(\log n)^a},\quad\forall n\geq 2
\end{equation}
with $a>1$ arbitrary but fixed, the Lyapunov
exponent $L(E)$ is H\"{o}lder continuous.
Then You and Zhang showed the similar result with more general $\omega$ in \cite{YZ}.
Observing that for the
Schr\"{o}dinger cocycle,  $\det A\equiv 1$ and $L_1(E)+L_2(E)=0$ for
any $E$, it makes that we can only study the Lyapunov exponent
defined by
\begin{equation}\label{lyaexp}
L(E):=\lim_{N\to\infty}\frac{1}{N}\int_{\mathbb{T}}\log \|A_N(x,E) \|dx .
\end{equation}
Recall that we say a function $f(E)$
is H\"{o}lder continuous, when there are nonnegative real constants
C, $\alpha$, such that
\[
|f(E_1)-f(E_2)|<C|E_1-E_2|^{\alpha}
\]
for all $E_1$ and $E_2$ in the domain of $f(E)$. The number
$\alpha$ is called the exponent of the H\"{o}lder condition or the
H\"{o}lder exponent. Compared with the Schr\"{o}dinger equation, the
Jacobi operator is more complicated with the cocycle
\[
 A(x,E)=\begin{pmatrix}
a(x)-E& -b(x) \\
b(x+\omega)& 0
\end{pmatrix}.
\]
The author in \cite{T} showed that the Lyapunov
exponent $L(E)$ defined in \eqref{lyaexp} is H\"{o}lder continuous,
and the H\"{o}lder exponent does not depend on the $L(E)$.
It is a better result than what in \cite{GS}, as the H\"{o}lder exponent
of the Lyapunov exponent in \cite{GS} depends on $L(E)$. Later,
for the $2\times 2$ analytic quasi-periodic cocycles
$A(x)\in C^{\omega}_r (\mathbb{T},M_2(\mathbb{C}))$
which have a holomorphic
extension to a neighborhood of the strip
$\mathbb{S}_r:=\{z\in \mathbb{C}:|\mathrm{Im} z|\leq r\}$ and is endowed with the norm
\[
\|A\|_r:=\sup_{z\in \mathbb{S}_r}\|A(z)\|,
\]
Jitomirskaya, Koslover and Schulteis \cite{JKS} proved that the
Lyapunov exponent $L(A)$, which is defined by
\begin{equation}\label{lyaexp2}
L(A):=\lim_{N\to\infty}\frac{1}{N}\int_{\mathbb{T}}
\log\|\prod_{j=N-1}^0 A(x+j\omega) \|dx ,
\end{equation}
is a continuous for the fixed determinant with Diophantine $\omega$.
The author gave a proof of this result for the High dimensional torus
in \cite{T1}. Similar problems for higher dimensional quasi-periodic cocycles have
been studied by Schlag \cite{S}. In that paper, the  H\"{o}lder continuity is proven
if  all Lyapunov exponents defined in Oseledets Theorem are unequal.


This article concerns certain quasi-periodic cocycles
$(\omega,A(x,E))$ defined as follow. The $\omega$ is defined as
\eqref{dio} and it is well know that almost every $\omega \in (0,1)$
satisfies this condition. The function $x\to A(x,E)$ is a element of
the Banach space $C^{\omega}_r (\mathbb{T},M_2(\mathbb{C}))$. The
variable $E$ in the matrix $A(x,E)$ is defined as a parameter, and
the parameter space $(\mathscr{E},\mathrm{d})$ is a compact metric
space.  The Lyapunov exponents $L_1(E)$ and $L_2(E)$ which are
concerned in this paper are defined by the Oseledets Theorem. The
purpose of this paper is to study some properties of these two
Lyapunov exponents which can be regarded as  functions of the
parameter $E$.  Well, we proof  the following main theorem:

\begin{theorem}\label{mainthm}
Let $A(x,E)$ be jointly continuous in $\mathbb{T}\times\mathscr{E}$,
analytic uniformly in $E\in \mathscr{E}$ as a function $x\mapsto A(x,E)$,
and H\"{o}lder continuous with H\"{o}lder exponent $\beta$
as a function $E\mapsto A(x,E)$.
Assume $L_1(E_0)-L_2(E_0):=2\tilde{L}(E_0)\not=0$. Then there exists
$\rho>0$ such that for any $E\in (E_0-\rho, E_0+\rho)$, $L_1(E)$ and
$L_2(E)$ are H\"{o}lder continuous with H\"{o}lder exponent
$\alpha$. Moreover, $\alpha=c\beta$, where $c$ is a positive and small
constant depending only on $A(x,E)$ but not on $\tilde{L}(E_0)$;
and the $\rho$ has an expression as following:
\[
\rho(A,\tilde L(E_0))=\Big[\tilde L(E_0)
\exp \Big(-\frac{C_{\rho,1}}{\tilde{L}^{C_{\rho,2}}(E_0)}\Big)
\Big]^{1/\beta},
\]
where $C_{\rho,1}, C_{\rho,2}$ are the
big constants depending only on $A(x,E)$.
\end{theorem}

\begin{remark}\rm
The parameter space $(\mathscr{E},\mathrm{d})$ here is
always being the spectral spaces of operators, such like the
Schr\"{o}dinger equation \cite{GS}, the extended Happer's model
\cite{JM} and the Jacobi operators \cite{T}. It is well known
that the cocycle is uniform hyperbolic when the energy $E$ is not in
the spectral. Thus, we assume that this parameter space is compact,
 as the discrete operators' spectrums are always bounded.
\end{remark}

   The study of the H\"{o}lder exponent of the Lyapunov exponents
is a hot spot in our field. For the almost Mathieu operator with the
cocycle
\[
A(x,E)=\begin{pmatrix}
\lambda\cos x-E& -1 \\
1& 0
\end{pmatrix},
\]
Avila and Jitomirskaya proved that the Lyapunov exponent defined
by \eqref{lyaexp} is H\"{o}lder continuous with H\"{o}lder exponent
$1/2$, provided $\omega$ is a Diophantine number \cite{AJ}. For
the general Schr\"{o}dinger operator with the cocycle
\[
A(x,E)=\begin{pmatrix}
v(x)-E& -1 \\
1& 0
\end{pmatrix},
\]
supposing  $v(x)$ is a small
perturbation of a trigonometric polynomial $v_0(x)$ of degree $k_0$
and  $\tilde{L}(E)>0$, Goldstein
and Schlag proved that Lyapunov exponent is H\"older
continuous with H\"older exponent $\frac{1}{2k_0}-\kappa$ for any $\kappa>0$
\cite{GS1}.
In this article, we study the continuity of the Lyapunov exponent from
a new perspective. We want to show the relationship between
the continuity of the function $E\to A(x,E)$ and the continuity of
the Lyapunov exponents. \cite{WY} showed that the assumption that
$E\mapsto A(x,E)$ is analytic, is necessary. Here only consider the
condition that $A(x,E)$ is a $2\times 2$ matrix, but we believe that
the main theorem also work if  $A(x,E)\in M_d(\mathbb{C})$.


This article is organized as follows. In section 2 we get the Large
Deviation Theorem and some upper bound estimate by using some
propositions of the subharmonic functions. In section 3 we apply
Avalanche principle twice to prove the sharp large deviation theorem.
The proof of the main theorem is presented in section 4.

Some other common senses about analytic functions which will be applied
in this article are presented as follows.
Because $\mathscr{E}$ is compact, there
exist $D(A)$, $C_2(A)$ and $C_{\rm max}(A)$ , such that for any
$E\in\mathscr{E}$,
\begin{gather*}
\|\log |\det A(\cdot,E)|\|_1\leq D(A),\\
\|\log \|A(\cdot,E)\|-\frac{1}{2}\log|\det
A(\cdot,E)|\|_2=C_2(A,E)\leq C_2(A)
\end{gather*}
 and for any $x\in\mathbb{T}$ and any $E\in\mathscr{E}$,
\[
\|A(x,E)\|\leq C_{\rm max}(A,E)\leq C_{\rm max}(A).
\]
Also, by the suppose that $E\mapsto A(x,E)$ is H\"{o}lder continuous
in $x\in \mathbb{T}$ with H\"older exponent $\beta$, it is easy to
see that
\[
|D(E)-D(E')|\leq C_D(A)|E-E'|^{\beta}, \quad \forall E,E'\in \mathscr{E},
\]
where $D(E):=\int_{\mathbb{T}}\log|\det A(x,E)|dx$.

\section{Large deviation theorem and the upper bound estimate}

It is obvious that we can define
$L_1(E)=\lim_{N\to\infty}\frac{1}{N}\int_{\mathbb{T}}\log
\|A_N(x,E)\|dx$. For easy notation, we replace $L_1(E)$ by $L(E)$.

 It is also convenient to replace
$A_N(x,E)$ by $A_N(e(x),E)$ (with $e(x)=e^{2\pi ix}$), where
$A_N(z,E)$ is analytic function in the annulus
$\mathscr{A}_\rho=\{z\in\mathbb{C}:1-\rho< |z|< 1+\rho\}$ uniformly
in $E\in\mathscr{E}$.

Set $u_n(z,E)=\frac{1}{n}\log \|A_N(z,E)\|$,
$d_N(z,E)=\frac{1}{n}\log |\det A_N(z,E)|$. Sometimes we use
$u_N(z)$ or $u_N$ for short, and the same for $d_N(z,E)$.
Let $L_{N,r}(E)=<u_N(re(\cdot))>$, $D_r(E)=<\log |\det A(re(\cdot))>$.
For $r=1$ we use notations $L_N(E)$ and $D(E)$.


Note that $u_N(z)$ and $d_N(z)$ are subharmonic functions in $\mathscr{A}_\rho$
and then  the following Large Deviation
Theorem for the subharmonic functions applies, provided $\omega$ satisfies \eqref{dio}.

\begin{theorem}[{\cite[Theorem 2.15, Remark 2.16]{T}}] \label{22006}
There exists $\check{N}(A,\omega)$ such that for any $N\geq \check{N}$,
any $1-\frac{\rho}{2}\leq r\leq 1+\frac{\rho}{2}$ and $\delta<1$ holds
\begin{gather*}
\operatorname{meas}(\{x:|u_N(re(x))-L_r |>\delta \})<\exp(-\check{c}\delta^2 N),\\
\operatorname{meas}(\{x:|d_N(re(x))-D_r |>\delta \})<\exp(-\check{c}\delta^2 N),
\end{gather*}
where $\check{c}=\check{c}(A)$.
\end{theorem}

\begin{remark}\label{29} \rm
(1) For the avalanche principle in the next section, we need to define
the unimodular matrix of $A_N(z,E)$:
\begin{equation} \label{26001}
\tilde{A}_N(z,E):=\frac{1}{|\det
A_N(z,E)|^{1/2}}A_N(z,E),
\end{equation}
and the unimodular function of $u_N(z,E)$
\[ %\label{26002}
\tilde{u}_N(z,E):=\frac{1}{N}\log\|\tilde{A}_N(z,E)\|=u_N(z,E)
-\frac{1}{2N}\log|\det(A_N(z,E))|.
\]
 By Theorem \ref{22006}, for any $N\geq \check{N}$
\begin{equation}\label{25001}
\operatorname{meas}(\{x:|\tilde{u}_N(re(x))
-\tilde{L}_{N,r}(E)|>\delta\})<\exp(-\tilde{c}\delta^2 N),
\end{equation}
where
$\tilde{L}_{N,r}(E)=<\tilde{u}_N(re(\cdot))>=L_{N,r}-\frac{D_r}{2}$.

(2) In the proof of Theorem \ref{22006}, the following lemma will be necessary
and apply in the later of this paper:

\begin{lemma}[{\cite[Lemma 2.12]{T}}] \label{lem2.3}
For any $1-\frac{\rho}{2}\leq r \leq 1+\frac{\rho}{2}$, $\delta$ and $K$,
\[
  \operatorname{meas}\Big\{x:|\sum_{k=1}^K u_N(re(x+k\omega))-K \tilde{L}_{N,r}  |
>\delta K \Big\} <\exp(-c\delta K),
\]
where $c=c(A,\omega)$.
\end{lemma}

(3) let us note that the constants $c,\tilde{c}$ here do not depend on
 $\delta$. In particular, one can
choose here $\delta$ depending on N.
\end{remark}

 In the next part of this section, we estimate the upper
bound of the subharmonic function $u_N(z,E)$ and some other
functions. Now we first review a lemma from \cite{GS1}.

\begin{lemma}[{\cite[Lemma 4.1]{GS1}}] \label{lem:liprad}
For any $r_1,r_2$ so that $1-\frac{\rho}{2}\leq r_1,r_2\leq
1+\frac{\rho}{2}$ one has
\[
u_N(re(\cdot),E)>-L_N(E)|\leq C_{\rho}|r-1|.
\]
\end{lemma}

\begin{lemma}\label{22010}
For any $N\geq \check{N}(A)$
\[
\frac{1}{N}\log\|A_N(e(x),E)\|\leq L_N(E)
+C_6\big(\frac{\log N}{N}\big)^{1/2},
\]
 where $C_6=C_6(A)$ and $\check{N} $ is as
in Theorem \ref{22006}.
\end{lemma}

\begin{proof}
Let $0<\delta<\frac{\rho}{4}$ be arbitrary. Note that
$e(x+iy)=e^{-2\pi y}e(x)$,
$1-\frac{\rho}{4}\leq e^{-2\pi y}\leq 1+\frac{\rho}{4}$, if
$|y|\leq \frac{\delta}{4\pi e C'_\rho}$, where $C'_\rho=\max(1,C_\rho)$ and
$e=\exp(1)$. By Lemma \ref{lem:liprad}, one has
\begin{equation}\label{25019}
|\langle u_N(re(\cdot))\rangle -L_N(E)|
\leq \delta,\quad \text{if } |y|\leq \frac{\delta}{4\pi e C'_\rho}.
\end{equation}
 Set
\[
\mathbb{B}_y:=\{x:|u_N(e(x+iy))-L_N(E)|>2\delta\}.
\]
It follows from \eqref{25019} that for $|y|\leq \frac{\delta}{4\pi e
C'_\rho}$, there holds
\[
\mathbb{B}_y\subseteq
\{x:|u_N(e(x+iy))-<u_N(e(\cdot+iy))>|>\delta\}.
\]
By Theorem \ref{22006} one obtains
 $\operatorname{meas}\mathbb{B}_y\leq \exp(-\check{c}\delta^2 N)$.
The function $u_N(e(x+iy))$ is
subharmonic, for $e(x+iy)\in \mathscr{A}_\rho$. Let $x_0$ be
arbitrary and $y_0=0$.
Then $e(x_0)\in \mathscr{A}_{\frac{\rho}{4}}$.
Due to subharmonicity one has for any
$t_0<\frac{\rho}{4}$,
\begin{align*}
u_N(e(x_0))-L_N(E)
&\leq \frac{1}{\pi t_0^2}\iint_{|(x,y)-(x_0,0)|\leq
t_0}[u_N(e(x+iy))-L_N]\,dx\,dy \\
&=\frac{1}{\pi t_0^2}\int_{|y|\leq t_0}\int_{|x- x_0|\leq
\sqrt{t_0^2-|y|^2}}[u_N(e(x+iy))-L_N]\,dx\,dy.
\end{align*}
Furthermore
\begin{align*}
&\int_{|x- x_0|\leq \sqrt{t_0^2-|y|^2}}[u_N(e(x+iy))-L_N]dx\\
&=\Big(\int_{\{|x- x_0|\leq
\sqrt{t_0^2-|y|^2}\}\bigcap \mathbb{B}_y}+\int_{\{|x- x_0|\leq
\sqrt{t_0^2-|y|^2}\}\setminus \mathbb{B}_y}
\Big)[u_N(e(x+iy))-L_N]dx.
\end{align*}
Note that
\[
|u_N(e(x+iy))-L_N|\leq 2\delta,\quad \text{if }
x\not\in \mathbb{B}_y \text{ and } y<\frac{\delta}{4e\pi C'_\rho}.
\]
So
\[
\big|\int_{\{|x-
x_0|\leq \sqrt{t_0^2-|y|^2}\}\setminus
\mathbb{B}_y}[u_N(e(x+iy))-L_N]dx\big|
\leq 2\delta\times(2\sqrt{t_0^2-|y|^2}).
\]
By Cauchy-Schwartz inequality,
\begin{align*}
&\big|\int_{\{|x-x_0|\leq \sqrt{t_0^2-|y|^2}\}\bigcap
\mathbb{B}_y}[u_N(e(x+iy))-L_N]dx\big|\\
&\leq \Big(\int_0^1|u_N(e(x+iy))-L_N|^2dx\Big)^{1/2}
(\operatorname{meas}\mathbb{B}_y)^{1/2}\\
&\leq C_7\exp(-\frac{\check{c}}{2}\delta^2 N).
\end{align*}
Set
$t_0=\delta/(4e\pi C'_\rho)$; then
\begin{align*}
u_N(e(x))-L_N
&\leq \frac{1}{\pi t_0^2}\int_{|y|\leq
t_0}[C_7\exp(-\frac{\check{c}}{2}\delta^2
N)+2\delta\times(2\sqrt{t_0^2-|y|^2})]dy \\
&\leq \frac{1}{\pi t_0^2}\times C_7\exp(-\frac{\check{c}}{2}\delta^2
N)\times (2t_0)+2\delta \\
&= \frac{8e C_7C'_\rho}{\delta}\exp(-\frac{\check{c}}{2}\delta^2
N)+2\delta .
\end{align*}
 Set $\delta=\big(\frac{C_8\log N}{N}\big)^{1/2}$, where
 $C_8>2/\check{c}$. Then
$\exp(-\frac{\check{c}}{2}C_8\log N)<\frac{1}{N}$, and
\[
u_N(e(x))\leq L_N +8e C_7C'_\rho\times(\frac{N}{C_8\log N})^{1/2}
 \frac{1}{N}+2(\frac{C_8\log N}{N})^{1/2}
\leq L_N +C_6(\frac{\log N}{N})^{1/2}.
\]
\end{proof}

\begin{lemma}\label{22021}
Set
\[
F_N(x,E):=\frac{1}{2N}\sum_{n=0}^{N-1} \big(\log|\det
A(e(x+n\omega),E)|-D(E)\big).
\]
Then
\begin{itemize}
\item[(1)] For any $N$ and any $k$ holds
\begin{align*}
&|\log\|\tilde{A}_N(e(x+k\omega),E)\|-\log\|\tilde{A}_N(e(x),E)\|\, |\\
&\leq 2k(\log C_{\rm max}(A,E)-\frac{1}{2}D(E))-kF_k(x)-kF_k(x+N\omega),
\end{align*}

\item[(2)] For any $N$  and any $k\geq \check{N}$ holds
\begin{align*}
&|\log\|\tilde{A}_N(e(x+k\omega),E)\|-\log\|\tilde{A}_N(e(x),E)\|\,|\\
&\leq 2k\tilde{L}_k(E)+2C_6(k\log k)^{1/2}-kF_k(x)-kF_k(x+N\omega).
\end{align*}
\end{itemize}
\end{lemma}

\begin{proof}
(1) Recall that any $N$,
$u_N(e(x),E)\leq \log C_{\rm max}(A,E)$. Then
\begin{align*}
&\log\|\tilde{A}_N(e(x),E)\|\\
&= Nu_N(e(x),E)-\frac{1}{2}\sum_{n=0}^{N-1}\log
|\det A(e(x+n\omega),E)| \\
&\leq N\log C_{\rm max}(A,E) -\frac{1}{2}\sum_{n=0}^{N-1}\log |\det
A(e(x+n\omega),E)| \\
&= N (\log C_{\rm max}(A,E)-\frac{1}{2}D(E)
)-\frac{1}{2}\sum_{n=0}^{N-1}(\log|\det A(e(x+n\omega),E)|-D(E)) \\
&= N (\log C_{\rm max}(A,E)-\frac{1}{2}D(E))-NF_N(x,E)
\end{align*}
One has if $\det M=1$, then $\|M\|=\|M^{-1}\|$ and
\[
\tilde{A}_N(e(x+k\omega),E)\tilde{A}_k(e(x),E)
=\tilde{A}_k(e(x+N\omega),E)\tilde{A}_N(e(x),E).
\]
Thus,
\begin{align*}
&|\log\|\tilde{A}_N(e(x+k\omega),E)\|-\log\|\tilde{A}_N(e(x))\|\,|\\
&\leq \log\|\tilde{A}_k(e(x),E)\|+\log\|\tilde{A}_k(e(x+N\omega),E)\| \\
&\leq  2k (\log C_{\rm max}(A,E)-\frac{1}{2}D(E)
)-kF_k(x,E)-kF_k(x+N\omega,E)
\end{align*}

(2) From the previous lemma, we know that for any
$N\geq \check{N}(A)$,
\[
\frac{1}{N}\log\|A_N(e(x),E)\|
\leq L_N(E) +C_6(\frac{\log N}{N})^{1/2}.
\]
Then the rest follows as in  part (1).
\end{proof}

\section{Avalanche principle and the sharp large deviation theorem}

For the rest of the paper, without special statement,
$N\geq \check{N}$ and $N\geq \check{K}$ from now on
($\delta$ in $\check{K}$ will be defined in
Lemma \ref{22012}). Furthermore, we do not use $e(x+iy)$ with $y\not=0$
and write $x$ instead of $e(x)$ in all expressions.

\begin{proposition} \label{prop:AP}
Let $A_1,\ldots,A_n$ be a sequence of  $2\times 2$-matrices whose determinants
satisfy
$ %\label{eq:detsmall}
\max_{1\le j\le n}|\det A_j|\le 1$.
Suppose that
\begin{gather*}
\min_{1\le j\le n}\|A_j\|\ge\mu>n, %\label{large} \\
 \max_{1\le j<n}[\log\|A_{j+1}\|+\log\|A_j\|-\log\|A_{j+1}A_{j}\|]
<\frac12\log\mu %\label{diff}.
\end{gather*}
 Then
\begin{equation}
\Big|\log\|A_n\cdot\ldots\cdot A_1\|+\sum_{j=2}^{n-1}
\log\|A_j\|-\sum_{j=1}^{n-1}\log\|A_{j+1}A_{j}\|\Big| <
C\frac{n}{\mu} \label{eq:AP}
\end{equation}
with some constant $C$.
\end{proposition}

\begin{proof}
This lemma is called the Avalanche Principle. For the proof, see \cite{GS}.
Recently, Schlag \cite{S} gave a general Avalanche Principle for $n\times n$
matrix.
\end{proof}

\begin{lemma}\label{62001}
Let $\tilde{c}$ be as in \eqref{25001}.
Let $\tilde{L}_N(E)>100\delta>0$, where $\delta<1$ is a constant not
depending on $N$, and
$\tilde{L}_{2N}(E)>\frac{9}{10}\tilde{L}_{N}(E)$. Let $N'=m N$,
$m\in \mathbb{N}$ and $ m\leq \exp(\frac{\tilde{c}}{4}\delta^2 N) $.
Then
\[
|\tilde{L}_{N'}(E)+\tilde{L}_{N}(E)-2\tilde{L}_{2N}(E)|
\leq \exp(-\tilde{c}'\delta^2 N)+\frac{2}{9m}L_N(E),
\]
where
$\tilde{c}'=\tilde{c}'(A)$. If
$\exp(\frac{\tilde{c}}{10}\delta^2 N) \leq m\leq \exp(\frac{\tilde{c}}{4}\delta^2 N)$,
 we have
\[
|\tilde{L}_{N'}(E)+\tilde{L}_{N}(E)-2\tilde{L}_{2N}(E)|
\leq \exp(-\hat{c}\delta^2 N),
\]
where $\hat{c}=\hat{c}(A)$. Furthermore,
if $\tilde{L}_{N_0}(E)>100\delta>0$,
$\tilde{L}_{2N_0}(E)>\frac{9}{10}\tilde{L}_{N_0}(E)$ and
$\exp(-\hat{c}\delta^2N_0)\leq \delta/12$, then there exists
$\tilde{N}_0=\tilde{N}_0(A,\delta,N_0)\leq
(\exp(\frac{\tilde{c}}{8}\delta^2 N_0)+1)N_0$ such that for any
$N\geq \tilde{N}_0$,
 \[
|\tilde{L}_N(E)+\tilde{L}_{N_0}(E)-2\tilde{L}_{2N_0}(E)|<\exp(-\bar{c}'\delta^2
N_0),
\]
where $\bar{c}'=\bar{c}'(A)$. Furthermore,
 \[
 |\tilde{L}(E)+\tilde{L}_{N_0}(E)-2\tilde{L}_{2N_0}(E)|< \exp(-\bar{c}\delta^2
 N_0),
\]
where $\bar{c}=\bar{c}(A)$.
\end{lemma}

\begin{proof}
In section three in \cite{T}, we proved this lemma for Jacobi operator,
 which is a special $2\times 2$ analytic matrix. It is easy to see that the
proof there is suitable in this general condition.
\end{proof}

\begin{lemma}\label{62003}
Assume $\tilde{L}(E_0)>0$. There exists
$\check{C}(A,\tilde{L}(E_0))$ such that with
$\rho_0'=\big[\frac{\tilde{L}(E_0)}{200N}
\exp\big(-\check{C}(A,\tilde{L}(E_0))N\big)\big]^{1/\beta}$, one
has
\[
|\tilde{L}_{N}(E_0)-\tilde{L}_{N}(E)|<\frac{\tilde{L}(E_0)}{100},
\]
for any
$|E-E_0|<\rho_0'(E_0,N)$ and any $N$.
\end{lemma}

\begin{proof}
Note that
\begin{equation} \label{70007}
\begin{aligned}
&|\,\|A_{N}(x,E_0)\|-\|A_{N}(x,E)\|\, |\\
&\leq \|A_{N}(x,E_0)-A_{N}(x,E)\| \\
&\leq \sum_{j=0}^{N-1}\big(\|A(x+(N-1)\omega,E_0)\dots
A(x+(j+1)\omega,E_0)\| \\
&\quad\times \|A(x+j\omega,E_0)- A(x+j\omega,E)\|\,
\|A(x+(j-1)\omega,E) \dots A(x,E)\|\big ) \\
&\leq NC_{\rm max}(A)^{N-1}\times C(A)|E_0-E|^{\beta} .
\end{aligned}
\end{equation}
 By \eqref{26001}, if $|\det A_N(x,E_0)|\leq |\det A_N(x,E)|$, one has
\begin{equation} \label{70008}
\begin{aligned}
|\,\|\tilde{A}_{N}(x,E_0)\|-\|\tilde{A}_{N}(x,E)\|\, |
&= \big|\frac{\|A_{N}(x,E_0)\|}{|\det A_N(x,E_0)|^{1/2}}
-\frac{\|A_{N}(x,E)\|}{|\det A_N(x,E)|^{1/2}}\big |\\
&\leq \frac{NC_{\rm max}(A)^{N-1} C(A)|E_0-E|^{\beta}}{|\det
A_N(x,E_0)|^{1/2}}
\end{aligned}
\end{equation}
Assume for instance that $\|\tilde{A}_{N}(x,E_0)\|\geq \|\tilde{A}_{N}(x,E)\|$.
Then
\begin{equation} \label{60002}
\begin{aligned}
& |\log\|\tilde{A}_{N}(x,E_0)\|-\log \|\tilde{A}_{N}(x,E)\| \,|\\
&= \log\frac{\|\tilde{A}_{N}(x,E_0)\|}{\|\tilde{A}_{N}(x,E)\|}
=\log (1+\frac{\|\tilde{A}_{N}(x,E_0)\|
 -\|\tilde{A}_{N}(x,E)\|}{\|\tilde{A}_{N}(x,E)\|})\\
&\leq \frac{\|\tilde{A}_{N}(x,E_0)\|-
\|\tilde{A}_{N}(x,E)\|}{\|\tilde{A}_{N}(x,E)\|}
\leq\|\tilde{A}_{N}(x,E_0)\|- \|\tilde{A}_{N}(x,E)\| \\
&\leq \frac{NC_{\rm max}(A)^{N-1} C(A)|E_0-E|^{\beta}}{|\det
A_N(x,E_0)|^{1/2}}.
\end{aligned}
\end{equation}
By  Theorem \ref{22006},  for any $\delta$ and any $K$
\[
\operatorname{meas}\Big\{x:|\sum_{k=1}^K\log
|\det A(x+k\omega)|-K\langle \log |\det A(\cdot)|\rangle |
>\delta K\Big\}<\exp(-c'\delta K).
\]
Thus if
$x\not\in\mathbb{B}_1$,
$\operatorname{meas}{\mathbb{B}_1}<\exp(-c'\times\frac{800C_2(A)}{\tilde{L}(E_0)c'}N)
=\exp(-\frac{800C_2(A)}{\tilde{L}(E_0)}N)$,
 then
\begin{align}
 |\log |\det A_N(x,E_0)|\, |
&< |\langle \log |\det A(\cdot,E_0)|\rangle
|N+\frac{800C_2(A)}{\tilde{L}(E_0)c'}N \nonumber \\
&=|D(E_0)|N+\frac{800C_2(A)}{\tilde{L}(E_0)c'}N  \nonumber \\
&\leq D(A)N+\frac{800C_2(A)}{\tilde{L}(E_0)c'}N
=\hat{C}(A,\tilde{L}(E_0))N. \label{30010}
\end{align}
It is obvious  that the same estimate holds in the other three
conditions.

So if $x\not\in\mathbb{B}_1$,
$\operatorname{meas}{\mathbb{B}_1}
<\exp(-\frac{800\tilde{C}(A)^{1/2}}{\tilde{L}(E_0)}N)$, then
\begin{align*}
|\tilde{u}_{N}(x,E_0)-\tilde{u}_{N}(x,E)|
&\leq C_{\rm max}(A)^{N-1}C(A)|E_0-E|^{\beta}\exp(\hat{C}(A,\tilde{L}(E_0))N)\\
&\leq \exp\left(\check{C}(A,\tilde{L}(E_0))N\right)|E_0-E|^{\beta}.
\end{align*}
Set $\rho'_0=\big[\frac{\tilde{L}(E_0)}{200}
\exp\left(-\check{C}(A,\tilde{L}(E_0))N\right )\big]^{1/\beta}$.
Then, if $|E-E_0|\leq \rho_0'$, we have
$\left|\tilde{u}_{N}(x,E_0)-\tilde{u}_{N}(x,E) \right|
<\frac{\tilde{L}(E_0)}{200}$, and if
$x\not\in\mathbb{B}_1,\operatorname{meas}{\mathbb{B}_1}
<\exp(-\frac{800\tilde{C}(p,q)^{1/2}}{\tilde{L}(E_0)}N)$, we also have
\begin{equation}\label{60012}
\Big|\int_{\mathbb{T}\backslash
\mathbb{B}_1}\tilde{u}_{N}(x,E_0)dx-\int_{\mathbb{T}\backslash
\mathbb{B}_1}\tilde{u}_{N}(x,E)dx \Big|
<\frac{\tilde{L}(E_0)}{200}.
\end{equation}
 By the Cauchy-Schwartz inequality,
\begin{align*}
\big|\int_{\mathbb{B}_1}\tilde{u}_{N}dx\big|
&=\big|\int_{\mathbb{T}}\tilde{u}_N1_{\mathbb{B}_1}dx\big|\\
&\leq \|\tilde{u}_N(E)\|_2 (\operatorname{meas}\mathbb{B}_1)^{1/2}\\
&\leq C_2(A)\exp(-\frac{400C_2(A)}{\tilde{L}(E_0)}N)
\end{align*}
for $E $ or $E_0$.
As $y\exp(-\xi y)\leq \xi^{-1}$ for any $y, \xi>0$. Thus
\begin{equation}\label{60013}
\big|\int_{\mathbb{B}_1}\tilde{u}_{N}dx\big|
\leq \frac{\tilde{L}(E_0)}{400N}
\leq \frac{\tilde{L}(E_0)}{400}
\end{equation}
for $E$ or $E_0$. Combining \eqref{60012} with \eqref{60013}, one has
\[
|\tilde{L}_{N}(E_0)-\tilde{L}_{N}(E)|
<\frac{\tilde{L}(E_0)}{200}+2\frac{\tilde{L}(E_0)}{400}
=\frac{\tilde{L}(E_0)}{100}.
\]
\end{proof}

\begin{lemma}\label{62006}
Assume $\tilde{L}(E_0)>0$. There exists $\rho=\rho(A,\tilde{L}(E_0))>0$
and $\tilde{N}_0=\tilde{N}_0(A,E_0)<+\infty$ such
that for any $N\geq\tilde{N}_0$ and any $|E-E_0|<\rho$
 \[
|\tilde{L}_{N}(E)-\tilde{L}(E)|<\frac{1}{20}\tilde{L}(E),\quad
\frac{11}{10}\tilde{L}(E_0)>\tilde{L}(E)>\frac{9}{10}\tilde{L}(E_0).
\]
\end{lemma}

\begin{proof}
One has $\lim_{n\to\infty}\tilde{L}(E_0)=\tilde{L}(E_0)$.
Therefore, there exists $N_0=N_0(A,E_0)$  such that the following
statements hold:
\begin{itemize}
\item[(1)]
$|\tilde{L}_n(E_0)-\tilde{L}(E_0)|<\frac{\tilde{L}(E_0)}{100}$ for
$n\geq N_0(A,E_0)$, which implies  that
$\tilde{L}_{N_0}(E_0)-\tilde{L}_{2N_0}(E_0)<\frac{\tilde{L}(E_0)}{100}$,
as $\tilde{L}(E_0)\leq \tilde{L}_{2N_0}(E_0)\leq
\tilde{L}_{N_0}(E_0)$;

\item[(2)]
$\exp(-\hat{c}\delta^2N_0)\leq \frac{\delta}{12}$,
$\exp(-\bar{c}\delta^2
 N_0))<\frac{1}{50}\tilde{L}(E_0)$, $\exp(-\bar{c}'\delta^2
 N_0))<\frac{1}{50}\tilde{L}(E_0)$ with
$\delta=\min(\frac{1}{200}\tilde{L}(E_0),\frac{1}{2})$, where
$\hat{c}$ , $\bar{c}$ and $\bar{c}'$ are as Lemma \ref{62001}.
\end{itemize}
Using
Lemma \ref{62003} applied to $N_0$ and $2N_0$. One has for
$|E-E_0|<\rho(A,\tilde{L}(E_0)):=\rho'_0(\tilde{L}(E_0),2N_0)$,
\begin{equation} \label{60033}
\begin{aligned}
\tilde{L}_{N_0}(E)
&\geq \tilde{L}(E_0)-
|\tilde{L}_{N_0}(E)-\tilde{L}_{N_0}(E_0)|-|\tilde{L}_{N_0}(E_0)-\tilde{L}(E_0)|\\
&> \tilde{L}(E_0)-\frac{\tilde{L}(E_0)}{100}-\frac{\tilde{L}(E_0)}{100}
=\frac{49}{50}\tilde{L}(E_0),
\end{aligned}
\end{equation}
and
\begin{equation} \label{60034}
\begin{aligned}
&|\tilde{L}_{N_0}(E)-\tilde{L}_{2N_0}(E)|\\
&\leq |\tilde{L}_{N_0}(E)-\tilde{L}_{N_0}(E_0)|+|\tilde{L}_{N_0}(E_0)
 -\tilde{L}_{2N_0}(E_0)|+|\tilde{L}_{2N_0}(E_0)-\tilde{L}_{2N_0}(E)|\\
&< \frac{\tilde{L}(E_0)}{100}+\frac{\tilde{L}(E_0)}{100}+\frac{\tilde{L}(E_0)}{100}
 =\frac{3}{100}\tilde{L}(E_0)<\frac{1}{10}\tilde{L}_{N_0}(E).
\end{aligned}
\end{equation}
Thus Lemma \ref{62001}
applies with $\tilde{L}_{N_0}(E)$, $\delta$, $N_0$ and $E$.
Then there exists a number $\tilde{N}_0=\tilde{N}_0(A,\delta,N_0)\leq
(\exp(\frac{\tilde{c}}{8}\delta^2 N_0)+1)N_0$ such that for any
$N\geq \tilde{N}_0$ there holds
\begin{gather}
|\tilde{L}_N(E)+\tilde{L}_{N_0}(E)-2\tilde{L}_{2N_0}(E)|<\exp(-\bar{c}'\delta^2
N_0), \nonumber \\
 \label{60035}
|\tilde{L}(E)+\tilde{L}_{N_0}(E)-2\tilde{L}_{2N_0}(E)|
< \exp(-\bar{c}\delta^2  N_0),
\end{gather}
 where $\bar{c}'=\bar{c}'(A)$ and $\bar{c}=\bar{c}(A)$ are as in Lemma
\ref{62001}. These imply
\begin{equation} \label{60036}
\begin{aligned}
|\tilde{L}(E)-\tilde{L}_{N}(E)|
&\leq\exp(-\bar{c}'\delta^2 N_0)+\exp(-\bar{c}\delta^2 N_0)\\
&<  \frac{1}{50}\tilde{L}(E_0)+\frac{1}{50}\tilde{L}(E_0)
=\frac{1}{25}\tilde{L}(E_0).
\end{aligned}
\end{equation}
Combining \eqref{60033}, \eqref{60034} with \eqref{60035}, one
obtains
\begin{align*}
&|\tilde{L}(E_0)-\tilde{L}(E)|\\
&\leq |\tilde{L}(E)+\tilde{L}_{\tilde{N}_0}(E)-2\tilde{L}_{2\tilde{N}_0}(E)|
+|\tilde{L}(E_0)-\tilde{L}_{\tilde{N}_0}(E)|+2|\tilde{L}_{\tilde{N}_0}(E)-
\tilde{L}_{2\tilde{N}_0}(E)| \\
&< \frac{1}{50}\tilde{L}(E_0)+\frac{1}{50}\tilde{L}(E_0)
+2\frac{3}{100}\tilde{L}(E_0)=\frac{1}{10}\tilde{L}(E_0).
 \end{align*}
It implies
\begin{gather*} %\label{60038}
\frac{11}{10}\tilde{L}(E_0)>\tilde{L}(E)>\frac{9}{10}\tilde{L}(E_0), \\
|\tilde{L}(E)-\tilde{L}_{N}(E)|
<\frac{1}{25}\tilde{L}(E_0)<\frac{1}{25}
(\frac{10}{9})\tilde{L}(E)
=\frac{2}{45}\tilde{L}(E)<\frac{1}{20}\tilde{L}(E).
\end{gather*}
\end{proof}

\begin{remark}\label{2} \rm
From \eqref{30010},
$\hat{C}(A,\tilde{L}(E_0))=D(A)+\frac{800\tilde{C}(A)^{1/2}}{\tilde{L}(E_0)c'}$.
It is obvious that $\hat{C}(A,\tilde{L}(E_0))$ is a positive
constant, which is smaller when $\tilde{L}(E_0)$ becomes larger.
Thus $\check{C}(A,\tilde{L}(E_0))=\hat{C}(A,\tilde{L}(E_0))+\log
C(A)$ is a positive and monotonically increasing function of
$\tilde{L}(E_0)$. Recall that
\begin{equation}\label{60020}
0<\rho(A,\tilde{L}(E_0))=\big[\frac{\tilde{L}(E_0)}{200}
\exp\big(-2\check{C}(A,\tilde{L}(E_0))N_0\big)\big]^{1/\beta}
< 1.
\end{equation}
$N_0$  depends on the rate of convergence of
$\tilde{L}_n(E_0)\to\tilde{L}(E_0)$
(see the proof of Lemma \ref{62006}). On the other hand,
in Lemma \ref{62001}, let
$m=\exp(\frac{\tilde{c}}{8}\delta^2 N)$, $N'=m N$, then
\[
|\tilde{L}_{N'}-\tilde{L}_{2N'}|<2\exp(-\hat{c}\delta^2
N)<\big(\frac{1}{N'}\big)^{C_1(A)}.
\]
Like the induction in the proof of
Lemma \ref{62001}, one gets
\begin{equation}\label{60021}
|\tilde{L}_{N}-\tilde{L}|<\big(\frac{1}{N}\big)^{C_2(A)}.
\end{equation}
Actually, in \cite{S}, the better estimate holds as
\[
|\tilde{L}_{N}-\tilde{L}|<\frac{C}{N},
\]
with the large constant $C$ depending on $\tilde{L}$.
 Finally, combining \eqref{60020},
\eqref{60021} with some assumptions in Lemma \ref{62001} and Lemma
\ref{62006}, we have
\[ %\label{60022}
\rho(A,\tilde{L}(E_0))\simeq\Big[\tilde{L}(E_0)
\exp\Big(-\frac{C_{\rho,1}}{(\tilde{L}(E_0))^{C_{\rho,2}}}\Big)\Big]^{1/\beta},
\]
where $C_{\rho,1}, C_{\rho,2}$ are the
big constants depending only on $A(x,E)$.
\end{remark}

   The first part of this section shows that the uniform property that
for any $E\in (E_0-\rho,E_0+\rho)$ and for the uniformly large $N$, the
distance between $ \tilde{L}_{N}(E)$ and $\tilde{L}(E)$ is
controlled by a uniform constant $\tilde{L}(E)$.
This  property  is very important and will be applied repeatedly in the
following part of this paper, such like the sharp Large Deviation Theorem
(Lemma \ref{22012}) and the proof of the main theorem (Section 4).

\begin{lemma}\label{22012}
Assume $\tilde{L}(E_0)>0$. There exist $N_1(A,E_0)$ such that for any
$N\geq N_1$ and any $E\in(E_0-\rho,E_0+\rho)$ holds
\[
\operatorname{meas}\{x:|\tilde{u}_N(x,E)-\tilde{L}(E)|
> \frac{\tilde{L}(E)}{10}\}<\exp(-c \tilde{L}(E) N),
\]
where constant $c$ depends only
on $A$, but does not depend on $E$ or $E_0$.
\end{lemma}

\begin{proof}
Choose $\bar{N}_0$ such that
 \begin{equation}\label{21118}
\begin{gathered}
\bar{N}_0>\max\Big(\frac{10^4C_6}{
\tilde{L}(E_0)},\max_{E\in\mathscr{E}}\frac{10^6(\log
C_{\rm max}(A,E)-\frac{1}{2}D(E))}{
\tilde{L}(E_0)},40,\tilde{N}_0\Big),\\
 \log \bar{N}_0<\bar{N}_0^{1/3}
\end{gathered}
\end{equation}
 where $(\log C_{\rm max}(A,E)-D(E))$ is as in Lemma \ref{22021},
$C_6$ is as in Lemma \ref{22010}, $\tilde{N}_0$ is as in Lemma \ref{62006}.

Take $N\geq \bar{N}_0^3$, $K:=\frac{1}{800}N\geq \bar{N}_0$.
Thus for any $E\in(E_0-\rho,E_0+\rho)$,
 \begin{equation} \label{21119}
\tilde{L}_{K}(E)<(1+\frac{1}{20})\tilde{L}(E).
\end{equation}
Using Lemma \ref{22021} one obtains
\begin{align*} %\label{21124}
&|\tilde{u}_N(x,E)-\frac{1}{K}\sum_{k=1}^K\tilde{u}_N(x+k\omega,E)|\\
&\leq \frac{1}{KN}\Big[ \sum_{k=1}^{\bar{N}_0}2k(\log C_{\rm max}(A,E)
 -\frac{1}{2}D(E))
 +\sum_{k=\bar{N}_0+1}^K 2 k\tilde{L}_k(E)\\
&\quad +\sum_{k=\bar{N}_0+1}^K 2C_6(k\log k)^{1/2}\Big]
 -\frac{1}{KN}\sum_{k=1}^K(kF_k(x)+kF_k(x+N\omega)) \\
&=I+II .
\end{align*}
By \eqref{21118}, \eqref{21119} and Lemma \ref{62006}, one
has
\begin{equation} \label{21122}
\begin{aligned}
I&<\frac{\bar{N}_0^2(\log C_{\rm max}(A,E)-\frac{1}{2}D(E))}{KN}
 +4\frac{21}{20}\tilde{L}(E)\frac{K}{N}++\frac{C_6K^{1/2}(\log K)^{1/2}}{N}\\
&< \frac{ \tilde{L}(E_0)}{320}+\frac{\tilde{L}(E)}{160}+\frac{
 \tilde{L}(E_0)}{320}=\frac{\tilde{L}(E)}{80}.
 \end{aligned}
\end{equation}
If $\sum_{k=1}^KkF_k(x,E)<-\frac{KN}{160} \tilde{L}(E)$, then
there is a $k$ such that
 $ kF_k(x,E)<-\frac{N}{160}\tilde{L}(E)$.
We know that
\[
\operatorname{meas}\{x:|kF_k(x)-k\langle F_k(x)\rangle |>k\delta\}
<\exp(-c\delta k),
\]
Since $\langle F_k\rangle =0$,
\begin{align*}
\operatorname{meas}\{x:kF_k(x)
&<-\frac{N}{160} \tilde{L}(E)\}\\
&\leq \operatorname{meas}\{x:|kF_k(x)|>\frac{N}{160}
\tilde{L}(E)\}\\
& <\exp(-c\frac{N \tilde{L}(E)}{160k}k)
=\exp(-c_2 N \tilde{L}(E)).
\end{align*}
So
\begin{equation} \label{21123}
\begin{aligned}
&\operatorname{meas}\Big\{x:\sum_{k=1}^K kF_k(x)<-\frac{KN}{160}\tilde{L}(E)\Big\}\\
&\leq K\exp(-c_2 N \tilde{L}(E)<\exp(-c'_2 \tilde{L}(E) N),
\end{aligned}
\end{equation}
if $N$ is large enough depending on $L(E_0)$ (see Lemma \ref{62006} and
\eqref{21118}). Combining \eqref{21122} with \eqref{21123} one has
\begin{equation}\label{21124}
\begin{aligned}
&\operatorname{meas}\Big\{x:|\tilde{u}_N(x,E)-\frac{1}{K}
\sum_{k=1}^K\tilde{u}_N(x+k\omega,E)|>\frac{\tilde{L}(E)}{40}\Big\}\\
&\leq 2\exp(-c'_2 N \tilde{L}(E))
<\exp(-c'' \tilde{L}(E)N).
\end{aligned}
\end{equation}
 On the other hand,
 recall that by Lemma \ref{lem2.3}, for any $K$,
\begin{gather*}
\operatorname{meas}\Big\{x:|\sum_{k=1}^Ku_N(x+k\omega)-K<u_N(\cdot)>|
>\delta K\Big\}
<\exp(-c\delta K), \\
\operatorname{meas}\Big\{x:|\sum_{k=1}^K\frac{1}{N}\log
|\det A(x+k\omega)|-KD|>\delta K\Big\}
<\exp(-c'\delta K).
\end{gather*}
By the definition of $\tilde{u}_N(x,E)$ and $\tilde{L}_N(E)$,
there exists $\check{K}=\check{K}(A, E_0)$ such that for any $K>\check{K}$
holds (with $\delta:=\frac{ \tilde{L}(E)}{40}$)
\[
\operatorname{meas}\Big\{x:|\sum_{k=1}^K\tilde{u}_N(x+k\omega,E)
-K\langle \tilde{u}_N(\cdot,E)\rangle|
>\frac{\tilde{L}(E)}{40} K\Big\}
\leq \exp(-\hat{c} \tilde{L}(E) K).
\]
Note that if
\[
\big|\frac{1}{K}\sum_{k=1}^K\tilde{u}_N(x+k\omega,E)-\langle \tilde{u}_N(\cdot,E)\rangle
\big|\leq\frac{ \tilde{L}(E)}{40},
\]
then
\begin{align*}
&|\frac{1}{K}\sum_{k=1}^K\tilde{u}_N(x+k\omega,E)-\tilde{L}(E)|\\
&\leq |\frac{1}{K}\sum_{k=1}^K\tilde{u}_N(x+k\omega,E)
 -\tilde{L}_N(E)|+|\tilde{L}_N(E)-\tilde{L}(E)| \\
&< \frac{1}{40}\tilde{L}(E)+\frac{1}{20}
\tilde{L}(E) = \frac{3}{40} \tilde{L}(E)\,.
\end{align*}
Therefore
\[
\operatorname{meas}\Big\{x:|\frac{1}{K}
\sum_{k=1}^K\tilde{u}_N(x+k\omega,E)-\tilde{L}(E)|>\frac{3}{40}
\tilde{L}(E)\Big\}<\exp(-\hat{c}  \tilde{L}(E) K).
\]
Combining this with \eqref{21124}, there exists $N_1=N_1(A,E_0)$
such that for any $N\geq N_1$ holds
\[
\operatorname{meas}\{x:|\tilde{u}_N(x,E)-\tilde{L}(E)|>\frac{ \tilde{L}(E)}{10}\}
<\exp(-c_0 L(E) N),
\] where $c_0$ depends
only on $A$. Here we replace  $c_0$ by $c$ for convenient notations.
\end{proof}


\begin{lemma}\label{62010}
Assume $\tilde{L}(E_0)>0$.   Let $N\geq N_1$ and
$E\in(E_0-\rho,E_0+\rho)$ be arbitrary. Then
\begin{equation}
|\tilde{L}(E)+\tilde{L}_{N}(E)-2\tilde{L}_{2N}(E)|<\exp(-c
\tilde{L}(E) N),
\end{equation}
where $c=c(A)$.
\end{lemma}

\begin{proof}
By the sharp Large Deviation Theorem, Lemma \ref{22012},
this lemma get better conclusion than Lemma \ref{62001},
 which comes from Theorem \ref{22006} and Remark \ref{29}, but the proof here is
the same as the later one.
\end{proof}


\section{proof of the main theorem}

Let $\tilde{L}(E_0)>0$. By Lemma \ref{62006} and \eqref{60036},
one has that for any $N\geq \tilde{N}_0(A,E_0)$ and
$E\in(E_0-\rho,E_0+\rho)$,
\begin{equation} \label{40001}
\tilde{L}_N(E)\leq
\tilde{L}(E)+\frac{1}{25}\tilde{L}(E_0)
<\frac{11}{10}\tilde{L}(E_0)+\frac{1}{25}\tilde{L}(E_0)
=\frac{57}{50}\tilde{L}(E_0).
\end{equation}
Let $E'\to E$ such that
$|D(E)-D(E')|\leq \frac{1}{5}\tilde{L}(E_0)$. Thus,
\begin{equation} \label{40002}
\begin{aligned}
L_N(E)-\frac{1}{2}D(E')
&\leq L_N(E)+\frac{1}{10}\tilde{L}(E_0)-\frac{1}{2}D(E)\\
&=\tilde{L}_N(E) +\frac{1}{10}\tilde{L}(E_0)
\leq \frac{62}{50}\tilde{L}(E_0).
\end{aligned}
\end{equation}
  Assume $\|\tilde{A}_N(x,E)\|\geq
\|\tilde{A}_N(x,E')\|$ and $|\det A_N(x,E)|\leq |\det A_N(x,E)|$.
By \eqref{70007}, \eqref{70008} and \eqref{60002},
\begin{equation} \label{40003}
\begin{aligned}
&|\log\|\tilde{A}_N(x,E)\|-\log \|\tilde{A}_N(x,E')\| \,|\\
&\leq \|\tilde{A}_N(x,E)\|-
\|\tilde{A}_N(x,E')\|=\frac{\|A_N(x,E)\|}{|\det A_N(x,E)|^{1/2}}
 -\frac{\|A_N(x,E')\|}{|\det A_N(x,E')|^{1/2}} \\
&\leq \frac{\|A_N(x,E)\|- \|A_N(x,E')\|}{|\det
A_N(x,E)|^{1/2}}\leq \frac{\|A_N(x,E)-A_N(x,E')\|}{|\det A_N(x,E)|^{1/2}} \\
&\leq C(A)|E-E'|^{\beta}\frac{\sum_{j=0}^{N-1}
\|\prod_{m=1}^{N-j}A(x+(N-m)\omega,E)\|\,
\|\prod_{m=j-1}^0A(x+m\omega,E')\|}{|\det
A_N(x,E)|^{1/2}} \\
&= C(A)|E-E'|^{\beta}\sum_{j=0}^{N-1}
\Big(\|\prod_{m=1}^{N-j}A(x+(N-m)\omega,E)\|\,
\|\prod_{m=j-1}^0A(x+m\omega,E')\|\Big) \\
&\quad \times \exp(-\frac{N}{2}D(E)) \exp(-N F_N(x,E));
\end{aligned}
\end{equation}
see Lemma \ref{22021} for the
definition of $F_N(x)$.

Let $N\geq N_2$, where
$N_2(A):=\max_{E_1,E_2\in\mathscr{E}}\frac{2\log
C_{\rm max}(A,E_1)-D(E_2)}{L(E_0)}\tilde{N}_0\geq 2\tilde{N}_0$. Then
\begin{align*}
\eqref{40003}
&\leq  \Big(\sum_{j=1}^{\tilde{N}_0}+\sum_{j=\tilde{N}_0+1}^{N-\tilde{N}_0}
 +\sum_{j=N-\tilde{N}_0+1}^{N}\Big) \|\prod_{m=1}^{N-j}A(x+(N-m)\omega,E)\|\\
& \quad  \times\|\prod_{m=j-1}^0A(x+m\omega,E')\| \exp(-\frac{N}{2}D)
 \exp(-N F_N(x,E)) C(A)|E-E'|^{\beta} \\
&:=I+ II +III .
\end{align*}
By Lemma \ref{22010} and Lemma \ref{22021}, one has
\begin{align*}
I,III &\leq  \sum_{j=1}^{\tilde{N}_0}\exp\Big(
L_{N-j}(E)(N-j)+C_6(N-j)^{1/2}\log^{1/2}(N-j)\\
&\quad +\log C_{\rm max}(A,E')(j-1)-\frac{N}{2}D(E)\Big)
 \exp(-N F_N(x,E))\times C(A)|E-E'|^{\beta} \\
&\leq  \sum_{j=1}^{\tilde{N}_0}
\exp\Big(\tilde{L}_{N-j}(E)(N-j)+C_6N^{1/2}\log^{1/2}N
+\big (\log C_{\rm max}(A,E')\\
&\quad -\frac{1}{2}D(E)\big )(j-1)
 -\frac{1}{2}D(E) \Big)  \exp(-N F_N(x,E))\times C(A)|E-E'|^{\beta} \\
&\leq  \sum_{j=1}^{\tilde{N}_0} \exp
\Big(\frac{1}{2}\tilde{L}(E_0)N+\frac{62}{50}\tilde{L}(E_0)N+C_6N^{1/2}\log^{1/2}N
 -\frac{1}{2}D\Big) \\
&\quad\times \exp(-N F_N(x,E))C(A)|E-E'|^{\beta},
\end{align*}
and
\begin{align*}
II &\leq  \sum_{j=\tilde{N}_0+1}^{N-\tilde{N}_0}\exp\Big(
L_{N-j}(E)(N-j)+C_6(N-j)^{1/2}\log^{1/2}(N-j)+L_{j-1}(E')(j-1)\\
&\quad +C_6(j-1)^{1/2}\log^{1/2}(j-1) -\frac{N}{2}D(E)\Big)
 \exp(-N F_N(x,E))\times C(A)|E-E'|^{\beta} \\
&\leq  \sum_{j=N-\tilde{N}_0+1}^{N}\exp\Big( \big(\log
C_{\rm max}(A,E) -\frac{1}{2}D(E)\big)(N-j)+ \Big(L_{j-1}(E')\\
&\quad -\frac{1}{2}D(E)\Big)(j-1)
  +C_6N^{1/2}\log^{1/2}N-\frac{1}{2}D(E)\Big)\\
&\quad\times \exp(-N F_N(x,E))\times C(A)|E-E'|^{\beta} \\
&\leq  \sum_{j=\tilde{N}_0+1}^{N-\tilde{N}_0}
\exp\Big(\frac{62}{50}\tilde{L}(E_0)N+C_6N^{1/2}\log^{1/2}N-\frac{1}{2}D
\Big) \\
&\quad\times \exp(-N F_N(x,E))C(A)|E-E'|^{\beta}.
\end{align*}
Combining these expressions, we have
\begin{align*}
\eqref{40003}
&\leq \sum_{j=1}^N \exp\Big(\frac{87}{50}\tilde{L}(E_0)N
+C_6N^{1/2}\log^{1/2}N-\frac{1}{2}D
\Big) \exp(-N F_N(x,E)) \\
&\quad \times C(A)|E-E'|^{\beta}.
\end{align*}
 There exists
$N_3=N_3(A,E_0)$ such that for any $N\geq N_3$ there holds
\[
C(A)\sum_{j=1}^N \exp\Big(\frac{87}{50}\tilde{L}(E_0)N+C_6N^{1/2}\log^{1/2}N
-\frac{1}{2}D \Big)
\leq \exp(2\tilde{L}(E_0)N).
\]
It is easy to see that there
are similar processes for the other three conditions. Thus
\[
\eqref{40003}\leq |E-E'|^{\beta}\exp(2\tilde{L}(E_0)N)
 \max\{\exp(-N F_N(x,E))\exp(-N F_N(x,E'))\}.
\]
Set
\[
\mathbb{B}(E):=\{x:NF_N(x,E)<-N\tilde{L}(E_0)\},
\]
then
\begin{align*}
\operatorname{meas}(\mathbb{B}(E))
&\leq \operatorname{meas}(\{x:|NF_N(x,E)-N\langle F_N(\cdot,E)\rangle|
>N\tilde{L}(E_0)\})\\
&<\exp(-c\tilde{L}(E_0)N),
\end{align*}
since $<F_N(\cdot,E)>=0$. By the Cauchy-Schwartz inequality, one has
\begin{align*}%\label{40004}
|\tilde{L}_N(E)-\tilde{L}_N(E')|
&= \int_{\mathbb{T}\backslash(\mathbb{B}(E)\bigcup\mathbb{B}(E'))}|\tilde{u}_N(x,E)
-\tilde{u}_N(x,E')|dx\\
&\quad +\int_{\mathbb{B}(E)\bigcup\mathbb{B}(E')}|\tilde{u}_N(x,E)
-\tilde{u}_N(x,E')|dx \\
&< |E-E'|^{\beta} \exp (3\tilde{L}(E_0)N)+4C_2(A)\exp(-\frac{c}{2}\tilde{L}(E_0)N) .
\end{align*}
Let $N\geq N_4:=\max(N_1,N_2,N_3)$, where $N_1$ is as in Lemma
\ref{22012}. By Lemma \ref{62010},
\begin{align*}
&|\tilde{L}(E)-\tilde{L}(E')|\\
&\leq |\tilde{L}(E)+\tilde{L}_N(E)-2\tilde{L}_{2N}(E)|
 +|\tilde{L}(E')+\tilde{L}_N(E')-2\tilde{L}_{2N}(E')| \\
&\quad +|\tilde{L}_N(E)-\tilde{L}_N(E')|+2|\tilde{L}_{2N}(E)-\tilde{L}_{2N}(E')| \\
&< 2\exp(-c \tilde{L}(E_0)N)+3|E-E'|^{\beta}
\exp (3\tilde{L}(E_0)N)+12C_2(A)\exp(-\frac{c}{2}\tilde{L}(E_0)N) \\
&< \exp(-c_7\tilde{L}(E_0)N)+3 \exp(3\tilde{L}(E_0)N)|E-E'|^{\beta},
 \end{align*}
where $c_7=c_7(A)$. Then when $E'\to E$, there exists $N\geq N_4$ such
that
\[
\exp\big(-(6+c_7)\tilde{L}(E_0)(N+1)\big)\leq |E-E'|^{\beta}
\leq \exp\big(-(6+c_7)\tilde{L}(E_0)N\big).
\]
It implies
\begin{align*}
|\tilde{L}(E)-\tilde{L}(E')|
&< 4\exp(-c_7\tilde{L}(E_0)N)\\
&=4\exp\big(-\frac{N}{N+1}c_7\tilde{L}(E_0)(N+1)\big)
 <\exp\big(-\frac{2c_7}{3}\tilde{L}(E_0)(N+1)\big) \\
&= \exp\big(-\frac{2c_7}{18+3c_7}\tilde{L}(E_0)N\big)
< |E-E'|^{\frac{2c_7\times\beta}{18+3c_7}}.
\end{align*}
 By the definition, one also has
\begin{align*} %\label{main}
|L_1(E)-L_1(E')|,\ |L_2(E)-L_2(E')|
&\leq |\tilde{L}(E)-\tilde{L}(E')|+\frac{1}{2}|D(E)-D(E')| \\
&< |E-E'|^{\frac{2c_7\times\beta}{18+3c_7}}+\frac{C_D(A)}{2}|E-E'|^{\beta} \\
&\leq (1+\frac{C(A)}{2})|E-E'|^{\frac{2c_7\times\beta}{18+3c_7}}.
\end{align*}

\subsection*{Acknowledgments}
The author was supported by the Fundamental Research Funds for the Central Universities (Grant 2013B01014) and by the National Nature Science Foundation
of China (Grant 11326133).

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\end{document}