Large Deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-R\"ussmann frequency

TL;DR

This paper establishes large deviation theorems for Dirichlet determinants of quasi-periodic Jacobi operators with Brjuno-Rüssmann frequencies, linking eigenvalue distribution to the irrationality of the frequency through the smallest deviation.

Contribution

It introduces large deviation theorems for these operators using a new approach based on the strong Birkhoff Ergodic Theorem for subharmonic functions with Brjuno-Rüssmann shifts.

Findings

Large deviation estimates depend on the smallest deviation related to the frequency's irrationality.

Eigenvalue distribution of Jacobi operators is characterized by the smallest deviation.

The Brjuno-Rüssmann function plays a key role in the deviation theorems.

Abstract

In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-R\"ussmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-R\"ussmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.

Equations

$$\left[H(x,\omega )\varphi \right](n)=-a(x+(n+1)\omega )\varphi (n+1)-\overline{a(x+n\omega )}\varphi (n-1)+v(x+n\omega )\varphi (n),n\in \mathbb{Z},$$

$$\left(\begin{array}[]{cc}\phi(n+1)\\ \phi(n)\\ \end{array}\right)=\frac{1}{a(x+(n+1)\omega)}\left(\begin{array}[]{cc}v(x+n\omega)-E&-\overline{a(x+n\omega)}\\ a(x+(n+1)\omega)&0\\ \end{array}\right)\left(\begin{array}[]{cc}\phi(n)\\ \phi(n-1)\\ \end{array}\right).$$

$$M(x,E,\omega):=\frac{1}{a(x+\omega)}\left(\begin{array}[]{cc}v(x)-E&-\overline{a(x)}\\ a(x+\omega)&0\\ \end{array}\right)$$

$$(\omega ,M):(x,\vec{v})\mapsto (x+\omega ,M(x)\vec{v})$$

$$M_n(x,E,\omega ):=\prod _{k=n}^{1}M(x+k\omega ,E)$$

$$L(E,\omega )=\underset{n\to \infty }{\lim }{L}_n(E,\omega )=\underset{n\to \infty }{\inf }{L}_n(E,\omega )\ge 0$$

$$L_n(E,\omega )=\frac{1}{n}{\int }_{\mathbb{T}}\log \Vert M_n(x,E,\omega )\Vert dx.$$

$$f^{a}_{[m,n]}(x,E,\omega)=f^{a}_{n-m+1}\bigl{(}x+(m-1)\omega,E,\omega\bigr{)},$$

$$\begin{aligned} f^{a}_{n}(x,E,\omega)&=\det\bigl{(}H_{n}(x,\omega)-E\bigr{)}\\ &=\begin{vmatrix}v\bigl{(}x+\omega\bigr{)}-E&-a\bigl{(}x+2\omega\bigr{)}&0&\cdots&\cdots&0\\[5.0pt] -\overline{a\bigl{(}x+2\omega\bigr{)}}&v\bigl{(}x+2\omega\bigr{)}-E&-a\bigl{(}x+3\omega\bigr{)}&0&\cdots&0\\[5.0pt] \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ &&&&&-a\bigl{(}x+n\omega\bigr{)}\\[5.0pt] 0&\dotfill&0&0&-\overline{a\bigl{(}x+n\omega\bigr{)}}&v\bigl{(}x+n\omega\bigr{)}-E\end{vmatrix}\end{aligned}.$$

$$\left[H^s(x,\omega )\varphi \right](n)=\varphi (n+1)+\varphi (n-1)+v(x+n\omega )\varphi (n),n\in \mathbb{Z}.$$

$$\Vert n\omega \Vert >\frac{C_{\omega }}{|n|{\left(\log |n|+1\right)}^{\alpha }}.$$

$$\Vert n\omega \Vert >\frac{C_{\omega }}{|n{|}^{\alpha }},$$

$$\mathrm{mes}\{x:\left|\frac{1}{n}\log \Vert M_n^s(x,E,\omega )\Vert -L_n^s(E,\omega )\right|>n^{-\sigma }\}<\exp \left(-n^{\sigma }\right);$$

$$\mathrm{mes}\{x:\left|\frac{1}{n}\log \Vert M_n^s(x,E,\omega )\Vert -L_n^s(E,\omega )\right|>\delta \}<\exp \left(-c{\delta }^{2}n\right).$$

$$\frac{1}{q_s(q_{s+1}+q_s)}<|\omega -\frac{p_s}{q_s}|<\frac{1}{q_s{q}_{s+1}}.$$

$$\beta (\omega ):=\underset{s}{\mathrm{limsup}}\frac{\log q_{s+1}}{q_s}\in [0,\infty ].$$

$$\left[H^m(x,\omega ,\lambda )\varphi \right](n)=\varphi (n+1)+\varphi (n-1)+2\lambda \cos \left(2\pi (x+n\omega )\right)\varphi (n).n\in \mathbb{Z}.$$

$$a(x)$$

$$v(x)$$

$$M_{n}^{s}(x,E,\omega)=\left(\begin{array}[]{cc}f_{n}^{s}(x,E,\omega)&f^{s}_{n-1}(x+\omega,E,\omega)\\ f^{s}_{n-1}(x,E,\omega)&f^{s}_{n-2}(x+\omega,E,\omega)\end{array}\right)$$

$$\mathrm{mes}\left\{x\in \mathbb{T}:\left|\log \left|f_n^s\left(x\right)\right|-⟨\log \left|f_n^s\right|⟩\right|>n\delta \right\}\le C\exp \left(-c\delta n({\delta }_0^n{)}^{-1}\right).$$

$$\Vert k\omega \Vert >\frac{C_{\omega }}{\Delta (k)},$$

$${\int }_1^{\infty }\frac{\log \Delta (t)}{t^2}<+\infty .$$

$$q_{s+1}<{\Gamma }_{\omega }(q_s)<q_s+q_{s+1},\text{mbox}and{\Gamma }_{\omega }(n)<{\Gamma }_{\omega }(q_s),\forall n\in (q_s,q_{s+1}).$$

$$\Delta (t)>t(\log t+1).$$

$$\mathrm{mes}\left\{x\in \mathbb{T}:\left|\log \left|f_n^a\left(x\right)\right|-⟨\log \left|f_n^a\right|⟩\right|>n\delta \right\}\le C\exp \left(-c\delta ({\delta }_{H.1}(n){)}^{-1}\right).$$

$$\Delta (t)<\exp \left(\frac{t}{\log t}\right).$$

$$\mathrm{mes}\left\{x\in \mathbb{T}:\left|\log \left|f_n^a\left(x\right)\right|-⟨\log \left|f_n^a\right|⟩\right|>n\delta \right\}\le C\exp \left(-c\delta ({\delta }_{H.2}(n){)}^{-1}\right).$$

$${\Delta }^{\prime }(t)\le \frac{\Delta (t)\log \Delta (t)}{t}.$$

$$\mathrm{mes}\left\{x\in \mathbb{T}:\left|\log \left|f_n^a\left(x\right)\right|-⟨\log \left|f_n^a\right|⟩\right|>n\delta \right\}\le C\exp \left(-c\delta ({\delta }_{H.3}(n){)}^{-1}\right).$$

$${\delta }_{H.1}(n)=\frac{\log \Delta (n)}{{\left({\Delta }^{-1}(C_{\omega }n)\right)}^{1-}}\gg {\delta }_{H.3}(n)=\frac{\log (C_{\omega }n)}{{\left[{\Delta }^{-1}(C_{\omega }n)\right]}^{1-}}.$$

Full text

The second author was supported by China Postdoctoral Science Foundation (Grant 2019M650094), the National Nature Science Foundation of China (Grant 11401166) and the Fundamental Research Funds for the Central Universities. He is the corresponding author.

1. Introduction

We study the following quasi-periodic analytic Jacobi operators $H(x,\omega)$ on $l^{2}(\mathbb{Z})$:

[TABLE]

where $v:\mathbb{T}\to\mathbb{R}$ is a real analytic function called potential, $a:\mathbb{T}\to\mathbb{C}$ is a complex analytic function and not identically zero. The characteristic equations $H(x,\omega)\phi=E\phi$ can be expressed as

[TABLE]

Define

[TABLE]

and call a map

[TABLE]

a Jacobi cocycle. Due to the fact that an analytic function only has finite zeros, $M(x,E,\omega)$ and the n-step transfer matrix

[TABLE]

make sense almost everywhere. By the Kingman’s subadditive ergodic theorem, the Lyapunov exponent

[TABLE]

always exists, where

[TABLE]

Let $H_{[m,n]}(x,\omega)$ be the Jaocbi operator defined by (1.1) on a finite interval $[m,n]$ with Dirichlet boundary conditions, $\phi(m-1)=0$ and $\phi(n+1)=0$. Let $f^{a}_{[m,n]}(x,E,\omega)=\det(H_{[m,n]}(x,\omega)-E)$ be its characteristic polynomial. One has

[TABLE]

where

[TABLE]

In this paper, the aim is to study the properties of $f_{N}^{a}(x,E,\omega)$. To state our conclusions, we first make some introductions to the background of our topic.

The operator (1.1) has the following important special case, which is called the Schrödinger operator and has been studied extensively:

[TABLE]

Then, $M_{n}^{s}(x,E,\omega)$, $L^{s}(E,\omega)$, $L^{s}_{n}(E,\omega)$ and $f^{s}_{n}(x,E,\omega)$ have the similar definitions. In [BG00], Bourgain and Goldstein proved that if $L^{s}(E,\omega)>0$, then for almost all $\omega$, the operator $H^{s}(0,\omega)$ has Anderson Localization, which means that it has pure-point spectrum with exponentially decaying eigenfunction. In [GS01], Goldstein and Schlag obtained the Hölder continuity of $L^{s}(E,\omega)$ in $E$ with the strong Diophantine $\omega$, i.e. for some $\alpha>1$ and any integer $n$,

[TABLE]

It is well known that for a fixed $\alpha>1$, almost every irrational $\omega$ satisfies (1.8). Obviously, if we define the Diophantine number as

[TABLE]

then it also has a full measure. In these two references, the key lemmas are the following called large deviation theorems (LDTs for short) for matrix $M_{n}^{s}(x,E,\omega)$ with these two frequencies: for the Diophantine $\omega$, it was proved in [BG00] that there exists $0<\sigma<1$ such that

[TABLE]

for the strong Diophantine $\omega$, it was proved in [GS01] that there exists $\delta_{0}^{n}=\frac{(\log n)^{A}}{n}$ such that for any $\delta>\delta_{0}^{n}$

[TABLE]

Here $\delta_{0}^{n}$ is called the smallest deviation in the LDT and very important in our paper.

Compared with the Schrödinger cocycle, one of the distinguishing features of the Jacobi cocycle is that it is not $SL(2,\mathbb{C})$. Then Jitomirskaya, Koslover and Schulteis [JKS09], and Jitomirskaya and Marx [JM11] proved that the LDT (1.10) for $M_{n}(x,E,\omega)$ and the weak Hölder continuity of the Lyapunov exponent of the analytic $GL(2,\mathbb{C})$ cocycles hold with the Diophantine frequency. In [T14], we showed that (1.11) can hold for $M_{n}(x,E,\omega)$ with the strong Diophantine $\omega$ and the continuity of the Lyapunov exponent of the Jacobi cocycles $L(E,\omega)$ can be Hölder in $E$.

For any irrational $\omega$, there exist its continued fraction approximates $\{\frac{p_{s}}{q_{s}}\}_{s=1}^{\infty}$, satisfying

[TABLE]

Define $\beta$ as the exponential growth exponent of $\{\frac{p_{s}}{q_{s}}\}_{s=1}^{\infty}$ as follows:

[TABLE]

Obviously, if $\omega$ is strong Diophantine or Diophantine, then $\beta(\omega)=0$. We say $\omega$ is the Liouville number, if $\beta(\omega)>0$. Recently, more and more attentions are paid to the question that what will happen to these operators with more generic $\omega$. So far, the most striking answers are mainly for the almost Mathieu operators (AMO for short), which is also a special case of the Jacobi ones

[TABLE]

The most famous one, the Ten Martini Problem, which was dubbed by Barry Simon and conjectures that for any irrational $\omega$, the spectrum of AMO is a Cantor set, was completely solved by Avila and Jitomirskaya [AJ09]. In that reference, they also proved that $H^{m}(x,\omega,\lambda)$ has Anderson Localization for almost every $x\in\mathbb{T}$ with $\lambda>e^{\frac{16}{9}\beta}$. In [AYZ17], Avila, You and Zhou improved it to $\lambda>e^{\beta}$.

While, the answers for the Schrödinger or Jacobi operators in the positive Lyapunov exponent regimes are mainly in the study of the continuity of the Lyapunov exponent. In [BJ02], they proved that the Lyapunov exponent is continuous in $E$ for any irrational $\omega$. The first result that the Hölder continuity holds for some weak Liouville frequency, which means that $\beta(\omega)<c$, where $c$ is a small constant depending only on the analytic potential $v(x)$, is [YZ14]. Recently, Han and Zhang [HZ18] ameliorated it to $\lambda>e^{C\beta}$ in the large coupling regimes, where the potential $v$ is of the form $\lambda v_{0}$ with a general analytic $v_{0}$ and $C$ is a positive constant also depending only on $v_{0}$. Our second author also proved the corresponding conclusion for the Jacobi operators in [T18]. These two results are optimal, because Avila, Last, Shamis and Zhou [ALSZ] showed that the continuity of the Lyapunov exponent of the almost Mathieu operators can’t be Hölder if $\beta>0$ and $e^{-\beta}<\lambda<e^{\beta}$.

Until now, we do not know much about the spectrum of the Schrödinger or Jacobi operators in the positive Lyapunov exponent regimes when the frequency is not strong Diophantine. The main reason is that we do not know much about the finite-volume determinant $f^{a}_{n}(x,E,\omega)$. While, for the almost Mathieu operators, it can be handled explicitly via the Lagrange interpolation for the trigonometric polynomial. This method can be applied for the following extend Harper’s operators, which also have the cosine potential, to obtain many spectral conclusions with the generic frequency, such as [AJM17] and [H18]:

[TABLE]

However, the Lagrange interpolation can not work for the Schrödinger or Jacobi operators, since their potentials both are generic analytic functions. Therefore, in [GS08], Goldstein and Schlag applied the LDT (1.11) and the relationship that

[TABLE]

to estimate the BMO norm of $f^{s}_{n}(x,E,\omega)$. Then they obtained the following LDT for $f^{s}_{n}(x,E,\omega)$ with the strong Diophantine $\omega$ by the John-Nirenberg inequality:

[TABLE]

This LDT was applied to get the Hölder exponent of the Hölder continuity of $L^{s}(E,\omega)$ in $E$ and the upper bound on the number of eigenvalues of $H^{s}_{n}(x,\omega)$ contained in an interval of size $n^{-C}$. What’s more, with its help, the estimation on the separation of the eigenvalues of $H^{s}_{n}(x,\omega)$ and the property that the spectrum of $H^{s}(x,\omega)$, denoted by $\mathcal{S}_{\omega}$, is a Cantor set were obtained in [GS11], and the homogeneity of $\mathcal{S}_{\omega}$ was proved in [GDSV18]. In [BV13] and [BV14], Binder and Voda applied this method to our analytic Jacobi operators (1.1). It must be noted that the above conclusions all hold only for the strong Diophantine $\omega$ and the LDTs for $f^{s}_{n}(x,E,\omega)$ and $f^{a}_{n}(x,E,\omega)$ are the key lemma in the method.

Now, we can declare that the concrete content of our main aim is to obtain the LDT for the finite-volume determinant $f^{a}_{n}(x,E,\omega)$ with more generic $\omega$. It is the preparation for the study of the spectrum problem for discrete quasiperiodic operators of second order in the future.

In this paper, we assume that the frequency $\omega$ is the Brjuno-Rüssmann number, which is a famous extension of the strong Diophantine number. It says that there exists a monotone increasing and continuous function $\Delta(t):[1,\infty)\to[1,\infty)$ such that $\Delta(1)=1$ and for any positive integer $k>0$,

[TABLE]

and

[TABLE]

For example, this Brjuno-Rüssmann function $\Delta(t)$ can be $t(\log t+1)^{\alpha}$, $t^{\alpha}$, $\exp\left((\log t)^{\alpha}\right)$, $\exp\left(t^{\frac{1}{\alpha}}\right)$ and $\exp\left(\frac{t}{(\log t)^{\alpha}}\right)$ with $\alpha>1$. Define $\Gamma_{\omega}(n)=\|n\omega\|^{-1}$. Due to (1.12), we have that

[TABLE]

Therefore, there exists another definition of the Brjuno-Rüssmann number as follow: There exists a function $\Psi_{\omega}(t)=\max\{\|k\omega\|^{-1},\ \forall 0<k\leq t,k\in\mathbb{Z}\}$ satisfying (1.17). Note that the denominator series $\{q_{s}\}_{s=1}^{\infty}$ and the function $\Psi_{\omega}$ depend on $\omega$. Thus, to make almost every irrational number satisfy (1.16), we assume that

[TABLE]

It implies that $\log\Delta(t)>\log t$ but it is false that $\frac{\Delta^{\prime}(t)}{\Delta(t)}=\left(\log\Delta(t)\right)^{\prime}>\left(\log t\right)^{\prime}=\frac{1}{t}$. However, there always exists another function $\tilde{\Delta}(t)$ which is larger than and close to $\Delta(t)$, and satisfies (1.17) and that $t\tilde{\Delta}^{\prime}(t)\geq\tilde{\Delta}(t)$ for any $t\geq 1$. So it is very reasonable for us to make the following hypothesis: 

Hypothesis H.1 $\Delta(t)>t(\log t+1)$ and $t\Delta^{\prime}(t)\geq\Delta(t)$ for any $t\geq 1$. 

Then, our first LDT for $f^{a}_{n}(x,E,\omega)$ is

Theorem 1**.**

Let $\omega$ be the Brjuno-Rüssmann number satisfying Hypothesis H.1 and $L(E,\omega)>0$. There exist constants $c=c(a,v,E,\omega)$ and $\breve{C}=\breve{C}(a,v,\omega)$, and absolute constant $C$ such that for any integer $n\geq 1$ and $\delta>\delta_{H.1}(n):=\frac{\breve{C}\log\Delta(n)}{\left(\Delta^{-1}(C_{\omega}n)\right)^{1-}}$,

[TABLE]

Remark 1.1*.*

In this paper, the notation $A^{1-}$ means $A^{1-\epsilon}$ for any small absolute $\epsilon>0$. And $A^{1+}$ has the similar definition.

Remark 1.2*.*

If we assume the potential $v$ is of the form $\lambda v_{0}$ with a general analytic $v_{0}$, then the second author [T18] proved that there exists $\lambda_{0}=\lambda_{0}(v_{0},a)$ such that the Lyapunov exponent $L(E,\omega)$ is always positive for any $E$ and any irrational $\omega$ under the condition $\lambda>\lambda_{0}$.

If $\Delta(t)=t(\log t+1)^{\alpha}$, then $\delta_{H.1}(n)=\frac{\breve{C}\log n}{n^{1-}}$ which is very close to the smallest deviation for the strong Diophantine number $\frac{(\log n)^{A}}{n}$. But if $\Delta(t)=\exp\left(t^{\frac{1}{\alpha}}\right)$, then $\delta_{H.1}(n)=\breve{C}n^{\frac{1}{\alpha}-}$ which is too large for us to apply Theorem 1 to the research of the spectrum of the analytic quasi-periodic operators (1.1) in our future work. Thus, we need to make some hypothesis to improve this smallest deviation when $\Delta(t)$ grows fast: 

Hypothesis H.2    $\omega$ satisfies Hypothesis H.1 and for any $t\geq 1$,

[TABLE]

From (1.17), $\Delta(t)$ has an upper bound of $\exp\left(\frac{t}{\log t}\right)$ generally. But it is possible that it grows very fast and exceeds this upper bound in some intervals, and in the rest it grows very slowly and makes the integral converge. Therefore, the aim of this hypothesis is to avoid this extreme possibility. Then, our second LDT for $f^{a}_{n}(x,E,\omega)$ is

Theorem 2**.**

Let $\omega$ be the Brjuno-Rüssmann number satisfying Hypothesis H.2 and $L(E,\omega)>0$. There exist constants $c=c(a,v,E,\omega)$ and $\breve{C}=\breve{C}(a,v,\omega)$, and absolute constant $C$ such that for any integer $n\geq 1$ and $\delta>\delta_{H.2}(n):=\frac{\breve{C}}{\left[\log(\Delta^{-1}(C_{\omega}n))\right]^{1-}}$,

[TABLE]

Remark 1.3*.*

With this hypothesis, no matter $\Delta(t)$ equals to $\exp\left(t^{\frac{1}{\alpha}}\right)$ or $\exp\left(\frac{t}{(\log t)^{\alpha}}\right)$, the smallest deviation $\delta_{H.2}(n)=\frac{\breve{C}}{\left[\log\log n\right]^{1-}}\ll 1$ which satisfies what we need for the study of the spectrum, such as Theorem 5.

As mentioned above, what we want to avoid is the case that $\Delta(t)$ grows faster than $\exp(t)$ in some intervals. But Hypothesis H.2 only requires that $\Delta(t)$ has an upper bound, but has no restriction on its derivative. Thus, we make the following hypothesis, which gives the mutual restriction between $\Delta(t)$ and $\Delta^{\prime}(t)$ and looks also very reasonable: 

Hypothesis H.3    $\omega$ satisfies Hypothesis H.1 and $\frac{\log\Delta(t)}{t}$ is non-increasing for any $t\geq 1$. 

Easy computation shows that this hypothesis is equivalent to the inequality

[TABLE]

Combined it with Hypothesis H.1, it shows that the bound of $t\Delta^{\prime}(t)$ is determined by $\Delta(t)$. Obviously, all examples of functions mentioned above satisfy this hypothesis. With its help, we improve Theorem 1 and 2 as follows:

Theorem 3**.**

Let $\omega$ be the Brjuno-Rüssmann number satisfying Hypothesis H.3 and $L(E,\omega)>0$. There exist constants $c=c(a,v,E,\omega)$ and $\breve{C}=\breve{C}(a,v,\omega)$, and absolute constant $C$ such that for any integer $n\geq 1$ and $\delta>\delta_{H.3}(n):=\frac{\breve{C}\log(C_{\omega}n)}{\left[\Delta^{-1}(C_{\omega}n)\right]^{1-}}$,

[TABLE]

Remark 1.4*.*

Since $t(\log t+1)<\Delta(t)$, it is obvious that

[TABLE]

On the other hand, due to the fact that $0<\Delta^{-1}(t)<t<\Delta(t)$, $n\gg\Delta\left(\log^{2}\Delta^{-1}(n)\right)$, we have that

[TABLE]

The key to prove these three LDTs for $f_{n}^{a}(x,E,\omega)$ is an ergodic theorem for the subharmonic function shifting on $\mathbb{T}$. Specifically, we know that if $T:X\to X$ is an ergodic transformation on a measurable space $(X,\Sigma,m)$ and $f$ is an $m-$integrable function, then the Birkhoff Ergodic Theorem tells that the time average functions $f_{n}(x)=\frac{1}{n}\sum_{k=0}^{n-1}f(T^{k}x)$ converge to the space average $\langle f\rangle=\frac{1}{m(X)}\int_{X}fdm$ for almost every $x\in X$. But it doesn’t tell us how fast do they converge? So, we call a theorem the strong Birkhoff Ergodic Theorem, if it gives the convergence rate. The following strong Birkhoff Ergodic Theorem for the subharmonic function shifting on $\mathbb{T}$ is the key which we just mentioned above:

Theorem 4**.**

Let $u:\Omega\to{\mathbb{R}}$ be a subharmonic function on a domain $\Omega\subset{\mathbb{C}}$ and $\omega$ be the Brjuno-Rüssman number satisfying Hypothesis H.1, or H.2, or H.3. Suppose that $\partial\Omega$ consists of finitely many piece-wise $C^{1}$ curves and $\mathbb{T}$ is contained in $\Omega^{\prime}\Subset\Omega$(i.e., $\Omega^{\prime}$ is a compactly contained subregion of $\Omega$). There exist constants $c=c(\omega,u)$ and $\breve{C}=\breve{C}(\Omega,u)$ such that for any positive $n$ and $\delta>\delta_{0}^{n}$,

[TABLE]

where

[TABLE]

A very interesting thing we find is that no matter the irrational frequency is, the convergence rate of the exceptional measure is always $\exp\left(-c\delta n\right)$. The only difference is the smallest deviation $\delta_{0}^{n}$. If $\beta(\omega)>0$, then our second author obtained in [T18] that $\delta_{0}^{n}=c\beta$ which is proved to be optimal in [ALSZ]; if $\beta(\omega)=0$, we obtain (1.22) which includes the result for the strong Diophantine number by Goldstein and Schlag. Correspondingly, the three LDTs we obtain in this paper can be unified into the following form:

[TABLE]

While, the exceptional measure in (1.23) will not converge when $\delta_{0}^{n}=c\beta$! The method created by Goldstein and Schlag and applied in this paper should be improved for the Liouville frequency. We think it is a good question for our further research in the future.

Here we need to emphasize that our paper is not weaker version of [T18]. That shows that the smallest deviation is $\delta_{0}^{n}=c\beta$, and then the strong Birkhoff ergodic theorem and the LDTs for matrices hold when the deviation is larger than $\delta_{0}^{n}$. Letting the positive Lyapunov exponent be this deviation, our second author obtained the Hölder continuity of the Lyapunov exponent. However, if we applied these results in our condition that $\beta=0$, then the smallest deviation is [math]! It is absurd! So, compared to [T18], the main aim of our second section is to find the smallest deviation when $\beta=0$. What’s more, we will find that in technology the key is to estimate $\sum_{j=1}^{2m-1}\frac{q_{s-j+1}}{q_{s-j}}\log q_{s-j+1}$. It is easy when $\beta>0$:

[TABLE]

While, when $\beta=0$, the fact that $\left\{\frac{\log q_{s+1}}{q_{s}}\right\}_{s=1}^{\infty}$ has different speeds, which depend on $\Delta(t)$, to converge to [math] makes this estimation much harder. On the other hand, the aims of our Section 3 and 4 are to obtain the LDT for $f_{n}^{a}$ and its applications, which are nonexistent in [T18]. In summary, the focus point of our paper is to show the importance of the smallest deviation of the strong Birkhoff ergodic theorem and calculate it when $\beta=0$. Of course, when we need the LDTs for matrices and the Hölder continuity of the Lyapunov exponent, such as Lemma 3.1 and 4.1, we can use the results from [T18] directly.

At last, we have an application of our LDTs, which estimates the upper bound on the number of eigenvalues of $H_{n}(x,\omega)$ contained in an interval of size $\left(\delta_{0}^{n}\right)^{\frac{1}{h}}$, where $h$ is the Hölder exponent of the Hölder continuity of $L(E,\omega)$, see Lemma 4.1. The distribution of the eigenvalues is very important in the further study of the spectrum problem for discrete quasiperiodic operators of second order. With fixed $x$ and $\omega$, the matrix $H_{n}(x,\omega)$ has $n$ eigenvalues. So we have an intuition that these eigenvalues have a more uniform distribution when the frequency $\omega$ is “more irrational”. For the Brjuno-Rüssman number, it means that $\Delta(t)$ grows more slowly and then $\delta_{0}^{n}$ is smaller. The following theorem verifies our intuition:

Theorem 5**.**

Let $\omega$ be the Brjuno-Rüssmann number satisfying Hypothesis H.1, or H.2, or H.3 and $L(E,\omega)>0$. Then, for any $x_{0}\in{\mathbb{T}}$ and $E_{0}\in\mathbb{R}$,

[TABLE]

We organize this paper as follows. In Section 2, we prove Theorem 4, the strong Birkhoff Ergodic theorem for the subharmonic function shifting on $\mathbb{T}$ with our Brjuno-Rüssmann frequency. We apply it to the analytic quasi-periodic Jacobi operator and obtain Theorem 1, Theorem 2 and Theorem 3 in Section 3, which are all the LDTs for $f_{n}^{a}(x,E,\omega)$ with different hypothesises. Then, we prove Theorem 5, an application of them, in the last section.

2. Strong Birkhoff Ergodic Theorem for Subharmonic Functions with the Brjuno-Rüssmann shift

Let $\{x\}=x-[x]$. For any positive integer $q$, complex number $\zeta=\xi+i\eta$ and $0\leq x<1$, define

[TABLE]

Let $|\{x+k_{0}\omega\}-\xi|=\min_{k=1}^{q_{s}}|\{x+k\omega\}-\xi|$, where $q_{s}$ is the denominator of the continued fraction approximants. In [GS01], Goldstein and Schlag proved Lemma 3.1 that for any irrational $\omega$, there exists an absolute constant $C$ such that

[TABLE]

Then

Lemma 2.1**.**

For any irrational $\omega$,

[TABLE]

where $D(x,\omega,n):=\min_{k=0}^{n-1}\{x+k\omega\}$.

Proof.

Define $x_{h}=x+hq_{s}\omega$ and $|\{x_{h}+k_{h}\omega\}-\xi|=\min_{k=0}^{q_{s}-1}|\{x_{h}+k\omega\}-\xi|$. Due to (2.25), we have

[TABLE]

We declare that if there exists $0\leq j<q_{s}$ such that $|\{x+j\omega\}-\xi|\leq\frac{1}{2q_{s}}-\frac{1}{q_{s+1}}$, then $j=k_{0}$. Actually, if $|\{x+j\omega\}-\xi|\leq\frac{1}{2q_{s}}-\frac{1}{q_{s+1}}$ and $j\not=k_{0}$, then $|\{x+k_{0}\omega\}-\xi|\leq|\{x+j\omega\}-\xi|\leq\frac{1}{2q_{s}}-\frac{1}{q_{s+1}}$, which implies

[TABLE]

Due to (1.12), it has

[TABLE]

Then,

[TABLE]

It is a contraction. Thus, there is at most one integer $0\leq k_{0}<q_{s}$ such that $|\{x+k_{0}\omega\}-\xi|<\frac{1}{2q_{s}}-\frac{1}{q_{s+1}}$ and

[TABLE]

Due to (1.12) again, it has

[TABLE]

Define $Q=[\frac{q_{s+1}}{q_{s}}]$ and let $j$ be the number such that $|\{x_{j}+k_{j}\omega\}-\xi|<\frac{1}{4q_{s+1}}$. Then by (2.30) and the above declaration, we have for any $j-2Q+1\leq h<j$ and $j<h\leq j+2Q-1$,

[TABLE]

Thus there are at most one point which is small than $\frac{1}{4q_{s+1}}$. Combining it with (2.27), we have

[TABLE]

∎

Lemma 2.2**.**

For any compact $\Omega\subset{\mathbb{C}}$, there exist constants $\tilde{c}=\tilde{c}(\omega)$ and $\tilde{C}=\tilde{C}(\Omega,\omega)$ such that for any $n\geq 1$, $\zeta\in\Omega$ and $0<\sigma\leq\tilde{c}$, we have

[TABLE]

where

[TABLE]

Proof.

We first apply Lemma 3.2 in [GS01]. It says that if $\Omega\subset\mathbb{T}$ is an arbitrary finite set, then for any $0<\sigma<1$,

[TABLE]

Set $\Omega=\{-m\omega:0\leq m<l_{s}q_{s}\}$. Then $\sharp\Omega=l_{s}q_{s}$ and $dist(x-\xi,\Omega)=D(x-\xi,-\omega,l_{s}q_{s})$. Thus, by (2.33),

[TABLE]

By Lemma 2.1, we have

[TABLE]

Now for any $n$, there exist $q_{s}$ and $q_{s+1}$ such that $q_{s}\leq n<q_{s+1}$. Let $n=l_{s}q_{s}+r_{s}$, where $l_{s}=[\frac{n}{q_{s}}]$, $0\leq r_{s}=n-l_{s}q_{s}<q_{s}$. Then,

[TABLE]

Let $r_{s-i+1}=l_{s-i}q_{s-i}+r_{s-i}$, where $l_{s-i}=[\frac{r_{s-i+1}}{q_{s-i}}]$, $0\leq r_{s-i}=r_{s-i+1}-l_{s-i}q_{s-i}<q_{s-i}$. Then

[TABLE]

Note that for any irrational $\omega$, the denominators of its continued fraction approximates satisfy

[TABLE]

Thus

[TABLE]

Therefore, if $\zeta\in\Omega^{\prime}$, where $\Omega^{\prime}$ is a compact subregion of $\mathbb{C}$, then

[TABLE]

where $C^{\prime}=C^{\prime}(\Omega)$.

Assume that $\omega$ satisfies Hypothesis H.1. Then, we assert that $\frac{t\log t}{\Delta^{-1}\left(C_{\omega}t\right)}$ is monotone increasing. Indeed, let $y=\Delta^{-1}\left(C_{\omega}t\right)$. Then, due to the hypothesis that

[TABLE]

it yields that

[TABLE]

Combining it with the fact that

[TABLE]

we have

[TABLE]

Now we finish the proof of the assertion as (2.35) shows that the numerator of the derivative of $\frac{t\log t}{\Delta^{-1}\left(C_{\omega}t\right)}$ is positive.

Due to (1.16) and (1.12),

[TABLE]

Therefore,

[TABLE]

We apply the assertion and obtain

[TABLE]

Recall that $\Delta(t)$ is monotone increasing and continuous. And so is $\Delta^{-1}(t)$. Combining it with (2.36), we have

[TABLE]

and for any $n>n_{0}(\omega)$,

[TABLE]

Choose $m\sim\log_{2}\Delta^{-1}(C_{\omega}n)$. Then

[TABLE]

Combining it with (2.34), (2.37), (2.39) and the assertion that $\frac{t\log t}{\Delta^{-1}\left(C_{\omega}t\right)}$ is monotone increasing, we have

[TABLE]

Assume that $\omega$ satisfies Hypothesis H.2 which implies that $\Delta(t)<\exp\left(\frac{t}{\log t}\right)$ holds. Then

[TABLE]

Combining them with (2.37) and (2.38),

[TABLE]

[TABLE]

Let $m\sim\log_{2}\log\Delta^{-1}(C_{\omega}n)$. Thus,

[TABLE]

and

[TABLE]

Assume the $\omega$ satisfies Hypothesis H.3 which implies that $\frac{\log\Delta(t)}{t}$ is non-increasing. Due to (2.38) and (2.39),

[TABLE]

By (2.37),

[TABLE]

Choose $m\sim\log_{2}(\Delta^{-1}(C_{\omega}n))$. Thus,

[TABLE]

Combining them, we have

[TABLE]

∎

Remark 2.1*.*

Note that $\log|x|$ is a subharmonic function. Thus, if $h$ is a 1-periodic harmonic function defined on a neighborhood of real axis, then for any positive $n$ and $0<\sigma\leq\tilde{c}$, we have

[TABLE]

To finish the proof of Theorem 4, we need the following Riesz’s theorem proved in [GS08]:

Lemma 2.3**.**

Let $u:\Omega\to{\mathbb{R}}$ be a subharmonic function on a domain $\Omega\subset{\mathbb{C}}$. Suppose that $\partial\Omega$ consists of finitely many piece-wise $C^{1}$ curves. There exists a positive measure $\mu$ on $\Omega$ such that for any $\Omega_{1}\Subset\Omega$ (i.e., $\Omega_{1}$ is a compactly contained subregion of $\Omega$),

[TABLE]

where $h$ is harmonic on $\Omega_{1}$ and $\mu$ is unique with this property. Moreover, $\mu$ and $h$ satisfy the bounds

[TABLE]

for any $\Omega_{2}\Subset\Omega_{1}$.

The proof of Theorem 4.

Notice that the ergodic measure for the shift on the Torus is the Lebesgue measure and $m(\mathbb{T})=1$. Then, $<u>=\int_{\mathbb{T}}u(x)dx$, and

[TABLE]

Recall that

[TABLE]

Then

[TABLE]

Since $\exp(\sigma\cdot)$ is a convex function, the Jensen’s inequality implies that

[TABLE]

Thus, combining it with (2.47), we have for any $0<\sigma\leq\frac{\tilde{c}}{\mu(\Omega^{\prime})}$,

[TABLE]

Recall the Markov’s inequality: For any measurable extended real-valued function $f(x)$ and $\epsilon>0$,we have

[TABLE]

Let $f(x)=\exp\left(\sigma|\sum_{k=1}^{n}u(x+k\omega)-n\langle u\rangle|\right)$ and $\epsilon=\exp(\sigma\breve{\delta}n)$, then

[TABLE]

We finish this proof by setting $\breve{C}=2\tilde{C}\mu(\Omega^{\prime})$. ∎

3. Large Deviation Theorems for $f_{n}^{a}(x,E,\omega)$

To apply Theorem 4, we first need to define some subharmonic functions. Let

[TABLE]

Note that a real function $f(x)$ on $\mathbb{T}$ has its complex analytic extension $f(z)$ on the complex strip $\mathbb{T}_{\rho}=\{z:|{\rm Im}z|<\rho\}$ and the complex analytic extension of $\bar{a}(x)$ should be defined on $\mathbb{T}_{\rho}$ by

[TABLE]

Then, the extension of $M^{a}_{n}(x,E,\omega)$ is

[TABLE]

where $z+\omega$ means $z\exp\left(2\pi i\omega\right)$ here. Moreover, simple computations yield that

[TABLE]

where

[TABLE]

Note that if ${\rm Im}z=0$, then $H_{n}(z,\omega)=H_{n}(x,\omega)$ is Hermitian. Now with fixed $E$ and $\omega$, the function $\frac{1}{N}\log\|M_{N}^{a}(z,E,\omega)\|$ is subharmonic. In this paper, we only need to consider $E\in\mathscr{E}$, where

[TABLE]

as the spectrum ${\mathcal{S}}_{\omega}\subset\mathscr{E}$. Thus, for any irrational $\omega$ and $1\leq n\in\mathbb{N}$,

[TABLE]

where

[TABLE]

We also need to define the unimodular matrix

[TABLE]

which makes sense a.e. $x\in\mathbb{T}$ and has the relationship

[TABLE]

Then, we have the LDTs for the matrices as follows:

Lemma 3.1**.**

Let $\omega$ be the Brjuno-Rüssmann number satisfying Hypothesis H.1, or H.2, or H.3 and $L(E,\omega)>0$. There exist $\hat{c}=\hat{c}(v,a,\omega)$ and $\check{c}=\check{c}(v,a,\omega)$ such that for any $n\geq 0$ and $\delta>\delta_{0}^{n}$,

[TABLE]

where $u(x,E,\omega)$ can be $\frac{1}{n}\log\|M_{n}^{a}(x,E,\omega)\|$, $\frac{1}{n}\log\|M_{n}(x,E,\omega)\|$ and $\frac{1}{n}\log\|M_{n}^{u}(x,E,\omega)\|$. What’s more, there exists $\bar{c}=\bar{c}(a,v,\omega)$ such that if $\delta=\kappa L(E,\omega)$ with $\kappa<\frac{1}{10}$, then the exception measure in (3.56) will be less than $\exp\left(-\bar{c}\kappa^{2}L(\omega,E)n\right)$.

Proof.

When $u=\frac{1}{n}\log\|M_{n}^{a}(x,E,\omega)\|$, The LDT (3.56) is about the analytic matrix; when $u=\frac{1}{n}\log\|M_{n}(x,E,\omega)\|$, The LDT (3.56) is about the Jacobi cocycles; when $u=\frac{1}{n}\log\|M^{u}_{n}(x,E,\omega)\|$, The LDT (3.56) is about the unimodular matrix to satisfy the hypothesises of Lemma 3.7, Lemma 3.8 and the Avalanche Principle(Proposition 3.1). In [T18], our second author obtained these LDTs with finite Liouville frequency, which means that $\beta(\omega)<\infty$. Due to the fact that $\beta(\omega)=0$ for any Brjuno-Rüssmann number, the proofs in that paper are also available here. ∎

What’s more, the following lemma shows that $L_{n}(E,\omega)$ and $L_{n}^{a}(E,\omega)=\langle\frac{1}{n}\log\|M_{n}^{a}(x,E,\omega)\|\rangle$ in the above LDTs can be exchanged by $L(E,\omega)$ and $L^{a}(E,\omega)$, respectively. Here,

[TABLE]

and

[TABLE]

Lemma 3.2**.**

Let $L(E,\omega)>0$. For any integer $n>1$, we have

[TABLE]

where $C_{0}=C_{0}\left(a,v,\omega,E\right)$.

Proof.

It is the same as Lemma 3.9 in [BV13], which was for the strong Diophantine $\omega$. They applied the same LDTs, whose $\delta_{0}^{n}=\frac{(\log n)^{A}}{n}$ with that frequency, to obtain its proof. It is available here, since it only need the fact, which our LDTs also satisfy, that $\delta_{0}^{n}$ is much less than the positive Lyapunov exponent. ∎

Although the details of the proof of Lemma 3.1 can be found in [T18], we still give a brief introduction here, to make the readers understand the methods we apply in this section to obtain Theorem 1-3. Easy computations show that the differences between $\frac{1}{N}\log\|M_{N}^{a}(x,E,\omega)\|$ and $\frac{1}{N}\log\|M_{N}(x,E,\omega)\|$ and between $\frac{1}{N}\log\|M_{N}^{a}(x,E,\omega)\|$ and $\frac{1}{N}\log\|M_{n}^{u}(x,E,\omega)\|$ are constructed by the combination of $\frac{1}{N}\sum_{j=1}^{N}\log|a(x+j\omega)|$ and $\frac{1}{N}\sum_{j=1}^{N}\log|\bar{a}(x+j\omega)|$, whose complex extensions can be estimated by our Theorem 4 easily. Therefore, we only need to prove the LDT for $\frac{1}{N}\log\|M_{n}^{a}(x,E,\omega)\|$, which also has a subharmonic extension. Due to this subharmonicity,

[TABLE]

On the other hand, for any $k\in\mathbb{Z}$,

[TABLE]

where $d(x)=\log|a(x+\omega)\bar{a}(x)|$. Obviously, it also can be solved by our Theorem 4. Now, we can explain why we apply the BMO norm and the John-Nirenberg inequality, not the method for (3.56), to obtain the LDTs for $f_{n}^{a}(x,E,\omega)$. The reason is that Theorem 4 holds for $f_{n}^{a}(x,E,\omega)$, but we can not handle the difference between $\frac{1}{n}\log|f_{n}^{a}(x,E,\omega)|$ and $\frac{1}{kn}\sum_{j=1}^{k}\log|f_{n}^{a}(x+j\omega,E,\omega)|$.

We will apply the analyticity of $f_{n}^{a}(x,E,\omega)$ and the subharmonicity of $\frac{1}{n}\log|f_{n}^{a}(x,E,\omega)|$ via the following lemmas in this paper.

Definition 3.1**.**

Let $H>1$. For any arbitrary subset ${\mathcal{B}}\subset{\mathcal{D}}(z_{0},1)\subset\mathbb{C}$ we say ${\mathcal{B}}\in Car_{1}(H,K)$ if ${\mathcal{B}}\subset\bigcup_{j=1}^{j_{0}}{\mathcal{D}}(z_{j},r_{j})$ with $j_{0}<K$, and

[TABLE]

Here $D(z,r)$ means the complex platform center at $z$ with radius $r$. If $d$ is a positive integer greater than one and ${\mathcal{B}}\subset\prod_{i=1}^{d}\subset{\mathbb{C}}^{d}$ then we define inductively that ${\mathcal{B}}\in Car_{d}(H,K)$ for any $z\in{\mathbb{C}}\backslash{\mathcal{B}}_{j}$, here ${\mathcal{B}}_{z}^{(j)}=\{(z_{1},\cdots,z_{d})\in{\mathcal{B}}:z_{j}=z\}.$

Lemma 3.3** (Cartan estimate, Lemma 2.4 in [GS11]).**

Let $\phi(z_{1},\cdots,z_{d})$ be an analytic function defined in a polydisk ${\mathcal{P}}=\prod_{j=1}^{d}{\mathcal{D}}(z_{j,0},1),\ z_{j,0}\in{\mathbb{C}}$. Let $M\geq\sup_{\underline{z}\in{\mathcal{P}}}\log|\phi(\underline{z})|,\ m\leq\log|\phi(\underline{z}_{0})|,\ \underline{z}_{0}=(z_{1,0},\cdots,z_{d,0})$. Given $H\gg 1$ there exists a set ${\mathcal{B}}\subset{\mathcal{P}},{\mathcal{B}}\in Car_{d}(H^{\frac{1}{d}},K),\ K=C_{d}H(M-m)$, such that

[TABLE]

for any $\underline{z}\in\prod_{j=1}^{d}{\mathcal{D}}(z_{j,0},\frac{1}{6})\backslash{\mathcal{B}}$.

Lemma 3.4** (Lemma 2.4 in [GS08]).**

Let $u$ be a subharmonic function defined on ${\mathcal{A}}_{\rho}$ such that $\sup_{{\mathcal{A}}_{\rho}}u\leq M$. There exist constants $C_{1}=C_{1}\left(\rho\right)$ and $C_{2}$ such that, if for some $0<\delta<1$ and some $L$ we have

[TABLE]

then

[TABLE]

Recalling the definitions of $M_{n}(x,E,\omega)$, $M_{n}^{a}(x,E,\omega)$, $M_{n}^{u}(x,E,\omega)$ and the expression (3.53), we have

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

Assume $L(E,\omega)=\gamma>0$. Then, we can obtain a particular deviation theorem as follow:

Lemma 3.5**.**

There exists $l_{0}=l_{0}\left(a,v,\gamma\right)$ such that

[TABLE]

for all $l\geq l_{0}$.

Proof.

It is the same as Lemma 4.2 in [BV13], which was for the strong Diophantine $\omega$. We have the same reason which we just stated in the proof of Lemma 3.2 to omit this proof. ∎

Note that in order to simplify the notation, we suppressed the dependence on $E$ and $\omega$. We will be doing this throughout this paper if there is no confusion. According to Lemma 3.4 and 3.5, we can have more choices of the deviation and the exceptional measure.

Lemma 3.6**.**

Let $\sigma>0$ and $g(n)>0$. There exist constants $l_{0}=l_{0}\left(a,v,\gamma\right)$ and $n_{0}=n_{0}\left(a,v,\gamma\right)$ such that

[TABLE]

for any $n\geq n_{0}$ and for any $l_{0}\leq l\lesssim g(n)$. The same result, but with possibly different $l_{0}$ and $n_{0}$, holds for $f^{u}_{l}$.

Proof.

Assume

[TABLE]

We have that

[TABLE]

on a set of measure greater than $\exp\left(-g(n)l^{-3}\right)$. By Lemma 3.4, it implies that for any $x\in\mathbb{T}$,

[TABLE]

Due to Theorem 4,

[TABLE]

Therefore, recalling (3.63), we have

[TABLE]

for all $x$ except for a set of measure less than $\exp\left(-cl\right)$. It contradicts with the previous lemma. At last, by (3.64), we can prove the result for $f^{u}_{l}$ by similar methods. ∎

Now we need some facts about stability of contracting and expanding directions of unimodular matrices. It follows from the polar decomposition that if $A\in SL\left(2,\mathbb{C}\right)$ then there exist unit vectors $u_{A}^{+}\perp u_{A}^{-}$ and $v_{A}^{+}\perp v_{A}^{-}$ such that $Au_{A}^{+}=\left\|A\right\|v_{A}^{+}$ and $Au_{A}^{-}=\left\|A\right\|^{-1}v_{A}^{-}$.

Lemma 3.7** (Lemma 2.5 in [GS08]).**

For any $A$, $B\in SL\left(2,\mathbb{C}\right)$ we have

[TABLE]

Lemma 3.8** (Lemma 4.5 in [BV13]).**

If $A\in SL\left(2,\mathbb{C}\right)$ and $w_{1}$, $w_{2}$, and $w_{3}$ are unit vectors in the plane then

[TABLE]

and

[TABLE]

Now, we can improve the Lemma 3.6, but the LDT is about three determinants.

Lemma 3.9**.**

There exist constants $0<\kappa=\kappa(\omega)<1$, $0<\tau=\tau(\omega)<1$, $l_{0}=l_{0}\left(a,v,\gamma\right)$ and $n_{0}=n_{0}\left(a,v,\gamma\right)$ such that

[TABLE]

for any $l_{0}\leq j_{1}\leq j_{1}+l_{0}\leq j_{2}\leq n^{\tau}$ and $n\geq n_{0}$.

Proof.

Here we assume $\delta_{0}^{n}\geq n^{-\frac{1}{3}}$, since the proof of Lemma 4.6 in [BV13] can be applied without any change when $\delta\leq n^{-\frac{1}{3}}$.

For any $1\leq j\leq n$, due to Lemma 3.1 and 3.2, choose the deviation $\delta=\frac{n}{j}\delta_{0}^{n}>\delta_{0}^{j}$ and then

[TABLE]

Let $\mathcal{G}_{n}$ be the set of points $x\in{\mathbb{T}}$ such that for any $1\leq j\leq n$ and $\left|l\right|\leq 2n$,

[TABLE]

and

[TABLE]

Due to (3.66) and Theorem 4 for $\log|a(x)|$, we have that

[TABLE]

Note that $\det M_{l}^{u}(x,E,\omega)\equiv 1$. Therefore, for any $x,\ E$ and $\omega$,

[TABLE]

Let $\left\{e_{1},e_{2}\right\}$ be the standard basis of $\mathbb{R}^{2}$ and for any integer $j$, $u_{j}^{+}$, $u_{j}^{1}$, $v_{j}^{+}$ and $v_{j}^{-}$ be the unit vectors satisfying $u_{j}^{+}\perp u_{j}^{-}$, $v_{j}^{+}\perp v_{j}^{-}$, $M_{j}^{u}u_{j}^{+}=\left\|M_{j}^{u}\right\|v_{j}^{+}$ and $M_{j}^{u}u_{j}^{-}=\left\|M_{j}^{u}\right\|^{-1}v_{j}^{-}$. Then

[TABLE]

If $\left|f^{u}_{n}\left(x\right)\right|\leq\exp\left(nL_{n}-100n\delta_{0}^{n}\right)$, then

[TABLE]

Due to Lemma 3.2, for any $x\in\mathcal{G}_{n}$,

[TABLE]

Hence,

[TABLE]

Suppose (3.65) fails. Let $\sigma<\kappa<1/2$. Recall $n\delta_{0}^{n}\geq n^{1-2\sigma}$ and set

[TABLE]

We have

[TABLE]

If $x\in\tilde{{\mathcal{G}}}_{n}$, then either $\left|u_{n}^{-}\left(x\right)\wedge e_{1}\right|\leq\exp\left(-40n\delta_{0}^{n}\right)$ or $\left|v_{n}^{+}\left(x\right)\wedge e_{2}\right|\leq\exp\left(-40n\delta_{0}^{n}\right)$ has to hold for two of the points $x$, $x+j_{1}\omega$, $x+j_{2}\omega$.

We first assume that

[TABLE]

From Lemma 3.8 and Lemma 3.7, we have that if $x\in{\mathcal{G}}_{n}$, then

[TABLE]

Combined it with Lemma 3.8 and (3.68), we obtain

[TABLE]

Due to the fact that

[TABLE]

and the setting of ${\mathcal{G}}_{n}$, we have

[TABLE]

Similarly, we can obtain

[TABLE]

if we assume that

[TABLE]

What’s more, the same type of estimates are obtained if we replace $\left(j_{1},j_{2}\right)$ in (3.68) and (3.69) with $\left(0,j_{1}\right)$ or $\left(0,j_{2}\right)$.

In conclusion

[TABLE]

for some choice of $l$ from $j_{1}-1$, $j_{2}-1$, $j_{2}-j_{1}-1$. However, choosing $g(n)=20n\delta_{0}^{n}$ in Lemma 3.6 and $\tau=\frac{\kappa-\sigma}{4}$ in the hypothesis of this lemma, we have

[TABLE]

Thus, we complete the proof by this contradiction. ∎

One of our methods to obtain a large deviation estimate for a single determinant is the $BMO(\mathbb{T})$ norm. $BMO(\mathbb{T})$ is the space of functions of bounded mean oscillation on $\mathbb{T}$. Identifying functions that differ only by an additive constant, then norm on $BMO(\mathbb{T})$ is given by

[TABLE]

where $\langle f\rangle_{I}:=\int_{I}f(x)dx$. Applying the previous lemma, we obtain the following lower bound of the mean value of $\frac{1}{N}\left|f^{u}_{N}\left(x\right)\right|$, which will help us estimate the BMO norm.

Lemma 3.10**.**

There exist constants $0<c_{0}=c_{0}(\omega)\leq 1$ and $n_{0}=n_{0}\left(a,v,\gamma\right)$ such that for $n\geq n_{0}$ we have

[TABLE]

Proof.

Set

[TABLE]

Then, $\mathop{\rm{mes}\,}\left(\mathbb{T}\setminus\Omega_{n}\right)\leq n^{\tau}\exp\left(-n^{1-\kappa}\right)<\exp\left(-\frac{1}{2}n^{1-\kappa}\right)$.

Define $\nu_{n}^{u}\left(x\right)=\log\left|f^{u}_{n}\left(x\right)\right|/n$ and set $M=\left[\frac{n^{\tau}}{l_{0}}\right]\geq n^{\frac{\tau}{2}}$ for large $n$. For any $x\in\Omega_{n}$ we have that $\nu_{n}^{u}\left(x+kl_{0}\omega\right)>L_{n}-100\delta_{0}^{n}-\frac{\log 3}{n}$ for all but at most two $k$’s, $1\leq k\leq M$. We have

[TABLE]

Define $\nu_{n}^{a}\left(x\right)=\log\left|f^{a}_{n}\left(x\right)\right|/n$. Note that $\nu_{n}^{a}(x)$ can be extended to the complex trip $\mathbb{T}_{\rho}$ where $\nu_{n}^{a}(z)$ is subharmonic. Due to (3.53), we have that

[TABLE]

Applying Cartan’s estimate, Lemma 3.3, to $f^{a}_{n}(z)$ with $M=Sn$, $m=<\nu_{n}^{a}>n$ and $H=n^{\frac{\tau}{4}}$, we have

[TABLE]

up to a set not exceeding $CM\exp\left(-n^{\frac{\tau}{4}}\right)$ in measure. Combining it with the relationship that

[TABLE]

and applying (3.72) and Theorem 4 for $\displaystyle\frac{1}{2n}\left(\sum_{j=1}^{n}+\sum_{j=0}^{n-1}\right)\log|a(x+j\omega)|$ with deviation $|D|$, we have

[TABLE]

up to a set $\mathcal{B}_{n}$ not exceeding $CM\exp\left(-n^{\frac{\tau}{4}}\right)+\exp(-\hat{c}|D|n)<\exp\left(-\frac{1}{2}n^{\frac{\tau}{4}}\right)$ in measure. Therefore,

[TABLE]

Let $g(n)=n^{3}$ in Lemma 3.6. Then simple calculations shows that $\left\|\nu_{n}^{u}\right\|_{L^{2}\left(\mathbb{T}\right)}\leq Cn^{3}$. Thus,

[TABLE]

Above all,

[TABLE]

∎

Remark 3.1*.*

Due to the setting of $\tau$ and (3.74), easy computations shows that

[TABLE]

We will show that the supermum of the subharmonic function $u_{n}^{a}(z,E,\omega)$ on $\mathbb{T}$ is closed to its mean value. Here, we will apply the property that a subharmonic function at a point is small than the its integration on the platform center at that point. From the proof of Theorem 4, it is easily seen that the sharp LDT for $u_{n}^{a}(x)$ can been extended to the complex region $\mathbb{T}_{\rho}$:

[TABLE]

where

[TABLE]

Lemma 4.1 in [GS08] proved that there exists $C_{0}=C_{0}\left(M_{0},\rho\right)$ such that for any $r_{1},r_{2}\in\left(1-\rho,1+\rho\right)$ we have

[TABLE]

Lemma 3.11**.**

For any integer $n>1$ we have that

[TABLE]

Proof.

Due to (3.75) with $\delta=\delta_{0}^{n}$, we have

[TABLE]

except for a set of measure less than $\exp\left(-\hat{c}n\delta_{0}^{n}\right)+\exp\left(-\check{c}\left(\delta_{0}^{n}\right)^{2}n\right)$. By the subharmonicity of $\log\left\|M_{n}^{a}\left(z\right)\right\|$ we have

[TABLE]

For $r\in\left(1-n^{-1},1+n^{-1}\right)$ we have

[TABLE]

∎

Then, we will use the following lemma proved by Bourgain, Goldstein and Schlag in [BGS01], not the definition, to calculate the BMO norm of subharmonic functions.

Lemma 3.12** (Lemma 2.3 in [BGS01]).**

Suppose u is subharmonic on $\mathbb{T}_{\rho}$, with $\mu(\mathbb{T}_{\rho})+\sup_{z\in\mathbb{T}_{\rho}}h(z)\leq n$ where $\mu(\mathbb{T}_{\rho})$ and $h(z)$ comes from Lemma 2.3. Furthermore, assume that $u=u_{0}+u_{1}$, where

[TABLE]

Then for some constant $C_{\rho}$ depending only on $\rho$,

[TABLE]

Lemma 3.13**.**

There exist constant $c_{1}=c_{1}(a,v,E,\rho,\gamma)$ and absolute constant $C$ such that for every integer $n$ and any $\delta>0$ we have

[TABLE]

where $c_{0}$ comes from Remark 3.1. The same estimate with possibly different $c_{1}$ holds for $f^{u}_{n}$.

Proof.

It is enough to establish the estimate for $n$ large enough. By Lemma 3.10 and Lemma 3.11,

[TABLE]

This implies that

[TABLE]

Due to Lemma 3.12 with setting $\epsilon_{0}=0$, we have

[TABLE]

Then, the well-known John-Nirenberg inequality tells us how to apply this MBO norm to obtain the large deviation theorem: Let $f$ be a function of bounded mean oscillation on $\mathbb{T}$. Then there exist the absolute constants $C$ and $c$ such that for any $\gamma>0$

[TABLE]

Thus,

[TABLE]

∎

Now, due to the above proof and Remark 3.74, to prove Theorem 1-3, the only thing we need to do is obtain $\left\|\frac{1}{n}\log|f_{n}^{a}|\right\|_{BMO}=O\left(\delta_{0}^{n}\right)$, when $\Delta(t)\sim t^{A}$ and $1<A<5$. In the following proof, we will use the Avalanche Principle to refine the previous estimation:

Proposition 3.1** (Avalanche Principle).**

Let $A_{1},\ldots,A_{n}$ be a sequence of $2\times 2$–matrices whose determinants satisfy

[TABLE]

Suppose that

[TABLE]

Then

[TABLE]

with some absolute constant $C$.

The Proof of Theorem 1 to 3.

Define

[TABLE]

and ${\mathcal{M}}_{n}^{a}$ analogously. Obviously, $\left|f_{n}^{a}\left(x\right)\right|=\left\|{\mathcal{M}}_{n}^{a}\left(x\right)\right\|$. Let $c^{\prime}$ be a small constant constant, $l\sim n^{c^{\prime}}$ be an integer and $n=l+\left(m-2\right)l+l^{\prime}$ with $2l\leq l^{\prime}\leq 3l$. Set $A_{j}^{u}\left(x\right)=M^{u}_{l}\left(x+\left(j-1\right)l\omega\right)$, $j=2,\ldots,m-1$,

[TABLE]

and

[TABLE]

The matrices $A^{a}_{j}$ have similar definitions. By Lemma 3.1, for any $j=2,\ldots,m-1$,

[TABLE]

And due to the fact that

[TABLE]

Lemma 3.13,3.10 and 3.1, we have

[TABLE]

and an analogous estimate for $\log\left\|A_{m}^{u}\right\|$. Now the hypothesis of Avalanche Principle are satisfied and hence

[TABLE]

up to a set of measure less than $3m\exp\left(-cL_{l}\left(\delta_{0}^{l}\right)^{-c_{0}}\right)$. By the definitions of $M_{n}^{u}$ and $M_{n}^{a}$, easy computations show that

[TABLE]

Thus, (3.84) also holds for ${\mathcal{M}}_{n}^{a}$. If we set

[TABLE]

then the previous relation can be rewritten as

[TABLE]

Similarly,for any $0\leq k<l-1$,

[TABLE]

where

[TABLE]

which means that we decrease the length of $A^{a}_{m}$ by $k$ and increase the length of $A^{a}_{1}$ by $k$. Adding these equations and dividing by $l$ yields

[TABLE]

up to a set of measure less than $3n\exp\left(-cL_{l}\left(\delta_{0}^{l}\right)^{-c_{0}}\right)$. For the functions $\displaystyle\sum_{j=l}^{\left(m-1\right)l-1}\frac{1}{l}\log\left\|M_{l}^{a}\left(x+j\omega\right)\right\|$ and $\displaystyle\sum_{j=l}^{\left(m-2\right)l-1}\frac{1}{l}\log\left\|M_{2l}^{a}\left(x+j\omega\right)\right\|$, Theorem 4 can be applied. Note that $ml\sim n$. So, the deviation $\delta$ is the smallest deviation $\delta_{0}^{n}$ we can choose here. Then,

[TABLE]

up to a set of measure less than $\exp\left(-cn\delta_{0}^{n}\right)$. Note that $u_{k}$, $k=0,\ldots,l-1$ have the subharmonic extensions. Therefore, for any $\frac{u_{k}}{l}$, Theorem 4 can be applied with $n=1$ and $\delta=\frac{n\delta_{0}^{n}}{l}$, and obtain that

[TABLE]

up to a set of measure less than $l\exp(-cn^{1-c^{\prime}}\delta_{0}^{n})$. Thus, combining these equations, we have that

[TABLE]

up to a set of measure less than $3n\exp\left(-cL_{l}\left(\delta_{0}^{l}\right)^{-c_{0}}\right)+l\exp(-cn^{1-c}\delta_{0}^{n})+\exp\left(-cn\delta_{0}^{n}\right)$. Recalling that $\delta_{0}^{n}=C_{\omega}n^{-\frac{1}{A}+}$, $c_{0}=\frac{1}{5}$ and $l\sim n^{c^{\prime}}$, we have

[TABLE]

where $c^{\prime\prime}$ is a small constant depending on $a,v,\omega$ and $E$. Integrating (3.85) and using the fact that $\left\|\log\left|f_{n}^{a}\right|\right\|_{L^{2}\left({\mathbb{T}}\right)}\leq Cn$, yields

[TABLE]

Combining it with (3.85), we have

[TABLE]

up to a set of measure less than $\exp\left(-c^{\prime\prime}n^{\frac{c^{\prime}}{2}}\right)$. Let ${\mathcal{B}}$ be this exceptional set and define

[TABLE]

where $u_{0}=0$ on ${\mathcal{B}}$ and $u_{1}=0$ on ${\mathbb{T}}\setminus{\mathcal{B}}$. Obviously, $\left\|u_{0}-\left\langle u_{0}\right\rangle\right\|_{L^{\infty}\left({\mathbb{T}}\right)}=O\left(\delta_{0}^{n}\right)$ and

[TABLE]

Due to Lemma 3.12,

[TABLE]

∎

Similar to Lemma 3.2, we also can prove that $\langle\frac{1}{n}\log|f_{n}^{a}|\rangle$ in Theorems 1-3 can be exchanged by $L^{a}$.

Lemma 3.14**.**

There exists a constant $C_{0}=C_{0}\left(a,v,E,\omega,\gamma\right)$ such that

[TABLE]

for all integers.

Proof.

Recall that

[TABLE]

up to a set of measure less than $3m\exp\left(-cL_{l}\left(\delta_{0}^{l}\right)^{-c_{0}}\right)$. Similarly,

[TABLE]

up to a set of measure less than $3m\exp\left(-cL_{l}(E)l\right)$. Subtracting these two expressions and then integrating, yields

[TABLE]

where

[TABLE]

Then, our conclusion is obtained by iterating this estimate. ∎

4. The proof of Theorem 5

We used the LDTs and the Avalanche Principle together in the above two proofs. As we have mentioned in the introduction, this method was first created in [GS01] to prove the Hölder continuity of Lyapunov exponent $L^{s}(E,\omega)$ in $E$ with the strong Diophantine $\omega$. Recently, our second author also applied it to obtain the same continuity of $L(E,\omega)$ with any irrational $\omega$ in [T18]. Our proof of Theorem 5 needs this result. Therefore, we list it as a lemma:

Lemma 4.1**.**

Assume $\beta(\omega)=0$ and $L(E_{0},\omega)>0$. There exists $r_{E}=r_{E}(a,v,E_{0},\omega)$ such that for any $|E-E_{0}|\leq r_{E}$,

[TABLE]

Furthermore, there exists a constant $h=h(a,v)$ called Hölder exponent such that for any $E_{1},E_{2}\in[E_{0}-r_{E},E_{0}+r_{E}]$,

[TABLE]

The proof of Theorem 5.

From Theorem 1-3 and Lemma 3.14, we have that for any $\delta>\delta_{0}^{n}$ and $\left(x,E\right)\in{\mathbb{T}}\times{\mathcal{E}}$ except for a set of measure $C\exp\left(-c\delta\left(\delta_{0}^{n}\right)^{-1}\right)$,

[TABLE]

Then, due to Fubini’s Theorem and Chebyshev’s inequality, there exists a set ${\mathcal{B}}_{n,\delta}\subset{\mathbb{T}}$ with $\mathop{\rm{mes}\,}{\mathcal{B}}_{n,\delta}<C\exp\left(-c\delta\left(\delta_{0}^{n}\right)^{-1}\right)$, such that for each $x\in{\mathbb{T}}\setminus{\mathcal{B}}_{n,\delta}$ there exists ${\mathcal{E}}_{n,\delta,x}\subset{\mathcal{D}}$, with $\mathop{\rm{mes}\,}{\mathcal{E}}_{n,\delta,x}<C\exp\left(-c\delta\left(\delta_{0}^{n}\right)^{-1}\right)$ such that (4.88) holds for any $E\in{\mathcal{E}}\setminus{\mathcal{E}}_{n,\delta,x}$. Therefore, there exist $x_{1},\,E_{1}$ satisfying

[TABLE]

and

[TABLE]

such that

[TABLE]

Define

[TABLE]

and

[TABLE]

The Jensen formula states that for any function $f$ analytic on a neighborhood of ${\mathcal{D}}(z_{0},R)$, see [L96],

[TABLE]

provided $f(z_{0})\neq 0$. Thus, we have that

[TABLE]

By Lemma 3.11, it yields

[TABLE]

Due to Lemma 4.1, if $|E_{1}-E_{2}|<4R$, then

[TABLE]

Combining it with Lemma 3.2 and the fact that $\delta_{0}^{n}\gg\frac{\left(\log n\right)^{2}}{n}$, we have

[TABLE]

Thus,

[TABLE]

Recalling that $|E_{0}-E_{1}|\ll R$, we have

[TABLE]

Note that $H_{n}(x,\omega)$ is Hermitian. Thus, by the Mean Value Theorem,

[TABLE]

Let $E_{j}^{\left(n\right)}\left(x,\omega\right)$, $j=1,\ldots,n$ be the eigenvalues of $H_{n}\left(x,\omega\right)$ ordered increasingly. Then,

[TABLE]

This implies that

[TABLE]

∎

Remark 4.1*.*

Similarly,

[TABLE]