# Pointwise modulus of continuity of the Lyapunov exponent and integrated
density of states for analytic multi-frequency quasiperiodic $M(2,\mathbb{C})$
cocycles

Matthew Powell  Department of Mathematics, Georgia Institute of Technology,
Atlanta GA, 30332

###### Abstract.

It is known that the Lyapunov exponent for multifrequency analytic cocycles is
weak-Hölder continuous in cocycle for certain Diophantine frequencies, and
that this implies certain regularity of the integrated density of states in
energy for Jacobi operators. In this paper, we establish the pointwise modulus
of continuity in both cocycle and frequency and obtain analogous regularity of
the integrated density of states in energy, potential, and frequency.

##  1\. Introduction

In this paper, we are interested in the regularity of the Lyapunov exponent
associated to multifrequency quasi-periodic cocycles. Let $M(2,\mathbb{C})$
denote the set of $2\times 2$ matrices with complex entries. Let
$\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ denote the $d$-dimensional
torus and let $S_{\omega}:\mathbb{T}^{d}\to\mathbb{T}^{d}$ denote the shift by
$\omega\in\mathbb{T}^{d}:$ $x\mapsto x+\omega.$ A $d$-dimensional quasi-
periodic cocycle is a pair $(A,\omega)\in
C(\mathbb{T}^{d},M(2,\mathbb{C}))\times\mathbb{R}$ understood as a linear skew
product
$(A,\omega):\mathbb{C}^{2}\times\mathbb{T}^{d}\to\mathbb{C}^{2}\times\mathbb{T}^{d}$
with

$(w,x)\mapsto(A(x)w,S_{\omega}x).$

Cocycles enjoy the property that they may be iterated, in the following sense:
the $N^{th}$ iterate of $(A,\omega)$ is

$A_{N}(x,\omega)=\prod_{j=N-1}^{0}A(S^{j}_{\omega}x).$

We are interested in analytic quasiperiodic cocycles, so we assume $A$ is an
analytic $M(2,\mathbb{C})$-valued function on $\mathbb{T}^{d}.$ It is known
that continuity of the Lyapunov exponent fails, in general, when $A$ is only
$C^{\infty}$ (c.f. [18] or [25]). Since we assume that $A$ is analytic on
$\mathbb{T}^{d},$ we may extend $A$ to some complex strip,
$|\Im(z_{j})|<\rho,j\leq d.$ We denote the space of such $A$ by
$C_{\rho}(\mathbb{T}^{d},M(2,\mathbb{C})),$ and imbue this with the natural
metric

$\left\|A\right\|_{\rho}:=\sup_{|\Im(z_{j})|<\rho/2}\left\|A(z_{1},...,z_{d})\right\|.$

######  Remark 1.

Note that we take the supremum over $|\Im(z_{j})|<\rho/2$ rather than over
$|\Im(z_{j})|<\rho$ to ensure that the supremum exists. We may do this because
$A$ is also an analytic cocycle over the closed set
$\left\\{z\in\mathbb{C}^{d}:|\Im(z_{j})|\leq\rho/2\right\\}.$

From this, we can inductively define a topology on the space of all analytic
cocycles, but since convergence in this topology is equivalent to convergence
in $(C_{\rho},\left\|\cdot\right\|_{\rho})$ for some $\rho>0,$ we will
restrict our attention to $C_{\rho}$ for a fixed $\rho.$

Cocycles of this form have been used extensively to study one-dimensional
discrete Jacobi operators:

$H_{x,\omega}:\ell^{2}(\mathbb{Z})\to\ell^{2}(\mathbb{Z})$

given by

(1)
$(H_{x,\omega}\psi)(n)=\overline{a(S^{n-1}_{\omega}x)}\psi(n-1)+a_{(}S^{n}_{\omega}x)\psi(n+1)+v(S^{n}_{\omega}x)\psi(n),$

where $a,v\in C(\mathbb{T}^{d},\mathbb{R}).$ Solutions to the eigenequation
$H_{x,\omega}\psi=E\psi$ may be recovered via the $N^{th}$ transfer matrix

$A_{N}(x,\omega,E)=\prod_{j=N-1}^{0}M_{j}(x,\omega,E),$

where

$M_{j}(x,\omega,E)=\begin{pmatrix}E-v(S^{j+1}_{\omega}x)&-\overline{a(S^{j}_{\omega}x)}\\\
a(S^{j+1}_{\omega}x)&0\end{pmatrix}.$

Indeed, any solution to $H_{x,\omega}\psi=E\psi$ satisfies

$\begin{pmatrix}\psi(N+1)\\\
\psi(N)\end{pmatrix}=A_{N}(x,\omega,E)\begin{pmatrix}\psi(1)\\\
\psi(0)\end{pmatrix}.$

Moreover, the transfer matrices is a classic example of an iterate of a
quasiperiodic cocycle, by taking $A(x,\omega)=M_{1}(x,\omega,E).$

Now observe that any $M(2,\mathbb{C})$ cocycle $A_{N}(x)$ for which
$\det(A(x))$ is not identically zero can be renormalized to form an
$SL(2,\mathbb{C})$ cocycle (see e.g.[17]), however the resulting cocycle will
lose pointwise boundedness if $\det(A(x))$ has zeros. This is precisely the
nature of difficulty when extending $SL(2,\mathbb{C})$ results to the
$M(2,\mathbb{C})$ case.

The (upper) Lyapunov exponent is defined as

(2)
$L^{\prime}(A,\omega)=\frac{1}{N}\int_{\mathbb{T}^{d}}\ln\left\|A_{N}(x,\omega)\right\|dx.$

Note that, while $L^{\prime}(A,\omega)$ need not be non-negative, the related
object

(3)
$L(A,\omega)=\lim_{N\to\infty}\int_{\mathbb{T}^{d}}L_{N}(\tilde{A},\omega,x)dx,$

is, where $\tilde{A}\in SL(2,\mathbb{C})$ is a renormalization of $A:$

(4) $\tilde{A}=\frac{1}{|\det A|^{1/2}}A.$

Moreover, $L_{N}$ and $L^{\prime}_{N}$ are related by the following relation:

(5)
$L_{N}(A,\omega)=L^{\prime}_{N}(A,\omega)-\frac{1}{2}\int_{\mathbb{T}^{d}}\ln|\det(A(x))|dx.$

It follows that, when $\ln|\det(A(x))|\in L^{1},$ both $L$ and $L^{\prime}$
share the same regularity properties.

It is often easier to deal with $L^{\prime}$ when proving general boundedness
and finite-scale continuity (see Sections 2 and 3), but it is easier to deal
with a non-negative quantity when proving and using our induction scheme (see
Sections 4 and 5). Thus both $L$ and $L^{\prime}$ play a role in this paper.

Returning to (1), an object related to the Lyapunov exponent for such
operators is the integrated density of states (IDS), which maybe generally
defined as in [7]. Let $E_{1}<E_{2}$ and define

$k(x,\omega,E_{1},E_{2})=\limsup_{N\to\infty}\frac{1}{2N+1}\left\\{\text{eigenvalues
of}R_{[-N,N]}H_{x,\omega}R_{[-N,N]}\text{ in }[E_{1},E_{2}]\right\\},$

where $R_{[-N,N]}$ denotes the projection onto the interval $[-N,N].$ In our
setting, this $\limsup$ is constant for Lebesgue a.e. $x.$ Clearly one may ask
about the regularity of this object in the various parameters,
$V,E_{1},E_{2},\omega.$ We obtain such a statement as a consequence of our
main theorem (see Corollary 1.1).

The continuity of the Lyapunov exponent (in both cocycle and frequency) in
this setting has been studied extensively in [20], where the author adapted an
argument of Bourgain originally used to study $SL(2,\mathbb{C})$ cocycles to
study singular cocycles.

The (pointwise) modulus of continuity of the Lyapunov exponent for one-
frequency quasi-periodic cocycles has been studied by many authors. While the
Lyapunov exponent is known to be continuous in a very general settings, the
(pointwise) modulus of continuity is a more delicate matter. This question was
first studied in [14] for one-frequency (the underlying torus is
$\mathbb{T}^{1}$) Schrödinger operators with fixed strongly diophantine
frequency $\omega.$ There, authors also obtained analogous modulus of
continuity for the IDS in the same setting. A key component of this proof was
an a priori positivity assumption: $L(E)>0.$ Shortly afterwards, the Lyapunov
exponent for Schrödinger cocycles was shown to be continuous in both energy
(for all frequencies) and frequency (at irrational frequencies) without
positivity or Diophantine assumptions [6]. This work was unable to obtain a
modulus of continuity, however. It turns out that any modulus of continuity
better than $\log$-Hölder requires both positivity and some arithmetic
(perhaps weak) assumption [4].

Many authors have worked to extend various results from [14] with some
success. The modulus of continuity of the Lyapunov exponent in the one-
frequency setting has been extended to singular cocycles and fixed weaker
diophantine frequencies [22]; the multifrequency case has proved more
delicate, and the known results still require some restrictive condition on
the frequency [11]. The modulus for both the Lyapunov exponent and IDS has
also been obtained and/or sharpened in a variety of settings closely related
to the Schrödinger case [1, 2, 8, 9, 10, 12, 13, 15, 16, 23, 24, 26]

We prove the following.

######  Theorem 1.1.

Let $A(x)\in C_{\rho}$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle
on $\mathbb{T}^{d}$ with a plurisubharmonic extension to the strip
$|\Im(z_{j})|<\rho.$ Suppose that $\omega\in\mathbb{T}^{d}$ is such that, for
some $\sigma\geq 1,$ $\left\|k\cdot\omega\right\|>\tau|k|^{-\sigma}>0$ for all
$k\in\mathbb{Z}^{d}$ with $0<|k|,$ and suppose that $L(A,\omega)>0.$ Then
there is $\delta=\delta(A)$ and $\gamma=\gamma(d,\sigma)\leq 1,$ such that,
for every $A^{\prime}\in C_{\rho}$ with
$\left\|A-A^{\prime}\right\|_{\rho}+\left\|\omega-\omega^{\prime}\right\|<\delta,$

(6)
$|L(A,\omega)-L(A^{\prime},\omega^{\prime})|<C(A)\exp\left\\{-c(A)\left(-\ln\left(\left\|A-A^{\prime}\right\|_{\rho}+\left\|\omega-\omega^{\prime}\right\|\right)\right)^{\gamma}\right\\}.$

We would like to make a few remarks at this point.

######  Remark 2.

Generally, a pointwise modulus of continuity of the form (6) is called
pointwise $\gamma$-weak-Hölder continuity.

######  Remark 3.

The exponent $\gamma$ above may be computed explicitly in terms of the
Diophantine parameter $\sigma$ and an exponent from a large deviation
estimate, which depends only on the dimension, $d,$ of the torus
$\mathbb{T}^{d}.$ In particular, improvement of the large deviation estimate
would lead to improvement of the exponent $\gamma,$ which has been remarked in
both [14] and [11].

######  Remark 4.

Notice that, while $\omega$ must satisfy $\omega\in\mathbb{T}^{d}$ is such
that $\left\|k\cdot\omega\right\|>\tau|k|^{-\sigma}>0$ for all
$k\in\mathbb{Z}^{d}$ with $0<|k|,$ the frequency $\omega^{\prime}$ need not.
This is a key improvement on [24], where $\omega^{\prime}$ had to also satisfy
the same condition, and is a consequence of our argument from [20].

In the remainder of this paper, when there can be no ambiguity, we will write
$\left\|A\right\|$ in place of
$\left\|A(x)\right\|=\left\|A(x)\right\|_{\rho}.$

As a corollary, we obtain analogous continuity for the IDS.

######  Corollary 1.1.

Let $H_{x,\alpha}$ be as in (1), with $V\in
C^{\omega}(\mathbb{T}^{d},\mathbb{R})$ a not identically singular analytic
function on $\mathbb{T}^{d}.$ Suppose that $\alpha\in\mathbb{T}^{d}$ is such
that $\left\|k\cdot\alpha\right\|>\tau|k|^{-\sigma}>0$ for all
$k\in\mathbb{Z}^{d}$ with $0<|k|,$ and suppose $L(E)>0.$ Let
$k(x,\alpha,E_{1},E_{2})$ be defined as above. Then for a.e. $x,$
$k(\alpha,E_{1},E_{2})$ obeys

$\displaystyle|k(\alpha,E_{1},E_{2})|<\exp\left\\{-\left(-\ln\left(|E_{1}-E_{2}|\right)\right)^{\gamma}\right\\}.$

This may be seen either as a consequence of Theorem 2.1 via an argument of
Bourgain [3] (and extended first by Schlag [21] and later by Liu [19] for any
Schrödinger operator with a large deviation estimate), or as a consequence of
Theorem 1.1 by an argument of Goldstein-Schlag using the Thouless formula,
which generalizes to multifrequency Jacobi operators. As these arguments are
standard and not new, we refer readers to those works for details.

The rest of this paper is organized as follows. In Section 2, we recall a few
well-known results for subharmonic functions and the finite-scale Lyapunov
exponents, which we use throughout. Then we prove a modulus of continuity for
the finite-scale Lyapunov exponent in Section 3. In Section 4, we provide an
inductive procedure to obtain a locally uniform rate of convergence for the
Lyapunov exponent when the frequency is fixed. In Section 5, we combine the
uniform rate of convergence with modulus of continuity for the finite scale
Lyapunov exponents to obtain corresponding modulus of continuity for the
Lyapunov exponent in both frequency and cocycle.

##  2\. Preliminaries: uniform bounds on Lyapunov exponents, large deviations,
and Avalanche Principle

Throughout this paper, we use $C$ and $C(\cdot)$ to denote large constants
which depend on uniform measurements of the cocycle and dimension, $c$ and
$c(\cdot)$ to denote small constants which depend on uniform measurements of
the cocycle and dimension, and $C_{\rho}$ to denote constants which depend on
the parameter $\rho.$ Unless otherwise stated, these constants may change by
multiplicative constants throughout their appearance in a proof, but will
remain finite, non-zero, and uniform in their respective parameters.

This section is devoted to a few essential preliminary results which will be
used in later sections. They may be found in a variety of other papers, and
are provided here without proof except where the proof introduces ideas useful
in the study of singular cocycles.

We begin with a uniform version of the Lojasiewicz inequality, which is used
throughout and may be of independent interest to readers.

######  Lemma 2.1 ([11] Lemma 6.1).

Let $f(x)\in C_{\rho}(\mathbb{T}^{d},\mathbb{C})$ be such that $f(x)$ is not
identically zero. Then there are constants $\delta=\delta(f)>0,S=S(f)<\infty,$
and $b=b(f)>0$ such that if $g(x)\in C_{\rho}(\mathbb{T}^{d},\mathbb{C})$ with
$\left\|g-f\right\|_{\rho}<\delta,$ then

(7) $\left|\left\\{x\in\mathbb{T}^{d}:|g(x)|<t\right\\}\right|<St^{b}$

for all $t>0.$

This has, as a consequence, uniform $L^{2}$-boundedness of $L_{N}(A,\omega)$
in $N,\omega,$ and locally in $A.$ This result may also be found in [11], but
our proof differs from the one therein.

######  Lemma 2.2.

Let $(A,\omega)$ be an analytic $M(2,\mathbb{C})$ cocycle for which $\det
A(x)$ is not identically zero. There is $\delta=\delta(A)>0$ and $C=C(A)$ such
that, for every cocycle $(B,\omega^{\prime})$ such that
$\left\|A-B\right\|_{\rho}+\left\|\omega-\omega^{\prime}\right\|<\delta,$

$\left\|L_{N}(B,\omega^{\prime})\right\|_{L^{2}}<C(A).$

###### Proof.

Note that it suffices to prove that $L_{N}^{\prime}(B,\omega^{\prime},x)$ and
$\ln|\det B(x)|$ obey this $L^{2}$-estimate. Clearly we have

$\displaystyle\int_{\mathbb{T}^{d}}\ln|\det B(x)|$ $\displaystyle=\int_{|\det
B(x)|\geq 1}+\sum_{j=0}^{\infty}\int_{2^{-j}>|\det B(x)|\geq 2^{-j-1}}.$

Moreover, since $A(x)$ is analytic, there is some $C(A)<\infty$ such that
$\left\|A(x)\right\|_{\rho}<C.$ Thus, for any $B$ such that
$\left\|A-B\right\|_{\rho}<C/2,$ we have

$\left\|B(x)\right\|_{\rho}<2C.$

It follows that

$|\det B(x)|<4C^{2}.$

On $\left\\{x:\epsilon\leq|\det B(x)|<1\right\\},$

$\left|\ln|\det B(x)|\right|<-\ln\epsilon.$

Hence, on $\left\\{x:2^{-j-1}\leq|\det B(x)|<2^{-j}\right\\},$

$\left|\ln|\det B(x)|\right|<(j+1)\ln 2.$

By Lemma 2.1, we have, for $\left\|A-B\right\|_{\rho}<\delta(A)$

$\left|\left\\{x:2^{-j-1}\leq|\det
B(x)|<2^{-j}\right\\}\right|<S(A)\left(2^{-j}\right)^{b}.$

Altogether, this yields

$\displaystyle\int_{\mathbb{T}^{d}}|\ln|\det B(x)||^{2}$ $\displaystyle\leq
4(\ln 4C(A))^{2}+S(A)\sum 2^{-bj}\left((j+1)\ln 2\right)^{2}$
$\displaystyle\leq(\ln 2C(A))^{2}+S(A)C(A)$ $\displaystyle<C(A).$

Now consider $L_{N}^{\prime}(B,\omega^{\prime},x).$ We claim that, for some
$C=C(A)$ and $\delta>0,$ and every $B$ such that
$\left\|A-B\right\|_{\rho}<\delta,$ we have

$|L_{N}^{\prime}(B,\omega^{\prime},x)|\leq
C+\frac{1}{N}\sum_{j=0}^{N-1}\left|\ln|\det(B(x+j\omega^{\prime}))|\right|.$

The desired conclusion quickly follows from this inequality and the above
estimate for $\ln|\det B(x)|,$ so we turn our attention to a proof.

Let $\delta>0$ be such that Lemma 2.1 holds for the analytic functions $\det
B(x)$ when $\left\|B-A\right\|_{\rho}<\delta.$ Observe that there is
$C=C(A)>0$ such that $\left\|A\right\|_{\rho}<e^{C},$ so
$\frac{1}{N}\ln\left\|A_{N}\right\|_{\rho}\leq C.$ By our assumptions on
$\left\|B-A\right\|_{\rho},$ it follows that

$\left\|B\right\|_{\rho}<e^{2C},$

and thus

$\frac{1}{N}\ln\left\|B_{N}(x)\right\|<2C.$

Now we consider two sets:

$F_{+}:=\left\\{x:\ln\left\|B_{N}(x)\right\|\geq 0\right\\}$

and

$F_{-}:=\left\\{x:\ln\left\|B_{N}(x)\right\|<0\right\\}.$

Clearly, for $x\in F_{+},$ the desired inequality holds. Consider $x\in
F_{-}.$ Recall that, for any $2\times 2$ matrix $A,$ we have

$|\det(A)|\leq 2\left\|A\right\|^{2},$

so it follows that

$\frac{1}{N}\ln\left\|B_{N}(x)\right\|\geq\frac{1}{2N}\ln|\det(B_{N}(x))|=\frac{1}{2N}\sum_{j=0}^{N-1}\ln|\det(B(x+j\omega))|.$

For $x\in F_{-},$ this implies

$|L_{N}^{\prime}(B,\omega,x)|\leq\left|\frac{1}{2N}\sum_{j=0}^{N-1}\ln|\det(B(x+j\omega))|\right|,$

which, after applying triangle inequality, is our desired bound.

∎

We conclude this section with two essential results which will be used in the
sequel. The first is a so-called large deviation estimate and the second is a
consequence of the Avalanche Principle which, in this form, was originally due
to Bourgain for $SL(2,\mathbb{C})$ cocycles, but extended to non-identically
singular $M(2,\mathbb{C})$ cocycles by us in [20]. We refer readers to either
[5] (for the $SL(2,\mathbb{C})$ case) or [20] (for the $M(2,\mathbb{C})$ case)
for proofs.

######  Theorem 2.1.

[[20] Theorem 2.2] Let $(A,\omega)$ be an analytic $M(2,\mathbb{C})$ cocycle,
with an analytic extension to $|\Im(z_{j})|<\rho,$ for which $\det A(x)$ is
not identically zero. Suppose $\omega\in\mathbb{T}^{d}$ is such that

$\left\|k\cdot\omega\right\|>\delta_{0}$

for all $0<|k|<K_{0}.$ Moreover, suppose

$N>K_{0}\delta_{0}^{-1}.$

Then there are $\delta=\delta(A)>0,$ $c=c(d)<1,$ and $C_{\rho}<\infty$ such
that for any $B$ with with an analytic extension to $|\Im(z_{j})|<\rho$
satisfying $\left\|B-A\right\|_{\rho}<\delta,$

(8)
$\left|\left\\{x\in\mathbb{T}^{d}:\left|L_{N}(B,x)-L_{N}(B)\right|>\rho^{-1}K_{0}^{-c}\right\\}\right|<e^{-C_{\rho}K_{0}^{c}}.$

######  Theorem 2.2.

[[20] Theorem 4.3] Let $(A,\omega)$ be an analytic $M(2,\mathbb{C})$ cocycle
with an analytic extension to $|\Im(z_{j})|<\rho.$ Fix $x\in\mathbb{T}^{d}$
and $\delta>0.$ Let $N>N_{0}(\delta)$ be sufficiently large and $N|N_{1}$ with
$N\leq N_{1}.$ Suppose that

(9) $\displaystyle L_{N}(A,x)$ $\displaystyle>\delta$ (10)
$\displaystyle\left|L_{N}(A,x)-L_{2N}(A,x)\right|$
$\displaystyle<\frac{1}{100}L_{N}(A,x)$ (11)
$\displaystyle\max_{n=N,2N}\left|L_{n}(A,x)-L_{n}(A,x+jN\omega)\right|$
$\displaystyle<\frac{\delta}{100},$

for all $j\leq N_{1}/N.$ Then

(12)
$\displaystyle\begin{split}\bigg{|}L_{N_{1}}(A,x)+\frac{1}{n}\sum_{j=0}^{n-1}L_{N}(A,x+&jN\omega)-\frac{2}{n}\sum_{j=0}^{n-1}L_{2N}(A,x+jN\omega)\bigg{|}\\\
&<\exp\left(-\frac{N}{4}L_{N}(x)\right)+CL_{N}(A,x)\frac{N}{N_{1}}.\end{split}$

Here $C$ is an absolute constant.

######  Remark 5.

In the above, $N_{0}$ is such that $N_{0}\delta>2.$

##  3\. Finite-scale weak-Hölder continuity

In the Schrödinger cocycle case (and the $SL(2,\mathbb{C})$ case more
generally), one of the key observations is that, for fixed $N,$
$L_{N}(A,\omega)$ is jointly continuous in $A$ and $\omega$ for any $\omega.$
This is a simple consequence of the everywhere invertibility of the cocycle,
$A.$ In this section, we prove an analogous result for non-identically-
singular cocycles. Indeed, our strategy hinges on this fact, as we observe
that, given a sequence of continuous functions (in this case,
$L_{N}(A,\omega)$), the continuity of the limiting object (in this case,
$L(A,\omega)$) may be obtained by quantitatively estimating the rate of
convergence ($L_{N}\to L$) uniformly in $A$ and $\omega.$ This argument does
not work if $L_{N}(A,\omega)$ is not continuous!

We do not claim that this result is novel; it is included for completeness,
and because the proof introduces some ideas which are useful for studying
singular cocycles.

######  Theorem 3.1.

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with an analytic extension to the strip $|\Im(z_{j})|<\rho.$
There are $\delta_{1}(A)>0$ and $b(A)>0$ such that for any $\alpha\geq 1,$
whenever $\left\|A-B\right\|<\delta_{1},$ we have

(13)
$\displaystyle\begin{split}|L_{N}^{\prime}(A,\omega)-L_{N}^{\prime}(B,\omega^{\prime})|&<C(A)e^{-2N^{\alpha-1}b(A)}\\\
&\quad+e^{2N^{\alpha}}\left(\left\|A-B\right\|+N\left\|\omega-\omega^{\prime}\right\|\right).\end{split}$

Moreover,

(14) $\displaystyle\left|\int_{\mathbb{T}^{d}}\ln|\det A(x)|-\ln|\det
B(x)|dx\right|$ $\displaystyle<C(A)\left\|A-B\right\|^{b/(1+b)}.$

######  Remark 6.

Note that this implies, by the definition of $L_{N}$ and $L_{N}^{\prime},$
that

(15)
$\displaystyle\begin{split}|L_{N}(A,\omega)-L_{N}(B,\omega^{\prime})|&<C(A)e^{-2N^{\alpha-1}b(A)}\\\
&\quad+e^{N^{\alpha}}\left(\left\|A-B\right\|+N\left\|\omega-\omega^{\prime}\right\|\right).\end{split}$

###### Proof.

We first prove the result for $L_{N}^{\prime}(A).$ The result for the
determinant follows from an analogous argument, and we will discuss how to
adjust the proof afterwards.

Let $C_{0}(A)$ be such that $\left\|A(x)\right\|\leq e^{C_{0}}.$ For
$\left\|A-B\right\|$ sufficiently small, depending only on $C_{0},$
$\left\|B(x)\right\|\leq e^{2C_{0}}.$ Fix $\delta_{0}>0$ and set

(16) $\displaystyle F_{A}(\delta_{0})$
$\displaystyle=\left\\{x\in\mathbb{T}^{d}:\left\|A_{N}\right\|<\delta_{0}\right\\}$
(17) $\displaystyle F_{B}(\delta_{0})$
$\displaystyle=\left\\{x\in\mathbb{T}^{d}:\left\|B_{N}\right\|<\delta_{0}\right\\}.$

Clearly

$\int_{\mathbb{T}^{d}}=\int_{F_{A}\cap F_{B}}+\int_{F_{A}^{c}\cap
F_{B}}+\int_{F_{A}\cap F_{B}^{c}}+\int_{F_{A}^{c}\cap F_{B}^{c}}.$

Note that $|\det(A_{N}(x))|\leq\left\|A_{N}(x)\right\|^{2}$ for all $x.$ Thus

$F_{A}(\delta_{0})\subset\left\\{x:|\det
A_{N}(x)|<\delta_{0}^{2}\right\\}\subset\bigcup_{j=0}^{N-1}\left\\{x:|\det
A(x_{j}\omega)|<\delta_{0}^{2/N}\right\\},$

and

$F_{b}(\delta_{0})\subset\left\\{x:|\det
B_{N}(x)|<\delta_{0}^{2}\right\\}\subset\bigcup_{j=0}^{N-1}\left\\{x:|\det
B(x+j\omega^{\prime})|<\delta_{0}^{2/N}\right\\}.$

Moreover, we know $\ln\left\|A_{N}(x)\right\|\in L^{p},1\leq p<\infty$ and,
for $\left\|A-B\right\|<\delta_{1}=\delta_{1}(A),$
$\ln\left\|B_{N}(x)\right\|\in L^{p},1\leq p<\infty$ and
$\left\|\ln\left\|B(x)\right\|\right\|_{p}<C(p)\left\|\ln\left\|A(x)\right\|\right\|_{p}.$
These all hold uniformly in $N.$ This implies that, for
$\left\|A-B\right\|<\delta_{1},$

$\int_{F_{A}\cap F_{B}}+\int_{F_{A}^{c}\cap F_{B}}+\int_{F_{A}\cap
F_{B}^{c}}\leq 2C(A)|F_{A}(\delta_{0})|+C(A)|F_{B}(\delta_{0})|,$

where $|F_{B}|$ denotes the Lebesgue measure. Moreover, by the Lojaciewz
inequality,

$|F_{A}(\delta_{0})|<C(A)N\delta_{0}^{2b(A)/N}$

and

$|F_{B}(\delta_{0})|<C(A)N\delta_{0}^{2b(A)/N}.$

Finally, for $x\in F_{A}^{c}\cap F_{B}^{c},$ we have

(18) $\displaystyle|\ln\left\|A_{N}(x)\right\|-\ln\left\|B_{N}(x)\right\||$
$\displaystyle\leq\max\left\\{\left\|A_{N}-B_{N}\right\|\left\|B_{N}\right\|^{-1},\left\|B_{N}-A_{N}\right\|\left\|A_{N}\right\|^{-1}\right\\}$
(19) $\displaystyle\leq\left\|A_{N}-B_{N}\right\|\delta_{0}^{-1}.$

Thus, we have

(20) $\displaystyle\begin{split}\int_{\mathbb{T}^{d}}&=\int_{F_{A}\cap
F_{B}}+\int_{F_{A}^{c}\cap F_{B}}+\int_{F_{A}\cap
F_{B}^{c}}+\int_{F_{A}^{c}\cap F_{B}^{c}}\\\ &\leq
C(A)\delta_{0}^{2b(A)/N}+N^{-1}\left\|A_{N}-B_{N}\right\|\delta_{0}^{-1}.\end{split}$

Moreover, a standard telescoping argument implies

(21) $\displaystyle\left\|A_{N}-B_{N}\right\|$ $\displaystyle\leq
N\left\|A\right\|^{2N}\left(\left\|A-B\right\|+\left\|N\omega-N\omega^{\prime}\right\|\right)$
(22) $\displaystyle\leq
N\left\|A\right\|^{2N}\left(\left\|A-B\right\|+N\left\|\omega-\omega^{\prime}\right\|\right).$

We thus have

(23)
$\displaystyle\begin{split}|L_{N}^{\prime}(A,\omega)-L_{N}^{\prime}(B,\omega^{\prime})|&<C(A)\delta_{0}^{2b(A)/N}\\\
&\quad+\left\|A\right\|^{2N}\left(\left\|A-B\right\|+N\left\|\omega-\omega^{\prime}\right\|\right)\delta_{0}^{-1}.\end{split}$

Finally, set

$\delta_{0}=e^{-N^{\alpha}},\quad\alpha>1.$

Recall that $\left\|A\right\|\leq e^{C_{0}},$ and consequently

$\displaystyle|L_{N}^{\prime}(A,\omega)-L_{N}^{\prime}(B,\omega^{\prime})|$
$\displaystyle<C(A)e^{-2N^{\alpha-1}b(A)}$
$\displaystyle\quad+e^{2C_{0}N}\left(\left\|A-B\right\|+N\left\|\omega-\omega^{\prime}\right\|\right)e^{N^{\alpha}}$
$\displaystyle\leq
C(A)e^{-2N^{\alpha-1}b(A)}+e^{2N^{\alpha}}\left(\left\|A-B\right\|+N\left\|\omega-\omega^{\prime}\right\|\right).$

To obtain the continuity for $\ln|\det(A)|,$ we use the above argument where
$F_{A}(\delta_{0})$ and $F_{B}(\delta_{0})$ are redefined as

(24) $\displaystyle F_{A}(\delta_{0})^{\prime}$
$\displaystyle:=\left\\{x:|\det(A(x))|<\delta_{0}\right\\}$ (25)
$\displaystyle F_{B}(\delta_{0})^{\prime}$
$\displaystyle:=\left\\{x:|\det(B(x))|<\delta_{0}\right\\}.$

At the end, we take $\delta_{0}=\left\|A-B\right\|^{1/(1+b)},$ where $b=b(a)$
is as in the Lojasiewicz inequality. ∎

##  4\. Lyapunov exponents: local uniform rate of convergence with fixed
Diophantine frequency

In this section, we will establish an inductive scheme to obtain a local
uniform rate of convergence for the Lyapunov exponents corresponding to fixed
Diophantine frequencies. We will break this up into multiple steps.

######  Lemma 4.1.

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with with an analytic extension to $|\Im(z_{j})|<\rho$ such
that $\det A(x)\not\equiv 0.$ Suppose that $\omega\in\mathbb{T}^{d}$ is such
that $\left\|k\cdot\omega\right\|>\delta_{0}>0$ for all $k\in\mathbb{Z}^{d}$
with $0<|k|\leq K_{0},$ where $K_{0}$ satisfies

(26) $\displaystyle K_{0}$ $\displaystyle>(\rho\kappa)^{-C}$ (27)
$\displaystyle N_{0}$ $\displaystyle>\kappa^{-C}\delta_{0}^{-1}K_{0}.$

Moreover, suppose that

(28) $\displaystyle L_{N_{0}}(A)$ $\displaystyle>10^{3}\kappa$ (29)
$\displaystyle|L_{N_{0}}-L_{2N_{0}}|$ $\displaystyle<\frac{1}{100}L_{N_{0}}.$

Then

(30) $|L_{N}+L_{N_{0}}-2L_{2N_{0}}|<C^{\prime}(A)e^{-C_{\rho}K_{0}^{c}}$

when $N_{0}|N,N=2^{j}N_{0}$ for some $j\geq 0,$ and

(31) $N<N_{0}e^{\frac{1}{2}\rho K_{0}^{c}}.$

In the above, $C$ is a sufficiently large absolute constant depending only on
$d$ which is defined in the proof, $c=C^{-1},$ and $C^{\prime}(A)$ is a
constant which is uniform in $A.$

###### Proof.

First, consider the set

$G=\left\\{x:|L_{N_{0}}(A,x)-L_{N_{0}}(A)|>\kappa\right\\}.$

Our large deviation theorem yields

$|G|<e^{-C_{\rho}K_{0}^{c}}.$

Thus, on $\mathbb{T}^{d}\backslash G,$

$L_{N_{0}}(A,x)>999\kappa,$

and

(32)
$\left|\int_{G}L_{N_{1}}(x)+L_{N_{0}}(x)-2L_{2N_{0}}(x)dx\right|<e^{-C_{\rho}K_{0}^{c}}.$

Let us now consider a further restriction to the set,
$F\subset\mathbb{T}^{d}\backslash G,$ consisting all of $x$ such that

$|L_{N_{0}}(A,x)-L_{N_{0}}(A,x+jN_{0}\omega)|<2\kappa$

for $1\leq j\leq N_{1}/N_{0}.$ Another application of our large deviation
theorem implies this set satisfies

$|\mathbb{T}^{d}\backslash
F|<\frac{N_{1}}{N_{0}}e^{-C_{\rho}K_{0}^{c}}<e^{-C_{\rho}K_{0}^{c}}$

for $N_{1}<N_{0}e^{C_{\rho}K_{0}^{c}}.$

Finally, one last application of our large deviation theorem, applied to (29),
implies the set

$H=\left\\{x\in\mathbb{T}^{d}\backslash
G:|L_{N_{0}}(x)-L_{2N_{0}}(x)|<\frac{1}{100}L_{N_{0}}(x)\right\\}$

obeys the measure estimate

$|\mathbb{T}^{d}\backslash H|<e^{-C_{\rho}K_{0}^{c}}.$

Now we see that every $x\in F\cap H$ satisfies the three hypotheses of Theorem
2.2, so on $F\cap H$ we obtain

(33)
$\displaystyle\begin{split}\bigg{|}L_{N_{1}}(A,x)+\frac{1}{n}\sum_{j=0}^{n-1}L_{N_{0}}(&A,x+jN_{0}\omega)-\frac{2}{n}\sum_{j=0}^{n-1}L_{2N_{0}}(A,x+jN_{0}\omega)\bigg{|}\\\
&<\exp\left(-\frac{N_{0}}{4}L_{N_{0}}(x)\right)+C^{\prime}L_{N_{0}}(A,x)\frac{N_{0}}{N_{1}}.\end{split}$

We may now integrate the left hand side of the above inequality, over
$\mathbb{T}^{d}:$

(34) $\int_{\mathbb{T}^{d}}=\int_{F\cap H}+\int_{F^{c}\cap H}+\int_{F\cap
H^{c}}.$

Using $L_{N_{0}}(x)>999\kappa$ on $F\cap G,$ our restriction on $N_{1},$ and
(33) we have

(35) $\int_{F\cap
H}<e^{-\frac{999}{4}N_{0}\kappa}+C^{\prime}(A)L_{N_{0}}(A)e^{-C_{\rho}K_{0}^{c}}.$

By the uniform $L^{2}$-boundedness of the integrand (i.e.
$\left\|L_{N}(A)\right\|_{2}<C(A)$) and our measure estimates on
$|\mathbb{T}^{d}\backslash F|$ and $|\mathbb{T}^{d}\backslash H|,$ we have

(36) $\displaystyle\int_{F^{c}\cap H}$ $\displaystyle\leq
C^{\prime}(A)e^{-C_{\rho}K_{0}^{c}}$ (37) $\displaystyle\int_{F\cap H^{c}}$
$\displaystyle\leq C^{\prime}(A)e^{-C_{\rho}K_{0}^{c}}.$

Combining everything yields our result.

∎

Next, we show that $L(A)>0$ ensures that $|L_{N_{0}}-L_{2N_{0}}|$ is always
small enough.

######  Lemma 4.2.

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with an analytic extension to the strip $|\Im(z_{j})|<\rho.$
Suppose that $\omega\in\mathbb{T}^{d}$ is such that
$\left\|k\cdot\omega\right\|>\delta_{0}>0$ for all $k\in\mathbb{Z}^{d}$ with
$0<|k|\leq K_{0},$ where $K_{0}$ satisfies

(38) $\displaystyle K_{0}$ $\displaystyle>(\rho^{1+c}\kappa)^{-C}$ (39)
$\displaystyle N_{0}$ $\displaystyle>\kappa^{-C}\delta_{0}^{-1}K_{0}.$

Moreover, suppose that $N_{0}=a2^{b},$ for some $a\in\mathbb{N}$ and
$b>-C\ln\kappa$ (here $C$ is the same as above) and

(40) $L(A)>10^{3}\kappa.$

Then

(41) $|L_{N_{0}}-L_{2N_{0}}|<\frac{1}{100}L_{N_{0}}.$

In the above, $C$ is a sufficiently large absolute constant depending only on
$d$ which is defined in the proof and $c=C^{-1}.$

######  Remark 7.

The condition $N_{0}=a2^{b}$ is not, in fact, necessary, but allowing for more
general scales requires a technical approximation argument. All we will need
for applications is $N_{0}=a2^{b},$ so that is what we will prove. We refer
readers to [20] Section 5 for a proof for arbitrary $N_{0}$ large.

###### Proof.

First, as before, we remark that large deviation implies

(42) $L_{N_{0}}(x)>L_{N_{0}}-\kappa\geq L-\kappa>999\kappa$

away from a set, call it $G,$ of measure $|G|<e^{-C_{\rho}K_{0}^{c}}.$ Since
$L_{N_{0}}(x)\in L^{2},$ we have

(43) $\left|\int_{\mathbb{T}^{d}\cap
G}L_{N_{0}}(x)-L_{2N_{0}}(x)dx\right|<Ce^{-C_{\rho}K_{0}^{c}}.$

Now consider the set $F$ consisting of all $x\in\mathbb{T}^{d}\backslash G$
such that

(44) $|L_{N}(x)-L_{N}(x+j\omega)|<\kappa$

for all $N=2^{-s}N_{0},$ with $0\leq s\leq s_{0}:=-C_{1}\ln\kappa,$ and $j\leq
2N_{0}$ such that $2^{-s}N_{0}|j$ for some $j.$

Note that $K_{0}^{-c}<\rho^{1+c}\kappa,$ by our assumption on $K_{0}.$ Thus,
for all such $N,$

$\left\\{x:|L_{N}(x)-L_{N}|>\kappa\right\\}\subset\left\\{x:|L_{N}(x)-L_{N}|>\rho^{-1-c}K_{0}^{-c}\right\\}.$

Moreover,

(45) $\displaystyle 2^{-s}N_{0}$
$\displaystyle>2^{-s}\kappa^{-C}\delta_{0}K_{0}$ (46)
$\displaystyle>\kappa^{C_{1}-C}\delta_{0}K_{0}.$

Taking $C\geq C_{1}$ ensures that the right hand side is at least
$\delta_{0}K_{0}.$ All of this together implies that we may apply our large
deviation to control the measure of $\mathbb{T}^{d}\backslash F$ as follows.

Large deviation, and our condition on $K_{0}$ implies

(47) $\displaystyle\left|\mathbb{T}^{d}\backslash F\right|$
$\displaystyle<\sum_{s=0}^{s_{0}}\sum_{j}\left|\left\\{|L_{N}(x)-L_{N}(x+j\omega)|>\kappa\right\\}\right|$
(48)
$\displaystyle\leq\sum_{s}\frac{2N_{0}2^{s}}{N_{0}}e^{-C_{\rho}K_{0}^{c}}$
(49)
$\displaystyle\leq(2^{s_{0}}-1)\frac{2N_{0}}{N_{0}}e^{-C_{\rho}K_{0}^{c}}$
(50)
$\displaystyle\leq\kappa^{-C_{1}}\frac{2N_{0}}{N_{0}}e^{-C_{\rho}K_{0}^{c}}$
(51) $\displaystyle\leq
K_{0}^{C_{1}/C}\frac{2N_{0}}{N_{0}}e^{-C_{\rho}K_{0}^{c}}$ (52)
$\displaystyle\leq\frac{2N_{0}}{N_{0}}e^{-C_{\rho}K_{0}^{c}}.$

Thus we have

$\left|\mathbb{T}^{d}\backslash F\right|<e^{-C_{\rho}K_{0}^{c}}<\kappa.$

Throughout this computation, we needed to take $K_{0}$ sufficiently large,
depending only on $\rho,C,$ and $C_{1}.$ Throughout, $\rho$ is fixed, and we
will later fix $C$ and $C_{1}$ in the proof independent of all other
parameters, so this will not pose an obstacle.

Since $L_{N}(A,x)$ is uniformly $L^{2}$-bounded in $N,$

$\int_{\mathbb{T}^{d}\backslash
F}\left|L_{N_{0}}(x)-L_{2N_{0}}(x)\right|dx<C(A)e^{-C_{\rho}K_{0}^{c}}<C(A)\kappa.$

Let us now define

$J:=\left\\{x\in F:L_{2^{-s_{0}}N_{0}}(x)<\kappa^{-1}\right\\}.$

By Chebyschev’s inequality, $|\mathbb{T}^{d}\backslash J|<C(A)\kappa,$ and
thus

$\int_{\mathbb{T}^{d}\backslash
J}\left|L_{N_{0}}(x)-L_{2N_{0}}(x)\right|dx<C(A)\kappa.$

Now consider $x\in J.$ By definition, $L_{N_{0}}(x)>999\kappa.$ Define scales
$N_{0s}$ inductively by $N_{01}=2^{-1}N_{0}$ and $N_{0(s+1)}=2^{-1}N_{0s}.$
Since $x\in F,$ we have

(53) $\displaystyle 999\kappa N_{0}\leq N_{0}L_{N_{0}}(x)$
$\displaystyle\leq\sum_{j=0}^{N_{0}/N_{0s}}N_{0s}L_{N_{0s}}(x+jN_{0s}\omega)$
(54)
$\displaystyle\leq\sum_{j=0}^{N_{0}/N_{0s}}(N_{0s}L_{N_{0s}}(x)+N_{0s}\kappa)$
(55) $\displaystyle=N_{0}L_{N_{0s}}(x)+N_{0}\kappa.$

By a similar computation, we have, for $1\leq s<s^{\prime}\leq s_{0}$

(56) $L_{N_{0s}}(x)\leq L_{N_{0s^{\prime}}}(x)+\kappa.$

For each such $x,$ we define $s(x)$ and new length scales
$N_{00}(x)=N_{0s(x)}=2^{-s(x)}N_{0}$ such that

(57) $\displaystyle N_{00}(x)$ $\displaystyle\leq\kappa N_{0}$ (58)
$\displaystyle|L_{N_{00}(x)}(x)-L_{2N_{00}(x)}|$
$\displaystyle<\frac{1}{100}L_{N_{00}(x)}.$

The first condition is achievable as long as $C_{1}>1.$ The second condition
may be rewritten as

$\frac{99}{100}L_{N_{00}(x)}(x)<L_{2N_{00}(x)}<\frac{101}{100}L_{N_{00}(x)}(x).$

The right inequality holds by $L_{2N_{00}(x)}<L_{N_{00}(x)}+\kappa.$ The left
inequality holds by taking $C_{1}$ sufficiently large (say $C_{1}>5$) and
using $L_{2^{-s_{0}}N_{0}}(x)<\kappa^{-1}.$

The hypotheses of Theorem 2.2 are now satisfied for $N_{00}(x)$ with
$\delta=100\kappa,$ so we have

(59)
$\displaystyle\begin{split}\bigg{|}L_{N}(x)&+\frac{N_{00}}{N}\sum_{j=0}^{\frac{N}{N_{00}}-1}L_{N_{00}(x)}(x+jN_{00}\omega)\\\
&\quad-\frac{2N_{00}}{N}\sum_{j=0}^{\frac{N}{N_{00}}-1}L_{2N_{00}(x)}(x+jN_{00}\omega)\bigg{|}\\\
&<e^{-cN_{00}L_{N_{00}}(x)}+CL_{N_{00}}(A,x)\frac{N_{00}(x)}{N}\end{split}$

for $N=N_{0},2N_{0}.$ Since $x\in J,$ we have
$L_{N_{00}}(x+jN_{00}\omega)=L_{N_{00}}(x)+O(\kappa)$ and
$L_{N_{00}}(x)>998\kappa.$ Moreover,
$L_{N_{00}}(A,x)<L_{2^{-s_{0}}N_{0}}(A,x).$ Consequently,

$|L_{N_{0}}(x)-L_{2N_{0}}(x)|<10\kappa+C(A)\frac{N_{00}(x)}{N_{0}}+C(A)\frac{N_{00}(x)}{2N_{0}}.$

Since $N_{00}(x)<\kappa N_{0},$ we have

$|L_{N_{0}}(x)-L_{2N_{0}}(x)|<C(A)\kappa.$

Hence, for $x\in F,$ we must have

$|L_{N_{0}}(x)-L_{2N_{0}}(x)|<\frac{1}{200}L_{N_{0}}(x).$

It now follows that

(60)
$|L_{N_{0}}-L_{2N_{0}}|<\frac{1}{200}L_{N_{0}}+C(A)e^{-C_{\rho}K_{0}^{c}}<\frac{1}{100}L_{N_{0}}.$

∎

The previous two lemmas immediately imply the following.

######  Theorem 4.1.

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with a plurisubharmonic extension to the strip
$|\Im(z_{j})|<\rho$ such that $\det A(x)\not\equiv 0.$ Suppose that
$\omega\in\mathbb{T}^{d}$ is such that
$\left\|k\cdot\omega\right\|>\delta_{0}>0$ for all $k\in\mathbb{Z}^{d}$ with
$0<|k|\leq K_{0},$ where $K_{0}$ satisfies

(61) $\displaystyle K_{0}$ $\displaystyle>(\rho^{1+c}\kappa)^{-C}$ (62)
$\displaystyle N_{0}$ $\displaystyle>\kappa^{-C}\delta_{0}^{-1}K_{0}.$

Moreover, suppose that $N_{0}=a2^{b},$ for some $a\in\mathbb{N}$ and
$b>-C\ln\kappa$ (here $C$ is the same as above) and

(63) $\displaystyle L(A)$ $\displaystyle>10^{3}\kappa.$

Then

(64) $|L_{N}+L_{N_{0}}-2L_{N_{0}}|<C(A)e^{-C_{\rho}K_{0}^{c}}$

when $N_{0}|N,N=2^{j}N_{0}$ for some $j\geq 0,$ and

(65) $N<N_{0}e^{C_{\rho}K_{0}^{c}}.$

Elementary iteration allows us to extend this result from
$N<N_{0}e^{C_{\rho}K_{0}^{c}}$ to $N<\exp(\exp(C_{\rho}K_{0}^{c})).$ This
formulation will be used to establish weak-Hölder continuity in frequency.

######  Corollary 4.1.

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with a plurisubharmonic extension to the strip
$|\Im(z_{j})|<\rho.$ Suppose that $\omega\in\mathbb{T}^{d}$ is such that
$\left\|k\cdot\omega\right\|>\delta_{0}>0$ for all $k\in\mathbb{Z}^{d}$ with
$0<|k|\leq K_{0},$ where $K_{0}$ satisfies

(66) $\displaystyle K_{0}$ $\displaystyle>(\rho^{1+c}\kappa)^{-C}$ (67)
$\displaystyle N_{0}$ $\displaystyle>\kappa^{-C}\delta_{0}^{-1}K_{0}.$

Moreover, suppose that $N_{0}=a2^{b},$ for some $a\in\mathbb{N}$ and
$b>-C\ln\kappa$ (here $C$ is the same as above) and

(68) $\displaystyle L(A)$ $\displaystyle>10^{3}\kappa.$

Then

(69) $|L_{N}+L_{N_{0}}-2L_{N_{0}}|<C(A)e^{-\rho K_{0}^{c}}$

when $N_{0}|N,N=2^{j}N_{0}$ for some $j\geq 0,$ and

(70) $N<\exp(\exp(\frac{1}{2}\rho K_{0}^{c})).$

By imposing a Diophantine condition on $\omega,$ we are able to perform a more
delicate iteration scheme to extend this result to the limit.

######  Theorem 4.2.

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with a plurisubharmonic extension to the strip
$|\Im(z_{j})|<\rho.$ Suppose that $\omega\in\mathbb{T}^{d}$ is such that
$\left\|k\cdot\omega\right\|>\tau|k|^{-\sigma}>0$ for all $k\in\mathbb{Z}^{d}$
with $0<|k|.$ Then for every $N_{0}=a2^{b}>N^{\prime},$ we have, for
$\eta=\eta(d,\sigma),$

(71)
$|L+L_{N_{0}}-2L_{2N_{0}}|<C(A)\exp\left\\{-C_{\rho}N_{0}^{\eta}\right\\}.$

Here $C$ is the same sufficiently large absolute constant from Lemma 4.2 and
$c=C^{-1}$ above. Here $N^{\prime}$ is a constant defined in our proof such
that $N_{0}$ satisfies the divisibility criterion needed to appeal to Lemma
4.2.

###### Proof.

For simplicity, we will assume $\rho=1,$ though the argument and result hold
for general $\rho,$ we would just have to adjust $N_{s},K_{s},$ and
$\kappa_{s}$ accordingly. We will begin by applying Theorem 4.1. We will
define the appropriate parameters first. Fix $c,C,C_{1}$ as in Theorem 4.1,
and let $K=K(c,C,C_{1},\sigma)$ be large enough such that

(72) $K^{C_{1}/C}e^{-K^{c}}<e^{-C_{\rho}K^{c}}$

and

(73) $e^{C_{\rho}K^{c/2}}>2K^{1+\sigma/2}.$

The first condition ensures that every $K_{0}>K$ satisfies the largeness
condition we imposed during the proof of Lemma 4.2, while the second condition
will ensure that our base step can be iterated (see below for details).

Take any $N_{0}=2^{b}>\tau^{-1}K^{\sigma+2}=N^{\prime}.$ We now define the
rest of our base parameters. Define $K_{0},\kappa_{0},$ and $\delta_{0}$ via

(74) $\displaystyle N_{0}$ $\displaystyle=:\tau^{-1}K_{0}^{\sigma+2},$ (75)
$\displaystyle K\leq K_{0}$ $\displaystyle=:\kappa_{0}^{-C},$ (76)
$\displaystyle\delta_{0}$ $\displaystyle:=\tau K_{0}^{-\sigma}.$

Lemma 4.2 is applicable with these parameters, and we have

$\left|L_{N_{1}}+L_{N_{0}}-2L_{N_{0}}\right|<C(A)e^{-C_{\rho}K_{0}^{c}}$

for $N_{0}|N_{1},N_{1}=2^{j}N_{0}<N_{0}e^{C_{\rho}K_{0}^{c}}.$

Now define

(77) $\displaystyle\kappa_{1}$ $\displaystyle=\kappa_{0}^{2},$ (78)
$\displaystyle K_{1}=\kappa_{1}^{-C}$
$\displaystyle=\kappa_{0}^{-2C}=K_{0}^{2},$

and

(79) $\delta_{1}=\tau K_{1}^{-\sigma}.$

Let $N_{1}=N_{0}e^{\frac{1}{2}K_{0}^{c}},$ where this is meant as: $N_{1}$ is
the largest integer multiple of $N_{0}$ no greater than
$N_{0}e^{C_{\rho}K_{0}^{c}}.$ We clearly have
$N_{1}\geq\frac{1}{2}N_{0}e^{C_{\rho}K_{0}^{c}}.$ Moreover, by our choice of
$\kappa_{1},K_{1},$ and $\delta_{1},$ we have

(80) $\displaystyle N_{1}$
$\displaystyle\geq\frac{1}{2}\kappa_{0}^{-C(\sigma+2)}e^{C_{\rho}K_{1}^{c/2}}$
(81)
$\displaystyle=\frac{1}{2}\kappa_{1}^{-C}\kappa_{1}^{-C\sigma/2}e^{C_{\rho}K_{1}^{c/2}}$
(82)
$\displaystyle=\frac{1}{2}\tau^{-1/2}\kappa_{1}^{-C}\delta_{1}^{-1/2}e^{C_{\rho}K_{1}^{c/2}}$
(83) $\displaystyle\geq\kappa_{1}^{-C}\delta_{1}^{-1}K_{1},$

for $K_{1}$ sufficiently large. In fact, we just need $K_{0}$ sufficiently
large so that $e^{C_{\rho}K_{0}^{c/2}}>2K_{0}^{1+\sigma/2},$ which is
certainly satisfied by our initial choice of $K.$

Consequently, Theorem 4.1 is applicable with $\kappa_{1},K_{1},\delta_{1},$
and $N_{1}$ as above:

(84) $\displaystyle|L_{N_{2}}+L_{N_{1}}-2L_{2N_{1}}|$
$\displaystyle<C(A)e^{-C_{\rho}K_{1}^{c}}$ (85)
$\displaystyle=C(A)e^{-C_{\rho}K_{0}^{2c}},$

for $N_{1}|N_{2},N_{2}=2^{j}N_{1}<N_{1}e^{C_{\rho}K_{1}^{c}}.$

Now we define

(87) $\displaystyle\kappa_{2}$ $\displaystyle:=\kappa_{1}^{2}=\kappa_{0}^{4},$
(88) $\displaystyle K_{2}$ $\displaystyle:=\kappa_{2}^{C}=\kappa_{0}^{4C},$
(89) $\displaystyle\delta_{2}$ $\displaystyle:=\tau K_{2}^{-\sigma},$

and

(90)
$N_{2}:=N_{1}e^{\frac{1}{2}K_{1}^{c}}\geq\frac{1}{2}N_{1}e^{C_{\rho}K_{1}^{c}}.$

Clearly, by our choice of $\kappa_{2},K_{2},\delta_{2},$ and what we know of
$N_{1},$ we have

(91) $\displaystyle N_{2}$
$\displaystyle\geq\frac{1}{2}\kappa_{1}^{-C}\delta_{1}^{-1}K_{1}e^{C_{\rho}K_{1}^{c}}$
(92)
$\displaystyle=\frac{1}{2}\tau^{-1}\kappa_{2}^{-C/2}K_{2}^{\sigma/2+1/2}e^{C_{\rho}K_{2}^{c/2}}$
(93)
$\displaystyle=\frac{1}{2}\tau^{-1/2}\kappa_{2}^{-C/2}\delta_{2}^{-1/2}K_{2}^{1/2}e^{C_{\rho}K_{2}^{c/2}}$
(94) $\displaystyle\geq\kappa_{2}^{-C}\delta_{2}^{-1}K_{2}$

whenever $K_{2}$ satisfies $2K_{2}^{1+\sigma/2}\leq e^{C_{\rho}K_{2}^{c/2}}.$
It is easy to see that our original condition on $K$ ensures that $K_{2}$ is
large enough.

We may now apply Lemma 4.2 with $\kappa_{2},K_{2},\delta_{2},$ and $N_{2}$ as
above to obtain

(95)
$|L_{N_{3}}+L_{N_{2}}-2L_{2N_{2}}|<C(A)e^{-C_{\rho}K_{2}^{c}}<C(A)e^{-C_{\rho}K_{0}^{4c}}.$

Continuing in this way, we inductively define

(96) $\displaystyle\kappa_{s}$ $\displaystyle:=\kappa_{s-1}^{2},$ (97)
$\displaystyle K_{s}$ $\displaystyle:=\kappa_{s}^{-C},$ (98)
$\displaystyle\delta_{s}$ $\displaystyle:=\tau K_{s}^{-\sigma},$

and

(99)
$N_{s}:=N_{s-1}e^{\frac{1}{2}K_{s-1}^{c}}\geq\frac{1}{2}N_{s-1}e^{C_{\rho}K_{s-1}^{c}}.$

Moreover, $N_{s-1}\geq\kappa_{s-1}^{-C}\delta_{s-1}^{-1}K_{s-1},$ and
$e^{C_{\rho}K_{s-1}^{c/2}}>2\tau^{1/2}K_{s-1}^{1+C/2}.$ Following the same
argument as before, we thus have

(100) $\displaystyle N_{s}$
$\displaystyle\geq\frac{1}{2}\kappa_{s-1}^{-C}\delta_{s-1}^{-1}K_{s-1}e^{C_{\rho}K_{s-1}^{c}}$
(101)
$\displaystyle=\frac{1}{2}\tau^{-1}\kappa_{s}^{-C/2}\delta_{s-1}^{-1}K_{s-1}e^{C_{\rho}K_{s-1}^{c}}$
(102)
$\displaystyle=\frac{1}{2}\tau^{-1/2}\kappa_{s}^{-C/2}\delta_{s}^{-1/2}K_{s}^{1/2}e^{C_{\rho}K_{s}^{c/2}}$
(103) $\displaystyle\geq\kappa_{s}^{-C}\delta_{s}^{-1}K_{s}.$

It follows that

(104)
$|L_{N_{s}}+L_{N_{s-1}}-2L_{2N_{s-1}}|<C(A)e^{-C_{\rho}K_{0}^{2^{s-1}c}}.$

At this point, we note that, at each step, we could have used $2N_{s},$ rather
than $N_{s},$ and obtained identical inequalities. Combining this fact with
the established inequalities, we obtain

(105)
$\left|L_{N_{s}}+L_{N_{0}}-2L_{2N_{0}}\right|<C(A)\sum_{j=0}^{s-1}e^{-C_{\rho}K_{0}^{2^{j}c}}.$

Altogether, this yields

(106) $\displaystyle|L_{N_{s}}+L_{N_{0}}-2L_{2N_{0}}|$
$\displaystyle<\sum_{j=0}^{\infty}C(A)e^{-C_{\rho}K_{0}^{2^{j}c}}$ (107)
$\displaystyle\leq 2C(A)e^{-C_{\rho}K_{0}^{c}}.$

Taking $s\to\infty,$ we obtain

(108) $|L+L_{N_{0}}-2L_{2N_{0}}|\leq 2C(A)e^{-C_{\rho}K_{0}^{c}}.$

Now, using our initial choice of $K_{0},$ we finally obtain

(109) $|L+L_{N_{0}}-2L_{2N_{0}}|\leq C(A)e^{-C_{\rho}N_{0}^{c/(\sigma+2)}}$

as desired, with $\eta=c(d)/(\sigma+2).$

∎

##  5\. Weak-H older continuity of the Lyapunov exponent

We begin this section by combining Theorem 4.2 and Theorem 3.1 to obtain weak-
Hölder continuity in $A.$ Then we will show that weak-Hölder continuity in
$\omega$ follows from Theorem 4.2 and the continuity of the Lyapunov exponent
in $\omega.$

######  Theorem 5.1.

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with an analytic extension to the strip $|\Im(z_{j})|<\rho.$
Suppose that $\omega\in\mathbb{T}^{d}$ is such that
$\left\|k\cdot\omega\right\|>\tau|k|^{-\sigma}>0$ for all $k\in\mathbb{Z}^{d}$
with $0<|k|.$ Then there is $\delta=\delta(A)$ and
$\gamma=\gamma(d,\sigma)\leq 1,$ such that, for every
$\left\|A-B\right\|<\delta,$

(110)
$|L(A,\omega)-L(B,\omega)|<C(A)\exp\left(-c(A)\left(-\ln\left\|A-B\right\|\right)^{\gamma}\right).$

###### Proof.

Since this result involves a fixed frequency, we will omit it in our notation.
Consider $\delta(A)>0$ such that Theorem 3.1 applies for
$\left\|A-B\right\|<\delta.$ Then we have, for every $N=a2^{b}>N^{\prime},$
where $N^{\prime}$ is the same absolute constant from Theorem 4.2, and every
$\alpha>1,$

$\displaystyle|L(A)-L(B)|$ $\displaystyle\leq|L(A)+L_{N}(A)-2L_{2N}(A)|$
$\displaystyle\quad+|L_{N}(B)+L_{N}(A)|+2|L_{2N}(B)-L_{2N}(A)|$
$\displaystyle\quad+|L(B)+L_{N}(B)-2L_{2N}(B)|$ $\displaystyle\leq
C(A)e^{-C_{\rho}N^{\eta}}$
$\displaystyle\quad+C(A)e^{-2N^{\alpha-1}b(A)}+e^{N^{\alpha}}\left\|A-B\right\|$
$\displaystyle\quad+C(B)e^{-2N^{\alpha-1}b(A)}+e^{N^{\alpha}}\left\|A-B\right\|$
$\displaystyle\quad+C(B)e^{-C_{\rho}N^{\eta}}.$

Moreover, the constants $C(A),C(B)$ in the above depend on uniform
measurements of $A$ and $B.$ In particular, the $L^{2}$-norms and constants
from the Lojaciewicz inequality. Since $A$ and $B$ are close in norm, these
constants are close, and we can, in fact, control them all by $2C(A).$

At this point, we set $N=\left(-\ln\left\|A-B\right\|\right)^{\beta},$ for
some $\beta$ to be defined later. This yields

$\displaystyle|L(A)-L(B)|$ $\displaystyle\leq
C(A)(e^{-C_{\rho}(-\ln\left\|A-B\right\|)^{\beta\eta}}+e^{-2(-\ln\left\|A-B\right\|)^{\beta(\alpha-1)}b(A)})$
$\displaystyle\quad+e^{(-\ln\left\|A-B\right\|)^{\beta\alpha}}\left\|A-B\right\|.$

Now, if $\beta<1/\alpha,$ then the $\left\|A-B\right\|$ term above is bounded
by $\left\|A-B\right\|^{1/2},$ and is thus irrelevant, so we will take
$\beta<1/\alpha.$ We have

$\displaystyle|L(A)-L(B)|$ $\displaystyle\leq
C(A)e^{-C_{\rho}(-\ln\left\|A-B\right\|)^{\beta\eta}}$
$\displaystyle\quad+C(A)e^{-2(-\ln\left\|A-B\right\|)^{\beta(\alpha-1)}b(A)}.$

Finally, setting $\alpha=\eta+1,$ we obtain our desired result with
$\gamma=\beta\eta.$ ∎

Weak-Hölder continuity in the frequency, at Diophantine frequencies, requires
an additional estimate from our proof that the Lyapunov exponent is continuous
at the frequencies with rationally independent components. We recall that
below without proof.

######  Lemma 5.1 ([20] Lemma 7.1).

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with an analytic extension to the strip $|\Im(z_{j})|<\rho.$
Assume $\omega\in\mathbb{T}^{d}$ such that

$\left\|k\cdot\omega\right\|>\delta\quad\text{for
}k\in\mathbb{Z}^{d_{2}},0<|k|\leq K.$

Moreover, suppose $\rho>K^{-c}.$ Then

$|L_{N}-L|<K^{-c}$

for all $N>K^{2}/\delta.$ Here $c=c(d)<1$ is the same constant from before.

Now we may obtain weak-Hölder continuity in $\omega.$

######  Theorem 5.2.

Let $A(x)$ be an analytic quasi-periodic $M(2,\mathbb{C})$-cocycle on
$\mathbb{T}^{d}$ with a plurisubharmonic extension to the strip
$|\Im(z_{j})|<\rho.$ Suppose that $\omega\in\mathbb{T}^{d}$ is such that
$\left\|k\cdot\omega\right\|>\tau|k|^{-\sigma}>0$ for all $k\in\mathbb{Z}^{d}$
with $0<|k|.$ Then there is $\delta>0$ and $\gamma=\gamma(d,\sigma)\leq 1,$
such that, for every $\left\|\omega-\omega^{\prime}\right\|<\delta,$

(111)
$|L(A,\omega)-L(A,\omega^{\prime})|<C(A)\exp\left\\{-c(A)\left(-\ln\left\|\omega-\omega^{\prime}\right\|\right)^{\gamma}\right\\}.$

######  Remark 8.

We note that $\delta$ arises in the form
$\left\|\omega-\omega^{\prime}\right\|<e^{-cN_{0}^{\beta}}\sim\delta,$ and
thus $\delta$ depends only on $N_{0},$ which needs to be sufficiently large to
apply Lemma 4.2.

###### Proof.

Since, this result involves a fixed cocycle, we will omit it in our notation.
We have, for every $N>N_{0},$ where $N_{0}$ is an absolute constant, and every
$\alpha>1,$

$\displaystyle|L(\omega)-L(\omega^{\prime})|$
$\displaystyle\leq|L(\omega^{\prime})+L_{N}(\omega^{\prime})-2L_{2N}(\omega^{\prime})|$
$\displaystyle\quad+|L_{N}(\omega^{\prime})+L_{N}(\omega)|+2|L_{2N}(\omega^{\prime})-L_{2N}(\omega)|$
$\displaystyle\quad+|L(\omega)+L_{N}(\omega)-2L_{2N}(\omega)|.$

Now, since $\omega$ satisfies the Diophantine condition
$\left\|k\cdot\omega\right\|>\tau|k|^{-\sigma},$ we may appeal to Theorem 4.2
to control the last term. Moreover, the middle two terms may be controlled
using Theorem 3.1, and we obtain

$\displaystyle|L(\omega)-L(\omega^{\prime})|$
$\displaystyle\leq|L(\omega^{\prime})+L_{N}(\omega^{\prime})-2L_{2N}(\omega^{\prime})|$
$\displaystyle\quad+C(A)e^{-2N^{\alpha-1}b(A)}+e^{N^{\alpha}}N\left\|\omega-\omega^{\prime}\right\|$
$\displaystyle\quad+C(A)e^{-2N^{\alpha-1}b(A)}+e^{N^{\alpha}}N\left\|\omega-\omega^{\prime}\right\|$
$\displaystyle\quad+C(A)e^{-C_{\rho}N^{\eta}}.$

Since $\omega^{\prime}$ need not be Diophantine, we are unable to appeal to
Theorem 4.2 directly, however, we may use Corollary 4.1 to obtain

$|L_{N^{\prime}}(\omega^{\prime})+L_{N}(\omega^{\prime})-2L_{2N}(\omega^{\prime})|<e^{-c^{\prime}N^{\eta}}$

for all $N^{\prime}\leq\exp(\exp(C_{\rho}N^{\eta})).$

Now, if we assume $\left\|\omega-\omega^{\prime}\right\|<e^{-N^{1/\beta}},$
then we have

$\left\|k\cdot\omega^{\prime}\right\|>|k|^{-\sigma}$

for all $0<|k|\leq e^{N^{1/\beta}/(1+\sigma)},$ and thus, for
$N^{\prime}=\exp(\exp(C_{\rho}N^{\eta})),$ we may appeal to Lemma 5.1 to
obtain

$|L_{N^{\prime}}(\omega^{\prime})-L(\omega^{\prime})|<e^{-cN^{1/\beta}}.$

It follows that

$|L(\omega^{\prime})+L_{N}(\omega^{\prime})-2L_{2N}(\omega^{\prime})|<e^{-cN^{1/\beta}}+e^{-c^{\prime}N^{\eta}}.$

Now we note that the condition on $\left\|\omega-\omega^{\prime}\right\|$ is
equivalent to

$N<(-\ln\left\|\omega-\omega^{\prime}\right\|)^{\beta},$

which is analogous to setting
$N=(-\ln\left\|\omega-\omega^{\prime}\right\|)^{\beta}$ for some $\beta>0.$
Taking everything together, we are in precisely the setting from the proof of
Theorem 5.1, and we may proceed as we did there to conclude. ∎

Theorem 1.1 now follows by combining Theorems 5.1 and 5.2.

## Acknowledgement

We would like to thank W. Liu for useful comments on an earlier version of
this work and S. Jitomirskaya for fruitful discussions. This research was
partially supported by NSF DMS-2052572, DMS-2052899, DMS-2155211, and Simons
681675.

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