Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 51, pp. 1–14.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NON-PERTURBATIVE POSITIVITY AND WEAK HÖLDER

CONTINUITY OF LYAPUNOV EXPONENT OF ANALYTIC

QUASI-PERIODIC JACOBI COCYCLES DEFINED ON A HIGH

DIMENSION TORUS

KAI TAO

Abstract. When analytic quasi-periodic cocycles are defined on a high di-
mension torus, their Lyapunov exponents have perturbative positivity and

continuity. In this article, we study a class of analytic quasi-periodic Jacobi co-

cycles defined on a two dimension torus. We show that in the non-perturbative
large coupling regimes, the Lyapunov exponent is positive for any frequency

and weak Hölder continuous for the full-measured frequency.

1. Introduction

We consider the quasi-periodic Jacobi operator Hx,ω,λv,a in `2(Z),(
Hx,ω,λv,aφ

)
(n) = −a(x2 + (n+ 1)ω2)φ(n+ 1)− ā(x2 + nω2)φ(n− 1)

+ λv(x1 + nω1)φ(n), n ∈ Z,
(1.1)

where v : T → R is a real analytic function called the potential, a : T → C is
a complex analytic function and not identically zero, λ is a real positive constant
called the coupling number, x = (x1, x2) is the phase, and ω = (ω1, ω2) is the
frequency. Their characteristic equations Hx,ω,λv,aφ = Eφ can be expressed as(

φ(n+ 1)
φ(n)

)
= M(x+ nω,E, λv, a)

(
φ(n)

φ(n− 1)

)
,

where

M(x+ nω,E, λv, a) =
1

a(x2 + (n+ 1)ω2)

(
λv(x1 + nω1)− E −ā(x2 + nω2)
a(x2 + (n+ 1)ω2) 0

)
.

In this article, we always fix the analytic functions v and a, and suppress them
from symbols. Then, we have the following analytic quasi-periodic Jacobi cocycles
(Mλ,E , ω) ∈ Cω(T2,M2(C)) × R2 where M2(C) is the set of 2 × 2 matrices with
complex entries:

(Mλ,E , ω) : C2 × T2 → C2 × T2 with (v, x)→ (Mλ,E(x)v, x+ ω),

2010 Mathematics Subject Classification. 37C55, 37F10.
Key words and phrases. Analytic quasi-periodic Jacobi cocycles; high dimension torus;

non-perturbative; positive Lyapunov exponent; weak Hölder continuous.
c©2020 Texas State University.

Submitted January 4, 2020. Published May 26, 2020.

1



2 K. TAO EJDE-2020/51

where

Mλ,E(x) =
1

a(x2 + ω2)

(
λv(x1)− E −ā(x2)
a(x2 + ω2) 0

)
.

Because the complex function a has only finite zero points in the complex plane,
the matrix Mλ,E and the Jacobi cocycles make sense almost everywhere.

Let M(x,E, λ) := Mλ,E(x) and define

Mn(x,E, ω, λ) =

0∏
j=n−1

M(x+ jω,E, λ)

=

0∏
j=n−1

1

a(x2 + (j + 1)ω2)

(
λv(x1 + jω1)− E −ā(x2 + jω2)
a(x2 + (j + 1)ω2) 0

)
,

which is called the transfer matrix of (1.1). Set

Ln(E,ω, λ) :=
1

n

∫
T2

log ‖Mn(x,E, ω, λ)‖dx.

From the Kingman’s subadditive ergodic theorem, we have

L(E,ω, λ) := lim
n→∞

Ln(E,ω, λ) = lim
n→∞

1

n
log ‖Mn(x,E, ω, λ)‖

for almost every x ∈ T2, which is called the Lyapunov exponent of (1.1).
Note that L(E,ω, λ) is non-negative, as∫

T2

log |detM(x,E, λ)|dx ≡ 0.

In this article, we first show that the Lyapunov exponent is always positive when
the coupling number is large.

Theorem 1.1. For any κ > 0, there exists λ0 = λ0(v, a, κ) > 0 such that for any
ω, if |λ| > λ0 and E is in the spectrum of (1.1), then

(1− κ) log |λ| < L(E,ω, λ) < (1 + κ) log |λ|.

Because of the uniform hyperbolicity, the Lyapunov exponent is always positive
when E is in the resolvent set.

Secondly, we study the continuity of L(E,ω, λ) in the energy E. It is well known
that L(E,ω, λ) is a C∞ function of E on the resolvent set. So we only need to
consider E ∈ E , which contains the spectrum and will be defined in (2.1). What’s
more, we need to assume that ω1 and ω2 are both the Diophantine number (DN for
short). Here when we say that a irrational number ω ∈ (0, 1) is the DN, it means
that ω satisfies the Diophantine condition

‖nω‖ ≥ Cω
|n|α

for all n 6= 0. (1.2)

It is well known that for a fixed α > 1, almost every ω ∈ T satisfies (1.2). Thus,
the set of ω we assumed has full measure in T2. Then, we obtain the weak Hölder
continuity of L(E,ω, λ) in E.

Theorem 1.2. Let E ∈ E , both ω1 and ω2 be the DN, and |λ| > λ0 where λ0

comes from Theorem 1.1 with κ = 1
100 . Then L(E,ω, λ) is a continuous function

of E with modulus of continuity

h(t) = exp (−c| log t|τ ) ,



EJDE-2020/51 NON-PERTURBATIVE POSITIVITY AND CONTINUITY 3

where τ = τ(α) and c = c(λv, a) are positive constants.

Remark 1.3. Actually, the d-dimension Diophantine number (DN) is always de-
fined by

‖n · ω‖ := ‖n1ω1 + · · ·+ ndωd‖ ≥
c

(|n1|+ · · ·+ |nd|)A
for all (n1, . . . , nd) ∈ Zd\{0} and A > d, which is also almost everywhere in Td.
Obviously, this 2-dimension DN is a subset of our frequency.

The research on Lyapunov exponents has been a hot topic in several fields for
a long time. In 2001 Goldstein and Schlag [8] developed two powerful techniques,
the Large Deviation Theorem and the Avalanche Principle. These two techniques
are widely applied in the literatures, to study the Schrödinger operator(

Hs
x,ω,λφ

)
(n) = φ(n+ 1) + φ(n− 1) + λv(x+ nω)φ(n), n ∈ Z,

where the potential v is a real analytic function on Td. Obviously, it is a special
case of (1.1) with a ≡ 1, and Ms

n(x,E, ω, λ), Ls(E,ω, λ) and Lsn(E,ω, λ) have the
corresponding definitions. When d = 1 and ω = ω this is the Strong DN:

‖nω‖ ≥ Cω
|n|(1 + log |n|)α

for all n 6= 0,

which is also almost everywhere in T for α > 1. They obtained that Ls(E,ω, λ) is
Hölder continuous in E in the positive Lyapunov exponent regimes. When d ≥ 2
and ω is the d-dimension DN, they obtained the perturbative result that there
exists a λ̃s0 := λ̃s0(v,A, ω) such that for any |λ| > λ̃s0, Ls(E,ω, λ) is positive for all
E and weak Hölder continuous in E. Readers may have doubts when the Lyapunov
exponent is positive for d = 1. Actually, Sorets-Spencer [14] proved in 1991 that
for any nonconstant real analytic potential v, there exists λs0 = λs0(v) such that
for any |λ| > λs0, the Lyapunov exponent is positive for any ω. In 2002, Bourgain-
Jitomirskaya [6] proved the joint continuity of Ls(E,ω, λ) in (E,ω) at every (E,ω0)
if ω0 is irrational and Ls(E,ω0, λ) is positive. Then in 2005, Bourgain [3] extended
this continuity and the result of the positive Lyapunov exponent in [14] from T to
Td.

All above results depend on the fact that the determinants of the Schrödinger
transfer matrices are always 1. For the analytic quasi-periodic GL(2,C) cocycles

M(x) =

(
v11(x) v12(x)
v21(x) v22(x)

)
,

where vij (i, j = 1, 2) are analytic function on Td, Jitomirskaya-Koslover-Schulteis
[10] and Jitomirskaya-Marx [11] proved the weak Hölder continuity of the Lyapunov
exponent in vij over the analytic category for 1-dimension Diophantine frequency.
Avila-Jitomirskaya-Sadel showed the continuity for any 1-dimension frequency in
[1]. The author extended it to d ≥ 2 for d-dimension Diophantine frequency in [16].
He also studied the following general analytic quasi-periodic Jacobi operators(
H̃x,ω,λv,aφ

)
(n) = −a(x+ (n+ 1)ω)φ(n+ 1)− ā(x+nω)φ(n− 1) +λv(x+nω)φ(n)

for n ∈ Z, and proved in [17] that when d = 1 and ω = ω is the strong DN, the
continuity of the Lyapunov exponent in E can be Hölder.

In summary, the Lyapunov exponent of the SL(2,C) cocycles is always positive
for any ω and any d in the large coupling regimes. But when the cocycles become
GL(2,C), we have the same result only for d = 1. Therefore, the first highlight of



4 K. TAO EJDE-2020/51

our paper is that it is the first conclusion of the positive Lyapunov exponents of a
class of GL(2,C) cocycles defined on T2 for any frequency. Secondly, we prove the
weak Hölder continuity for the more generic full-measured frequency (see Remark
1.3). Furthermore, both results are non-perturbative.

We organize this article as follows. In Section 2, we develop Bourgain-Goldstein’s
method, which was applied to the quasi-periodic Schrödinger equations in [4], to
prove Theorem 1.1. With its help, we obtain the large deviation theorem and
Theorem 1.2 in Section 3.

2. Positive Lyapunov exponent

It is well known that if v is real analytic function on T, then there exists some
ρv > 0 such that

v(x) =
∑
k∈Z

v̂(k)e2πikx, with |v̂(k)| . e−ρv|k|.

So, it has a holomorphic extension

v(z) =
∑
k∈Z

v̂(k)e2πikz

on the strip |=z| < ρv
10 , satisfying

|v(z)| ≤
∑
k∈Z
|v̂(k)|e2π|k||=zv| <

∑
k∈Z

e−ρv|k|eρv|k|
π
10 < Cv.

Easy computations show that the spectrum of our operators must be in the interval

E := [−2 max
x∈T
|a| − |λ|Cv, 2 max

x∈T
|a|+ |λ|Cv]. (2.1)

In the rest paper, we always fix the frequency ω and suppress it for ease from
now on. Define the analytic transfer matrix

Ma
n(x,E, λ) :=

0∏
j=n−1

Ma(x+ jω,E, λ),

where

Ma(x,E, λ) := a(x2 + ω2)M(x,E, λ) =

(
λv(x1)− E −ā(x2)
a(x2 + ω2) 0

)
.

Then for fixed λ,E and x2, the function

uan(·, x2, E, λ) =
1

n
log ‖Ma

n(x,E, λ)‖

has a subharmonic extension uan(z, x2, E, λ) (uan(z) for short) on |=z| < ρv
10 , which

is bounded by log (4 maxx∈T |a|+ 2|λ|Cv) for any E ∈ E . If we choose

Cmax = 4 max
x∈T
|a|+ 2Cv,

then for any x1, x2, ω and E ∈ E , it holds, for any |λ| ≥ 1,

uan(x1) ≤ logCmax|λ|. (2.2)

Set

Lan(E, λ) :=
1

n

∫
T2

log ‖Ma
n(x,E, λ)‖dx,

La(E, λ) := lim
n→∞

Lan(E, λ),



EJDE-2020/51 NON-PERTURBATIVE POSITIVITY AND CONTINUITY 5

which also exists by the Kingman’s subharmonic ergodic theorem. It is straightfor-
ward to check that

log ‖Ma
n(x,E, λ)‖ = log ‖Mn(x,E, λ)‖+

n∑
j=1

log |a(x2 + (j + 1)ω2)|,

Lan(E, λ) = Ln(E, λ) +D,

La(E, λ) = L(E, λ) +D,

where

D :=

∫
T

log |a(x)| dx =

∫
T

log |ā(x)| dx, (2.3)

which exists by the analyticity of a. Obviously, to obtain Theorem 1.1, we only
need to prove that for any |λ| > λ0(v, a, κ),(

1− κ

2

)
log |λ| < La(λ,E) <

(
1 +

κ

2

)
log |λ|.

Actually, the second inequality is trivial by (2.2) with large |λ|.
Now, we start the proof of the first inequality. First, we recall the following

lemmas from [4] and [18].

Lemma 2.1 ([4, Lemma 14.5]). For every 0 < δ < ρ, there is an ε such that

inf
E1

sup
δ/2<y<δ

inf
x∈[0,1]

|v(x+ iy)− E1| > ε.

Lemma 2.2 ([18, Corollary 2]). Let u : Ω → [−∞,+∞) be an upper semicontin-
uous function. Then u(z) is a subharmonic function on Ω, if and only if for any
Jordan subdomain Ω′ satisfying Ω′ ⊂ Ω and any z ∈ Ω′, it satisfies

u(z) ≤
∫
∂Ω′

u(ζ)dµζ(z, ∂Ω′,Ω′),

where µ(z, ∂Ω′,Ω′) is the harmonic measure of ∂Ω′ at z ∈ Ω′.

Remark 2.3. Here we emphasize that this harmonic measure depends only on the
region Ω′ and the point z, not on the subharmonic function u(z). It is the key of
our method applied in this section.

Without loss of generality, we assume λ > 0. Fix 0 < δ � ρ and ε satisfying
Lemma 2.1. Define

λ̃0 = 200Cmaxε
−100/κ > 0

and let λ > λ̃0 > 0. Then, for any fixed E, there is δ/2 < y1 < δ such that

inf
x1∈[0,1]

∣∣v(x1 + iy1)− E

λ

∣∣ > ε.

Therefore,

inf
x1∈T

|λv(x1 + iy1)− E| > λε > 200Cmaxε
− 100

κ +1 > 200Cmax. (2.4)

For n ≥ 1 we define

Ma
n−1(iy1, x2, E, λ)

(
1
0

)
=

(
wn−1

1

wn−1
2

)
. (2.5)



6 K. TAO EJDE-2020/51

Then (
wn1
wn2

)
=

(
λv(iy1 + nω1)− E −ā(x2 + nω2)
a(x2 + (n+ 1)ω2) 0

)(
wn−1

1

wn−1
2

)
=

((
λv(iy1 + nω1)− E

)
wn−1

1 − ā(x2 + nω2)wn−1
2

a
(
x2 + (n+ 1)ω2

)
wn−1

1

)
.

(2.6)

Now we use induction to show that for any n ≥ 1,

|wn1 | ≥ |wn2 |, and |wn1 | ≥ (λε− 2Cmax)|wn−1
1 | ≥ (λε− 2Cmax)n. (2.7)

From definition (2.5), we have w0
1 = 1 and w0

2 = 0. Then

|w1
1| = |λv(iy1 + ω1)− E| > λε > 200Cmax, and |w1

2| < Cmax < |w1
1|,

which satisfy (2.7) for n = 1. Let n = t with

|wt1| ≥ |wt2|, and |wt1| > (λε− 2Cmax)|wt−1
1 | > (λε− 2Cmax)t. (2.8)

From (2.6) and (2.8), we have

|wt+1
1 | ≥ (λε− 2Cmax)wt1 > (λε− 2Cmax)t+1,

|wt+1
2 | ≤ 2Cmax|wt1| < 198Cmax|wt1| ≤ (λε− 2Cmax)|wt1| ≤ |wt+1

1 |,

which also satisfy (2.7) for n = t+ 1. Thus, (2.7) holds for any n ≥ 1. Then

‖Ma
n(iy1, x2, E, λ)‖ > (λε− 2Cmax)n and uan(iy1) > log(λε− 2Cmax).

We denote by H = {z : =z > 0} and Hρ = {z = x + iy : 0 < y < ρ
2} strips of the

complex plane. We denote by µ(z, E ,H) the harmonic measure of E at z ∈ H and
µs(iy1, Es,Hρ) the harmonic measure of Es at iy1 ∈ Hρ, where E ⊂ ∂H = R and
Es ⊂ ∂Hρ = R ∪ [y = ρ

2 ]. Note that ψ(z) = exp
(

2π
ρ z
)

is a conformal map from Hρ
onto H. From [7], we have

µs(iy1, Es,Hρ) ≡ µ(ψ(iy1), ψ(Es),H),

µ(z = x+ iy, E ,H) =

∫
E

y

(t− x)2 + y2

dt

π
.

Easy computations show that

µs[y =
ρ

10
] =

10πy1

πρ
<

10δ

ρ
and

dµs(x)

dx

∣∣
y=0

<
y1

x2 + y2
1

.

So, the subharmonicity and Lemma 2.2 yield

log(λε− 2Cmax) < uan(iy1) ≤
∫

[y1=0]∪[y1= ρ
10 ]

uan(z1)µs(dz1)

=

∫
y1=0

uan(x1)µs(dx1) +

∫
y1= ρ

10

uan(x1 + iy1)µs(dx1)

≤
∫
y1=0

uan(x1)µs(dx1) +
10δ

ρ

[
sup
y1= ρ

10

uan(x1 + iy1)
]

≤
∫
y1=0

uan(x1)µs(dx1) +
10(1 + κ)δ

ρ
log λ.



EJDE-2020/51 NON-PERTURBATIVE POSITIVITY AND CONTINUITY 7

Hence, by the definition of λ̃0 and δ � ρ, we have∫
R
uan(x1)µs(dx1) ≥ log(λε− 2Cmax)− 10(1 + κ)δ

ρ
log λ

≥
(
1− 10(1 + κ)δ

ρ

)
log λ+ log ε

>
(
1− κ

2

)
log λ.

(2.9)

Set
(uan)h(x1) = uan(x1 + h), h ∈ T.

Then, from Remark 2.3, and (2.4), it is easy to see that (2.9) also holds for
(uan)h(x1). So, for any h ∈ T, we have∫

R
uan(x1 + h)µs(dx1) >

(
1− κ

2

)
log λ. (2.10)

Define

Lan(x2, E, λ) :=

∫
T

1

n
log ‖Ma

n(x1, x2, E, λ)‖dx1.

Using (2.10) and integrating for h ∈ T, we obtain

Lan(x2, E, λ) =

∫ 1

0

uan(x1 + h)dh

≥
(∫

R
µs(dx1)

)(∫ 1

0

uan(x1 + h)dh
)

=

∫ 1

0

∫
R
uan(x1 + h)µs(dx1)dh

>
(
1− κ

2

)
log λ, ∀n ≥ 0.

(2.11)

Therefore,

Lan(E, λ) =

∫
T
Lan(x2, E, λ)dx2 >

(
1− κ

2

)
log λ, ∀n ≥ 0, (2.12)

which completes the proof as n→ +∞.

3. Large deviation theorems

As mentioned in the introduction, Goldstein and Schlag [8] introduced the large
deviation theorem and the avalanche principle. These two methods are standard
tools to study the continuity of the Lyapunov exponent. The avalanche principle
read as follows.

Proposition 3.1 ([8, Proposition 2.2]). Let A1, . . . , An be a sequence of 2 × 2-
matrices whose determinants satisfy

max
1≤j≤n

|detAj | ≤ 1. (3.1)

Suppose that

min
1≤j≤n

‖Aj‖ ≥ γ > n, (3.2)

max
1≤j<n

[log ‖Aj+1‖+ log ‖Aj‖ − log ‖Aj+1Aj‖] <
1

2
log γ. (3.3)



8 K. TAO EJDE-2020/51

Then ∣∣∣log ‖An · · ·A1‖+

n−1∑
j=2

log ‖Aj‖ −
n−1∑
j=1

log ‖Aj+1Aj‖
∣∣∣ < C

n

γ
(3.4)

where C =
∑∞
n=1 4n/n!.

Because of assumption (3.1), the key in the references mentioned in Section 1 is to
obtain a suitable large deviation theorem for some SL(2,C) matrices related to the
cocycles studied. In this paper, we also prove the corresponding statement, Lemma
3.2. Then, the other part of the proof of the weak Hölder continuity, including how
to apply the large deviation theorem and the avalanche principle, can be found in
[10, 16].

To state our large deviation theorem for the SL(2,C) matrices, we define the
unimodular matrices

Mu
n (x,E, λ) :=

Mn(x,E, λ)

|detMn(x,E, λ)|1/2
=

Ma
n(x,E, λ)

|detMa
n(x,E, λ)|1/2

.

Because of the analyticity, this definition makes sense almost everywhere. It is
straightforward to check that

detMa
n(x,E, λ) =

n−1∏
j=0

ā(x2 + jω2)a(x2 + (j + 1)ω2),

and
log ‖Mu

n (x,E, λ)‖

= log ‖Ma
n(x,E, λ)‖ − 1

2

n−1∑
j=0

log |ā(x2 + jω2)a(x2 + (j + 1)ω2)|.
(3.5)

Our desired large deviation theorem reads as follows.

Lemma 3.2 (Large Deviation Theorem). Let E ∈ E , both ω1 and ω2 be the DN,
and |λ| > λ0 where λ0 comes from Theorem 1.1 with κ = 1

100 . Then there exists an
n0 = n0(λv, a, ω) such that for any n ≥ n0,

meas
{
x ∈ T2 : | 1

n
log ‖Mu

n (x,E, λ)‖ − 〈 1
n

log ‖Mu
n (·, E, λ)‖〉| > 1

10
log λ

}
≤ C exp(−c log λn

σ
10 ).

where

C =

∞∑
n=1

4n

n!
, c = 2−2 log 2

which are called the absolute constants, and σ = σ(α) is positive.

To prove this lemma, we use the subharmonicity, which comes from the ana-
lyticity of v and a, and is the most important hypothesis in the following four
lemmas.

Lemma 3.3 ([9, Lemma 2.1]). Let u : Ω → R be a subharmonic function on a
domain Ω ⊂ C. Suppose that ∂Ω consists of finitely many piece-wise C1 curves.
Then there exists a positive measure µ on Ω such that for any Ω1 b Ω (i.e., Ω1 is
a compactly contained subregion of Ω)

u(z) =

∫
Ω1

log |z − ζ|dµ(ζ) + h(z),



EJDE-2020/51 NON-PERTURBATIVE POSITIVITY AND CONTINUITY 9

where h is harmonic on Ω1 and µ is unique with property. Moreover, µ and h
satisfy

µ(Ω1) ≤ C(Ω,Ω1)(sup
Ω
u− sup

Ω1

u),

‖h− sup
Ω1

u‖L∞(Ω2) ≤ C(Ω,Ω1,Ω2)(sup
Ω
u− sup

Ω1

u),

for any Ω2 b Ω1.

Lemma 3.4 ([2, Corollary 4.7]). Let u be a subharmonic function defined in the
annulus Aρ = {z : |=z| < ρ}. Suppose furthermore that u(x) =

∫
log |x− ζ|dµ(ζ) +

h(x) with ‖µ‖+ ‖h‖L∞ ≤ Č. Then, the fourier coefficient of u satisfies

|û(k)| . Č

|k|
.

Lemma 3.5 ([5, Lemma 2.3]). Suppose u is subharmonic on Aρ with supAρ
|u| ≤ n.

Furthermore, assume that u = u0 + u1, where

‖u0 − 〈u0〉‖L∞(T) ≤ ε0 and ‖u1‖L1(T) ≤ ε1.
Then for some constant Cρ depending only on ρ,

‖u‖BMO(T) ≤ Cρ
(
ε0 log(

n

ε1
) +
√
nε1

)
.

Remark 3.6. BMO(T) is the space of functions of bounded mean oscillation on
T, see [13]. Identifying functions that differ only by an additive constant, the norm
on BMO(T) is

‖f‖BMO := sup
I⊂T

1

|I|

∫
I

|f − 〈f〉I |dx,

where

〈f〉I =
1

|I|

∫
I

f(x)dx.

Lemma 3.7 ([17, Theorem 2.7]). Let u be a subharmonic function defined in the
annulus Aρ. Suppose furthermore that u(x) =

∫
log |x− ζ|dµ(ζ) + h(x) with ‖µ‖+

‖h‖L∞ ≤ Č. Then for any DN ω, we have

meas
{
x : |

n∑
j=1

u(x+ jω)− n〈u(·)〉| > δn
}
< exp(−cδn), (3.6)

where c = c(Č, ω).

Remark 3.8. It is obvious that the subharmonic function log |ā(z)a(z+ω2)| has a
upper bound Ca on the annulus Aa = {z : |=z| ≤ ρa}, and then Lemma 3.7 holds
because

meas
{
x : |

n∑
j=1

log |ā(x2 + jω2)a(x2(j + 1)ω2)| − 2nD| > δn
}
< exp(−cδn), (3.7)

where D is defined by (2.3). Combining this with (3.5), we obtain that the following
large deviation theorem for uan, which is the sufficient condition for Lemma 3.2:

meas
{
x ∈ T2 : | 1

n
log ‖Ma

n(x,E, λ)‖ − 〈 1
n

log ‖Ma
n(·, E, λ)‖〉| > 1

20
log λ

}
≤ C exp(−c log λn

σ
10 ).

(3.8)



10 K. TAO EJDE-2020/51

Now, we start the proof of (3.8). Fixing x2, E ∈ E and λ > λ0 with κ = 1
100 ,

we expand uan into its Fourier series of x1 and denote the Fourier coefficient as
ûan(k, x2, E, λ), i.e.,

uan(x,E, λ) =
∑
k∈Z

ûan(k, x2, E, λ)e2πikx1 ,

ûan(k, x2, E, λ) =

∫
x1∈T

uan(x1, x2, E, λ)e−2πikx1dx1.

Combining Lemmas 3.3 and 3.4, we obtain that there exists a C ′max such that

sup
x2∈T

|ûan(k, x2)| ≤ C ′max

|k|
, ∀k 6= 0. (3.9)

Here we suppress the fixed λ > λ0 and E ∈ E from symbols for ease, if there is no
doubt. Note that

uan(x1 + jω1, x2 + jω2) = 〈uan(·, x2 + jω2)〉+
∑
k 6=0

ûan (k, x2 + jω2) e2πik(x1+jω1).

Then

1

N

∣∣ N∑
j=1

[uan(x1 + jω1, x2 + jω2)− 〈uan(·, x2 + jω2)〉]
∣∣

=
1

N

∣∣ N∑
j=1

∑
k∈Z\{0}

ûan(k, x2 + jω2)e2πik(x1+jω1)
∣∣

≤ 1

N

∣∣ N∑
j=1

∑
0<|k|<K

ûan(k, x2 + jω2)e2πik(x1+jω1)
∣∣

+
1

N

∣∣ N∑
j=1

∑
|k|>K

ûan(k, x2 + jω2)e2πik(x1+jω1)
∣∣ := (a) + (b)

From (3.9), we have

‖(b)‖22 ≤
∑
|k|>K

sup
j
|û(k, x2 + jω)|2 ≤ (C ′max)2K−1.

On the other hand, by the Cauchy inequality,

|(a)|2 ≤ N−2
∣∣∣ N∑
j=1

∑
0<|k|<K

ûan(k, x2 + jω2)e2πik(x1+jω1)
∣∣∣2

≤ N−2
( N∑
j=1

∑
0<|k|<K

|ûan(k, x2 + jω2)|2
)∣∣∣ N∑

j=1

∑
0<|k|<K

e4πik(x1+jω1)
∣∣∣

≤ N−2
(
N sup

j

∑
0<|k|<K

|ûan(k, x2 + jω2)|2
)∣∣∣ ∑

0<|k|<K

N∑
j=1

e4πik(x1+jω1)
∣∣∣

≤ N−1(C ′max)2
∣∣∣ ∑

0<|k|<K

N∑
j=1

e4πik(x1+jω1)
∣∣∣.



EJDE-2020/51 NON-PERTURBATIVE POSITIVITY AND CONTINUITY 11

Easy computations show that∣∣ N∑
j=1

e2πijkω
∣∣ =

∣∣exp(2πikω) · (1− exp(2πiNkω))

1− exp(2πikω)

∣∣ ≤ 1

2‖kω‖
.

Combining this with (1.2), we have

|(a)|2 ≤ N−1 (C ′max)
2
∑

0<|k|<K

1

2‖2kω‖
≤ CK

α+1

N
.

Set K = Nσ where σ = 1/(2(α + 1)). Then |(a)|2 ≤ CN−
1
2 and ‖(b)‖22 ≤

(C ′max)
2
N−σ. For any fixed x2 ∈ T, we define

|u(x1)| =
N∑
j=1

[uan (x1 + jω1, x2 + jω2)− 〈uan(·, x2 + jω2)〉].

Then
meas{x1 ∈ T : |u(x1)| > N1−σ3 } < N−

σ
3 . (3.10)

We define B as the exceptional set for (3.10). Let u(x1) = u0(x1) + u1(x1) where
u0(x1) = 0 on B and u1(x1) = 0 on T\B. Thus

‖u0(x1)‖L∞(T) ≤ N1−σ3 and ‖u1(x1)‖L1(T) ≤ N1−σ3 .

From Lemma 3.5, we have

‖u‖BMO(T) ≤ CρN1−σ7 .

Recall the John-Nirenberg inequality ([13]),

meas{x ∈ T : |u(x)− < u > | > γ} ≤ C exp
(
− cγ

‖u‖BMO

)
,

with the absolute constants C
∑∞
n=1

4n

n! . c = 2−2 log 2. Let γ = 1
100N log λ.

Lemma 3.9. There exists an N0 := N0(λv, a) such that for any N > N0, E ∈ E ,
x2 ∈ T and DN ω1, it holds

meas
{
x1 ∈ T :

1

N
|
N∑
j=1

[uan(x1 + jω1, x2 + jω2)− 〈uan(·, x2 + jω2)〉]| > 1

100
log λ

}
≤ C exp(−cN σ

7 log λ).

Obviously, comparing this with our desired (3.8), we need to obtain the following
lemma, which studies the deviation between uan and its shifts.

Lemma 3.10. There exists a constant C̃2 := C̃2(λv, a) such for any C2 ≤ C̃2,
δ > 1 and N = C2δn, it holds

sup
x1∈T

meas
{
x2 ∈ T :

1

N
|
N∑
j=1

[uan(x+ jω)− uan(x)]| > δ
}
≤ 2N exp

(
− cδn

4

)
. (3.11)

Proof. Since detMa(x) = a(x2 + ω2)ā(x2), it follows that

(Ma)−1(x) =
1

a(x2 + ω2)ā(x2)

(
0 ā(x2)

−a(x2 + ω2) λv(x1 + ω1)− E

)
,

sup
x1∈T

‖(Ma)−1(x)‖ ≤ Cmax

|a(x2 + ω2)ā(x2)|
.



12 K. TAO EJDE-2020/51

So,

‖Ma
n(x+ ω)‖ ≤ ‖Ma(x+ nω)‖ ‖Ma

n(x)‖ ‖(Ma)−1(x)‖

≤ Cmax‖Ma
n(x)‖ Cmax

|ā(x2)a(x2 + ω2)|
,

and

‖Ma
n(x)‖ ≤ Cmax‖Ma

n(x+ ω)‖ Cmax

|ā(x2 + (n− 1)ω2)a(x2 + nω2)|
.

Therefore,

−C1 + log |ā(x2)a(x2 + ω2)| ≤ log ‖Ma
n(x)‖ − log ‖Ma

n(x+ ω)‖
≤ C1 − log |ā(x2 + (n− 1)ω2)a(x2 + nω2)|,

where C1 = 2 logCmax. Similarly,

− kC1

n
+

k−1∑
j=0

1

n
log |a(x2 + (j + 1)ω2)ā(x2 + jω2)|

≤ uan(x)− uan(x+ kω)

≤ kC1

n
−
k−1∑
j=0

1

n
log |ā(x2 + (n+ j − 1)ω2)a(x2 + (n+ j)ω2)|.

(3.12)

Let

Y −k =
{
x2 ∈ T : −kC1

n
+

k−1∑
j=0

1

n
log |a(x2 + (j + 1)ω2)ā(x2 + jω2)| < −δ

}
and N = C2δn where C1C2 ≤ 1

2 . Then, for any 1 ≤ k ≤ N ,

Y −k ⊂ Y −k
′

:=
{
x2 ∈ T :

k−1∑
j=0

log |a(x2 + (j + 1)ω2)ā(x2 + jω2)| < −δn
2

= − N

2C2

}
.

For D > 0,

Y −k
′ ⊂ Y −k

′′
:=
{
x2 ∈ T :

k−1∑
j=0

log |a(x2 + (j + 1)ω2)ā(x2 + jω2)| − 2kD < −δn
2

}
.

From (3.7), we have

meas Y −k ≤ meas Y −k
′′

≤ meas
{
x2 ∈ T : |

k−1∑
j=0

log |a(x2 + jω2)a(x2 + (j + 1)ω2)| − 2kD| > δn

2

}
≤ exp

(
− cδn

2k
· k
)

= exp
(
− cδn

2

)
.

For D < 0, let 8C2|D| < 1, to make 1
8C2

+D > 0. This implies that N
4C2

+ 2kD > 0
for 1 ≤ k ≤ N and

Y −k
′ ⊂ Y −k

′′′



EJDE-2020/51 NON-PERTURBATIVE POSITIVITY AND CONTINUITY 13

:=
{
x2 ∈ T :

k−1∑
j=0

log |a(x2 + (j + 1)ω2)ā(x2 + jω2)| − 2kD < − N

4C2
= −δn

4

}
.

From (3.7) again, it follows that meas Y −k ≤ exp(− cδn4 ) . Above all, there exists a

constant C̃2 := C̃2(λv, a) such for any C2 < C̃2 and 1 ≤ k ≤ N = C2δn,

meas Y −k ≤ exp
(
− cδn

4

)
. (3.13)

Similar calculations show that for the set

Y +
k :=

{
x2 ∈ T :

kC1

n
−
k−1∑
j=0

1

n
log |a(x2 + (n+ j − 1)ω2)a(x2 + (n+ j)ω2)| > δ

}
we have

meas Y +
k < exp

(
− cδn

4

)
.

Combining this with (3.12) and (3.13), we have that for any 1 ≤ k ≤ N ,

meas
{
x2 ∈ T : |uan(x+ kω)− uan(x)| > δ

}
≤ 2 exp

(
− cδn

4

)
.

Then, this lemma is obtained by the drawer principle:{
x2 ∈ T :

1

N
|
N∑
j=1

[uan(x+ jω)− uan(x)]| > δ
}

⊂ ∪Nj=1{x2 ∈ T : |uan(x+ jω)− uan(x)| > δ}. �

Remark 3.11. Obviously, we also obtain the deviation between the integrations
of x1 for uan and its shifts: there exists a constant C̃2 := C̃2(λv, a) such for any

C2 < C̃2, δ > 1 and N = C2δn,

meas
{
x2 ∈ T :

1

N
|
N∑
j=1

[〈uan(·, x2 + jω2)〉 − 〈uan(·, x2)〉]| > δ
}
≤ 2N exp

(
− cδn

4

)
.

Now, combining Lemmas 3.9 and 3.10, and Remark 3.11, there exists an N0 :=
N0(λv, a) such that for any N = C̃2

log λ
100 n > N0, E ∈ E and DN ω1, we have

meas
{
x ∈ T2 : |uan(x)− 〈uan(·, x2)〉| > 1

25
log λ

}
≤ C exp(−cN σ

7 log λ) + 4N exp
(
− c

4

log λ

100
n
)

< C exp(−cN σ
10 log λ).

(3.14)

At last, we need to exchange 〈uan(·, x2)〉 by 〈uan(·)〉. It comes from Section Two. By
(2.2), (2.11) and (2.12), for any λ > λ0

(
v, a, 1

100

)
, we have

199

200
log λ ≤ 〈uan(·, x2)〉 ≤ 210

200
log λ,

199

200
log λ ≤ 〈uan(·)〉 ≤ 210

200
log λ.

Therefore,

|〈uan(·, x2)〉 − 〈uan(·)〉| ≤ 1

100
log λ,

and then we obtain (3.8) by combining this with (3.14).



14 K. TAO EJDE-2020/51

Acknowledgments. This research was supported by the China Postdoctoral Sci-
ence Foundation (Grant 2019M650094).

References

[1] A. Avila, S. Jitomirskaya, C. Sadel; Complex one-frequency cocycles. J. Eur. Math. Soc., 16,
(2014), 1915–1935.

[2] J. Bourgain; Green ’s Function Estimates for Lattice Schrödinger Operators and Applica-

tions. Ann. Math. Stud., Princeton, NJ: Princeton University Press, 158, (2004),
[3] J. Bourgain; Positivity and Continuity of the Lyapunov exponent for Shifts on Td with

arbitrary frequency vector and real analytic potential. Journal D’analyse Mathématique,
96(2005), 313-355.

[4] J. Bourgain, M. Goldstein; On nonperturbative localization with quasi-periodic potential.

Ann. of Math., (2) 152, (2000), no. 3, 835-879.
[5] J. Bourgain, M. Goldstein, W. Schlag; Anderson localization for Schrödinger operators on Z2

with potentials given by the skew-shift. Comm. Math. Phys., 220 (2001), no. 3, 583-621.

[6] J. Bourgain, S. Jitomirskaya; Continuity of the Lyapunov exponent for quasiperiodic opera-
tors with analytic potential, J. Stat. Phys., 108(2002), 1028-1218.

[7] J. Garnett, D. Marshall; Harmonic Measure. Cambridge University Press, 2005.

[8] M. Goldstein, W. Schlag; Hölder continuity of the integrated density of states for quasi-
periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of

Math., 154, (2001), 155-203 .

[9] M. Goldstein, W. Schlag; Fine properties of the integrated density of states and a quantitative
separation property of the Dirichlet eigenvalues. Geom. Funct. Analysis, 18, (2008), 755-869.

[10] S. Jitomirskaya, D. A. Koslover, M. S. Schulteis; Continuity of the Lyapunov exponent for
analytic quasiperiodic cocycles. Ergodic Theory Dynam. Systems, 29 (2009),1881-1905.

[11] S. Jitomirskaya, C. A. Marx; Continuity of the Lyapunov Exponent for analytic quasi-perodic

cocycles with singularities. Journal of Fixed Point Theory and Applications, 10, (2011), 129-
146.

[12] S. Jitomirskaya, C. A. Marx; Dynamics and spectral theory of quasiperiodic Schrödinger-type

operators. Ergodic Theory Dynam. Systems, 37(2017), 2353-2393.
[13] E. Stein; Harmonic analysis. Princeton Mathematical Series 43, Princeton, NJ: Princeton

University Press,1993.

[14] E. Sorets, T. Spencer; Positive Lyapounov exponents for Schrödinger operators with quasi-
periodic potential. Comm. Math. Phys., 142(1991), 543-566.

[15] T. Ransford; Potential Theory in the Complex Plane, London Mathematical Society Student

Texts 28, Cambridge University Press, 2000.
[16] K. Tao; Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-

dimensional torus, Front. Math. China, 7(2012), 521-542.
[17] K. Tao; Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators. Bulletin

de la SMF, 142 (2014), 635-671.

[18] K. Tao; positive Lyapunov exponent of discrete analytic Jacobi operator. Elect. J. Diff. Equ.,
2017 no. 339 (2017), 1-12.

Kai Tao

Mathematics department, Southeast University, Jiulonghu Campus, Jiangning District,

Nanjing, Jiangsu Province 211189, China
Email address: ktao@hhu.edu.cn, tao.nju@gmail.com


	1. Introduction
	2. Positive Lyapunov exponent
	3. Large deviation theorems
	Acknowledgments

	References