Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications

Abstract

In this paper, we prove that any analytic quasi-periodic cocycle close to
constant is the Poincar\'{e} map of an analytic quasi-periodic linear system
close to constant. With this local embedding theorem, we get fruitful new
results. We show that the almost reducibility of an analytic quasi-periodic
linear system is equivalent to the almost reducibility of its corresponding
Poincar\'e cocycle.
By the local embedding theorem and the equivalence, we transfer the recent
local almost reducibility results of quasi-periodic linear systems \cite{HoY}
to quasi-periodic cocycles, and the global reducibility results of
quasi-periodic cocycles \cite{A,AFK} to quasi-periodic linear systems. Finally,
we give a positive answer to a question of \cite{AFK} and use it to prove
Anderson localization results for long-range quasi-periodic operator with
Liouvillean frequency, which gives a new proof of \cite{AJ05,AJ08,BJ02}. The
method developed in our paper can also be used to prove some nonlinear local
embedding results.

Full text

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arXiv:1202.2911v2 [math.DS] 16 Feb 2012
EMBEDDING OF ANALYTIC QUASI-PERIODIC
COCYCLES INTO ANALYTIC QUASI-PERIODIC LINEAR
SYSTEMS AND ITS APPLICATIONS
JIANGONG YOU AND QI ZHOU
Abstract. In this paper, we prove that any analytic quasi-periodic co-
cycle close to constant is the Poincar´ e map of an analytic quasi-periodic
linear system close to constant. With this local embedding theorem, we
get fruitful new results. We show that the almost reducibility of an ana-
lytic quasi-periodic linear system is equivalent to the almost reducibility
of its corresponding Poincar´ e cocycle. By the local embedding theorem
and the equivalence, we transfer the recent local almost reducibility re-
sults of quasi-periodic linear systems [21] to quasi-periodic cocycles, and
the global reducibility results of quasi-periodic cocycles [4, 5] to quasi-
periodic linear systems. Finally, we give a positive answer to a question
of [5] and use it to prove Anderson localization results for long-range
quasi-periodic operator with Liouvillean frequency, which gives a new
proof of [6, 7, 12]. The method developed in our paper can also be used
to prove some nonlinear local embedding results.
1. Motivations and main results
We are concerned with smooth quasi-periodic linear systems
?
where x ∈ R2, θ ∈ Td, ω ∈ Rdis rational independent and A ∈ Cr(Td,sl(2,R)),
r ∈ N ∪ {∞,ω}, we denote it by (ω,A). Typical examples are Schr¨ odinger
systems where
?
The time discrete counterparts of the quasi-periodic linear systems are
smooth quasi-periodic SL(2,R) cocycles:
˙ x = A(θ)x
˙θ = ω,
(1.1)
A(θ) = VE,q(θ) =
01
0
q(θ) − E
?
∈ sl(2,R).
(µ,A) :
Td−1× R2→ Td−1× R2
(θ,v) ?→ (θ + µ,A(θ) · v),
Date: February 17, 2012.
1

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2 JIANGONG YOU AND QI ZHOU
where µ ∈ Td−1with (1,µ) being rational independent, A ∈ Cr(Td−1,SL(2,R)).
Typical examples are quasi-periodic Schr¨ odinger cocycles where
?
They are related to one-dimensional quasi-periodic Schr¨ odinger operators
on l2(Z):
A(θ) = SV
E(θ) =
V (θ) − E
1
−1
0
?
∈ SL(2,R).
(1.2)(HV,µ,φx)n= xn+1+ xn−1+ V (nµ + φ)xn= Exn.
We denote by Cr
that are homotopic to the identity.
If d = 2, it is also interesting to consider the dual operator of (1.2), the
Long-range quasi-periodic operator:
?
where α ∈ R\Q, Vkare fourier coefficients of V (θ) ∈ Cr(T,R). These two
operators are closely related by Aubry duality (consult section 2.3 for more
details).
We say that (ω,A) is Crreducible, if there exist B ∈ Cr(2Td,SL(2,R))
and A∗∈ sl(2,R) such that B conjugates (ω,A) to (ω,A∗). It is clear that
the concept of reducibility defined above for Liouvillean frequencies is too
restrictive since in general even a R-valued cocycle is not reducible. So we
need to introduce the weaker concept almost reducibility. A system (ω,A) is
Cralmost reducible (resp. almost rotations reducible) if there exist sequences
of Bn∈ Cr(2Td,SL(2,R)), An∈ sl(2,R)(resp.An∈ Cr(Td,so(2,R))) and
Fn∈ Cr(Td,sl(2,R)), such that Bnconjugate (ω,A) to (ω,An+Fn), where
Fnis Crconverging to zero. Another useful concept is rotations reducibil-
ity. We say that (ω,A) is Crrotations reducible, if there exist B ∈ Cr(2Td,
SL(2,R)) and A∗∈ Cr(Td,so(2,R)) such that B conjugates (ω,A) to (ω,A∗).
These concepts can be defined similarly for cocycles A ∈ Cr
We say that the operator LV,α,ϕ, (resp.HV,α,φ) displays Anderson localiza-
tion, if it has pure point spectrum with exponentially decaying eigenfunc-
tions. We remark that reducibility, almost reducibility and Anderson local-
ization are important issues in the study of quasi-periodic linear systems
and the spectral theory of Schr¨ odinger operators [4, 7].
0(Td−1,SL(2,R)) the set of maps A ∈ Cr(Td−1,SL(2,R))
(1.3)(LV,α,ϕψ)n=
k∈Z
Vkψn−k+ 2cos2π(ϕ + nα)ψn,
0(Td−1,SL(2,R)).
1.1. Classical results review.
1.1.1. Reducibility of analytic quasi-periodic linear systems. The earliest re-
sult of local reducibility was due to Dinaburg and Sinai [15], who showed
that (ω,VE,q(θ)) is reducible for “most” sufficiently large E, where ω is
assumed to satisfy the classical Diophantine condition:
|?k,ω?| ≥γ−1
|k|τ,
0 ?= k ∈ Zd,

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LOCAL EMBEDDING THEOREM3
and γ,τ > 1 are fixed positive constants. Here (γ,τ) are called the Diophan-
tine constants of ω. Later, Eliasson [16] obtained the full measure reducibility
and the almost reducibility for (ω,VE,q(θ)) with Diophantine ω. The proof is
based on a crucial “resonance-cancellation” technique which was initially de-
veloped by Moser and P¨ oschel [30]. It should be noted that these results are
perturbative. Stronger concept is non-perburbative reducibility, which means
that the smallness of the perturbation does not depend on the Diophantine
constants (γ,τ).
However, few results for quasi-periodic linear systems were obtained since
[16] until [21]. Based on a unified approach (KAM theory and Floquet the-
ory), Hou and You [21] proved that any two-frequencies quasi-periodic linear
system close to constant is always almost reducible and non-perturbative re-
ducible. They further proved that it is analytically rotations reducible if the
rotation number is Diophantine w.r.t ω.
1.1.2. Anderson localization. Since non-perturbative reducibility results for
quasi-periodic Schr¨ odinger cocycles strongly depends on Anderson localiza-
tion for the dual operator, we review the Anderson localization results first.
For the almost Mathieu operator Hλcos,α,φ, Jitomirskaya [22] proved that if
α is Diophantine, λ > 2, then for a.e. φ, Hλcos,α,φhas Anderson localization.
In the sequel, Avila and Jitomirskaya [6] further proved that if λ > 2e16β/9,1
then Hλcos,α,φhas Anderson localization. Similar results for long-range op-
erator LλV,α,ϕwere considered by Bourgain and Jitomirskaya [12], Avila and
Jitomirskaya [7], and we refer to [10, 11] for results on Schr¨ odinger operators.
1.1.3. Reducibility of analytic quasi-periodic cocycles. Combining Aubry du-
ality [1] with Anderson localization results [12], Puig [31] obtained a non-
perturbative extension of Eliasson’s results. Avila and Jitomirskaya [7] fur-
ther developed a quantitative Aubry duality to prove that almost localization
implies almost reducibility for the dual model.
Different from the continuous case, global reducibility of quasi-periodic co-
cycles (reducibility of a cocycle which is not necessarily close to a constant)
is being developed rapidly. Based on Kotani theory [24, 33] and renormaliza-
tion scheme [25], Avila and Krikorian [8] proved that: if α is recurrent Dio-
phantine, A ∈ Cω
is either analytic reducible or non-uniformly hyperbolic. With respect to
Liouvillean frequency, Fayad and Krikorian [18] first obtained that: if α is
irrational, A ∈ C∞
is either C∞almost rotations reducible or non-uniformly hyperbolic. Their
main techniques in the proof were “algebraic conjugacy trick” and renor-
malization scheme [8]. Later, Avila, Fayad and Krikorian [5] developed this
method and obtained a local positive measure rotations reducibility result.
0(T,SL(2,R)), then for Lebesgue a.e.ϕ ∈ [0,1], (α,RϕA)2
0(T,SL(2,R)), then for Lebesgue a.e.ϕ ∈ [0,1], (α,RϕA)
1The definition of β can be found in section 2.1.
?
sin2πϕ cos2πϕ
2Rϕ :=
cos2πϕ
−sin2πϕ
?
.

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4 JIANGONG YOU AND QI ZHOU
Subsequently, they further proved that for Lebesgue a.e.ϕ ∈ [0,1], (α,RϕA)
is either analytic rotations reducible or non-uniformly hyperbolic.
Recently, Avila [3] proposed an authentic “global theory” of one-frequency
SL(2,R) cocycles. Cocycles which are not uniformly hyperbolic are classified
into three classes: supercritical, subcritical and critical. A central issue
in the reducibility theory is his Almost Reducible Conjecture: “subcritical
implies almost reducibility”. If α is exponentially Liouvillean, Avila [4]
proved the conjecture and obtained a corollary: any one-frequency analytic
quasi-periodic cocycle close to constant is almost reducible.
1.1.4. Comparison of continuous case and discrete case. In the continuous
case, local almost reducibility result is completely established recently by
Hou and You [21], while there is no result for global reducibility. Since there
is no Aubry duality [1] in the continuous case, reducibility can not be ob-
tained by proving localization or almost localization of the dual system as in
the discrete case [7, 31]. However, there is a uniform way to prove the local
almost reducibility results directly in the continuous case. Unfortunately,
the methods developed in [21] can not be applied to quasi-periodic cocycles
directly. Global reducibility results are completely missing for continuous
systems since there is no suitable renormalization scheme as in the discrete
case. It is still an interesting question whether there is corresponding renor-
malization scheme for the quasi-periodic linear systems. We note that [14]
also used renormalization technique to get reducibility results, however, the
method can only be applied to the local situation and restricted to Brjuno
frequency.
In the discrete case, although various global reducibility results [4, 5, 8,
18, 25] were obtained, local almost reducibility results are not satisfactory.
Firstly, there is no unified and direct approach to deal with the almost re-
ducibility problem for the quasi-periodic cocycles even in the local regime.
The existing approach which highly depends on the localization results for
the dual model, works only for Schr¨ odinger cocycles. Secondly, as pointed
by Avila, Fayad and Krikorian [5], the study of parabolic behavior was miss-
ing, also there was no generalization of Eliasson’s results [16]. Readers may
refer to the section 1.1 of [5] for more discussions.
In this paper, we shall establish a local embedding theorem, which serves
as a bridge between analytic quasi-periodic linear systems and quasi-periodic
cocycles. With this powerful tool, we can deduce fruitful new results. For
example, one can exchange the almost reducibility results of quasi-periodic
linear systems and quasi-periodic cocycles for free and then get many missing
results both for the continuous systems and discrete cocycles. Furthermore,
combining Aubry duality, we deduce Anderson localization results for the
long-range operator with Liouvillean frequency. The proof of the local em-
bedding theorem is interesting in itself and has further generalizations.
1.2. Embedding theorem.

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LOCAL EMBEDDING THEOREM5
1.2.1. A local embedding theorem. Let G = sl(n,R),sp(2n,R),o(n),u(n),so(n),
and let G be the corresponding Lie groups. We consider the following Cr
quasi-periodic linear system:
?
where A ∈ Cr(Td,G), µ ∈ Td−1with (1,µ) being rational independent,
θ = (θ1,?θ),?θ = (θ2,··· ,θd). Denote by Φtthe flow of (1.4) defined on
introduce A(·) = Φ1(0,·) which is clearly defined on Td−1and called the
corresponding Poincar´ e cocycle defined by (1.4). What we are interested in
is the converse, whether we can embed a given Crquasi-periodic cocycle
into a Crquasi-periodic flow? Apparently such a cocycle must be homotopy
to the identity.
If r ?= ω, using the method of suspension flow, Chavaudret [13] proved
that any Crsmooth G−exponential cocycle3can be Crembedded into
the flow. Rychlik [32] proved that any Crsmooth SU(2) cocycle can be
Crembedded into a quasi-periodic flow, the proof strongly depends on the
fact that SU(2) is simply connected. Both methods can’t be applied to the
analytic case.
The aim of this paper is to provide a new method to prove the local
embedding of an analytic quasi-periodic G− valued cocycle into an analytic
quasi-periodic linear system. For a bounded analytic function F defined on
|Imθ| < h, let ?F?h= sup|Imθ|<h?F?. We denote by Cω
all these ∗-valued functions (∗ will usually denote R, G). The main theorem
is the following:
˙ x = A(θ)x
˙θ = ω = (1,µ),
(1.4)
Td× Rnwhich is of the form: Φt(θ1,?θ,y) = (θ1+ t,?θ + tµ,Φt(θ1,?θ)y). We
h(Td,∗) the set of
Theorem 1.1. Let h > 0, µ ∈ Td−1with (1,µ) being rational independent,
A ∈ G, G ∈ Cω
0 such that the quasi-periodic cocycle (µ,eAeG(·)) can be analytically embed-
ded into a quasi-periodic linear system provided that ?G?h= ε < ǫ. More
precisely, there exist? A ∈ G, F ∈ Cω
?
Remark 1.1. The selection of? A and precise estimate on F in Theorems 1.1
cε if A is diagonalizable.
h(Td−1,G). There exist ǫ = ǫ(A,h,|µ|) > 0, c = c(A,h,|µ|) >
h/1+|µ|(Td,G) with ?F?h/1+|µ|≤ cε1/2,
such that (µ,eAeG(·)) is the Poincar´ e map of
˙ x = (? A + F(θ))x
˙θ = (1,µ).
will be given explicitly in the proof. We emphasize that? A = A, ?F?h/1+|µ|≤
Remark 1.2. The Cr(r ?= ω) version of the local embedding theorem is
also true. Different from [13], the local structure is preserved with precise
estimates.
3It means that the cocycle can be written as eA(·).