Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

Bulletin

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Publié avec le concours du Centre national de la recherche scientifique

de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Tome 141 Fascicule 1

2013

STRONG ALMOST REDUCIBILITY FOR ANALYTIC AND GEVREY

QUASI-PERIODIC COCYCLES

Claire Chavaudret

STRONG ALMOST REDUCIBILITY FOR ANALYTIC AND GEVREY QUASI-PERIODIC COCYCLES

by Claire Chavaudret

Abstract. — This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density, of reducible cocycles near a constant. Some algebraic structure can also be preserved, by doubling the period if needed.

Résumé(Presque réductibilité forte pour les cocycles quasi-périodiques de classe ana- lytique et Gevrey)

Cet article traite de la presque-réductibillité des cocycles quasi-périodiques à fré- quence diophantienne qui sont proches d’un cocycle constant. Nous démontrons un résultat de presque-réductibilité forte des cocycles analytiques et Gevrey, c’est-à-dire que le changement de variables obtenu pour conjuguer le cocycle initial à un cocycle proche d’une constante est dans une classe analytique ou Gevrey qui est indépendante de la proximité à la constante; ceci généralise certains résultats antérieurs de L.H.

Eliasson. Ce résultat a pour corollaire un théorème de densité ou de quasi-densité des cocycles réductibles au voisinage d’une constante. Il est possible de préserver certaines caractéristiques algébriques du cocycle initial en doublant la période.

Texte reçu le 16 juin 2010, révisé le 5 mai 2011, accepté le 14 décembre 2011.

Claire Chavaudret, Laboratoire J.-A. Dieudonné, Université de Nice Sophia Antipolis 2010 Mathematics Subject Classification. — 37C55, 37E45, 37F50, 37J40.

Key words and phrases. — Small divisors, small denominators, quasiperiodic skew-product, quasiperiodic cocycles, Lyapunov exponent, Floquet theory.

BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037-9484/2013/47/$ 5.00

©Société Mathématique de France

1. Introduction

We are concerned with quasi-periodic cocycles, that is, solutions of equations of the form

(1) ∀(θ, t)∈2T^d×R, d

dtX^t(θ) =A(θ+tω)X^t(θ); X⁰(θ) =Id

where A∈C⁰(2T^d,

G

^)and

G

is a linear Lie algebra. HereT^d =R^d/Z^d stands for the d-torus, d≥ 1, and 2T^d =R^d/(2Z^d)stands for the double torus. We will assume in this article that ω ∈R^d satisfies some diophantine conditions.

The solution of (1) is called the quasi-periodic cocycle associated toA and is defined on 2T^d×Rwith values in the connected component of the identity of a Lie group Gwhose associated Lie algebra is

G

. Terminology is explained by the fact thatAis the envelope of a quasi-periodic function, sincet7→A(θ+tω) is a quasi-periodic function for all θ∈2T^d. We sayX is a constant cocycle if A is constant. A constant cocycle is always of the formt7→e^tA.

A cocycle is said to be reducible if it is conjugated to a constant cocycle, in a sense that will be defined later on. The problem of reducibility of cocy- cles has been thoroughly studied and is of interest because the dynamics of reducible cocycles is well understood and because this problem has links with the spectral theory of Schrödinger cocycles and with the problem of lower di- mensional invariant tori in hamiltonian systems. In the periodic case (d= 1), Floquet theory tells that every cocycle is reducible modulo a loss of periodicity.

However, the problem is far more difficult if d is greater than 1 and it is not true that every cocycle is then reducible. The question becomes whether every cocycle is close, up to a conjugacy, to a reducible one; from this question comes the notion of almost-reducibility.

For any functional class

C

, a cocycle is said to bealmost-reducible in

C

^{if it}

can be conjugated to a cocycle which is arbitrarily close in the topology of

C

to a reducible one, with the conjugacy also in

C

. Reductibility implies almost reducibility, however the reverse is not true: there are non reducible cocycles even close to a constant cocycle (see [3]). Almost reducibility is an interesting notion since the dynamics of an almost reducible cocycle are quite well known on a very long time.

We first focus on cocycles generated by functions which are analytic on a neighbourhood of the torus, i.e real analytic functions which are periodic in the direction of the real axis (recall that they are matrix-valued). For such a functionF, we will let

|F |r= sup

|Imθ|≤r

||F(θ)||

where ||.|| stands for the operator norm.

The aim of this paper is to show that for

G=GL(n,C), GL(n,R), SL(2,C), SL(n,R), Sp(n,R)⁽¹⁾, O(n), U(n) in the neighbourhood of a constant cocycle, i.e under a smallness condition on the non-constant part of the cocycle, every cocycle which is analytic on an r-neighbourhood of the torus and G-valued is almost reducible in C_r^ω0(2T^d, G) for all0< r⁰ < r≤¹₂, in the sense defined above. The smallness condition only depends on the dimensionsn, d, on the diophantine class ofω, on the constant cocycle and on the loss of analyticityr−r⁰.

More precisely, we shall prove the following theorem, forGamong the groups cited above and

G

the Lie algebra associated toG:

Theorem 1.1. — Let0< r⁰< r≤ ¹₂,A∈

G

^{,F ∈C}r^ω(T^d,

G

). There is0<1 depending only on n, d, ω, A, r−r⁰ such that if

|F|r≤0

then for all >0, there exists A¯,F¯∈C_r^ω0(2T^d,

G

^{), Ψ}, Z∈C_r^ω0(2T^d, G)and A∈

G

such that for allθ∈2T^d,

∂ωZ(θ) = (A+F(θ))Z(θ)−Z(θ)( ¯A(θ) + ¯F(θ)) with

1. ∂ωΨ= ¯AΨ−ΨA, 2. |F¯|r⁰ ≤,

3. |Ψ|r⁰≤⁻¹⁸, 4. and|Z−Id|r⁰ ≤2

1 2

0.

Moreover, if G ⊂ GL(2,C) or if G = GL(n,C) or U(n), Z,A¯,F¯ are in C_r^ω0(T^d).

Property 1 states the reducibility of A¯. Theorem 1.1 immediately entails the following:

Theorem 1.2. — Let0< r⁰< r≤ ¹₂,A∈

G

^{,F ∈C}r^ω(T^d,

G

). There is0<1 depending only on n, d, ω, A, r−r⁰ such that if

|F|r≤0

then for all >0, there existsF∈C_r^ω0(2T^d,

G

^),Z∈C_r^ω0(2T^d, G)andA∈

G

such that for all θ∈2T^d,

∂_ωZ(θ) = (A+F(θ))Z(θ)−Z(θ)(A+F(θ)) with |F|r⁰ ≤.

(1)Withneven.

Note that in Theorem 1.2, we do not have any good estimate of Z. Theo- rem1.1 also holds if one choosesF in a class which is bigger thanC_r^ω(T^d,

G

^),

i.e the class of functions in C_r^ω(2T^d,

G

⁾satisfying some “nice periodicity prop- erties” with respect to the matrixA.

There is a loss of analyticity in this result, but it is arbitrarily small. A result close to Theorem1.1in the case whenG=GL(n,R)had already been proven in [5] by L.H. Eliasson:

Theorem [Eliasson] . — Let A∈gl(n,R) andF ∈C_r^ω(T^d, gl(n,R)). There is 0 < 1 depending only on n, d, κ, τ,||A||, r such that if |F|r ≤ 0, then for all >0, there exists 0 < r < r, Z ∈ C_r^ω(2T^d, GL(n,R))such that for all θ∈2T^d,

∂ωZ(θ) = (A+F(θ))Z(θ)−Z(θ)(A+F(θ)) with A∈gl(n,R),F∈C_r^ω(2T^d, gl(n,R))and|F|r≤.

Eliasson’s theorem merely states almost reducibility in∪r⁰>0C_r^ω0(2T^d, GL(n,R)), since the sequence (r)might well tend to 0. The achievement of Theorem1.1 is to state almost reducibility in a more general algebraic framework, but also, and mostly, to show that almost reducibility holds in a fixed neighbourhood of a torus even when this torus has dimension greater than 1. This is almost reducibility in a strong sense.

Note that, as was the case in [5], one cannot avoid to lose periodicity in Theorem1.1ifGis a real group with dimension greater than 2. The notion of

“nice periodicity properties” that will be given aims at limiting this loss to a period doubling. In comparison with the real framework, the symplectic frame- work does not introduce any new constraints in the elimination of resonances (Section2.2); therefore there is no more loss of periodicity here than in the case when G=GL(n,R). As before in [2], a single period doubling is sufficient in the case when Gis a real symplectic group.

The second part of this paper is dedicated to showing that the same method gives an analogous result for cocycles which are in a Gevrey class (Theorem3.1);

denoting byC_r^G,βthe class of Gevrey functions with exponentβand parameter r (so thatC_r^G,1 is the class of analytic functions), and by||.||β,r their norm, we have the following:

Theorem 1.3. — Let 0 < r⁰ < r ≤ ¹₂, A ∈

G

^{, F ∈ C}r^G,β(T^d,

G

). There is ₀<1 depending only onn, d, κ, τ, A, r−r⁰ such that if

||F||β,r≤0

then for all >0, there existsA¯,F¯∈C_r^G,β0 (2T^d,

G

^),Ψ, Z∈C_r^G,β0 (2T^d, G) andA∈

G

such that for allθ∈2T^d,

∂_ωZ(θ) = (A+F(θ))Z(θ)−Z(θ)( ¯A(θ) + ¯F(θ))

with

– A¯ conjugated toA by Ψ, – ||F¯||β,r⁰ ≤,

– ||Ψ||β,r⁰≤⁻¹⁸,

– and||Z−Id||β,r⁰ ≤2₀¹². Moreover,

– ifG⊂GL(2,C)or ifG=GL(n,C)orU(n),Z,A¯,F¯are inC_r^G,β0 (T^d);

– If

G

^{iso(n)or u(n), then}0 does not depend onA.

Ifn= 2or if

G

^isgl(n,C)oru(n), these results can be rephrased as density of reducible cocycles in the neighbourhood of constant cocycles:

Theorem 1.4. — Let

G

^{= gl(n,}C), u(n), gl(2,R), sl(2,R) or o(2). Let 0 <

r⁰ < r ≤ ¹₂ and A ∈

G

^{, F ∈ C}r^ω(T^d,

G

). There is 0 depending only on r− r⁰, n, d, ω, Asuch that if

|F|r≤0

then for all >0 there existsH ∈C_r^ω0(T^d,

G

⁾ which is reducible inC_r^ω0(T^d,

G

⁾

and such that

|A+F−H|r⁰ ≤.

A similar result, for smooth cocycles with values in compact Lie groups, was obtained by R. Krikorian in [7] (th.5.1.1). For cocycles over a rotation on the circle, analyticity is far better controlled (see for instance [1]) since it is then possible to use global methods. In this article, we are considering the case of a torus of arbitrary dimension. The KAM-type method that is being used here had already given way to full-measure reducibility results for cocycles with values inSL(2,R)([3], [6]).

Sketch of the proof and organization of the paper. — The proof of Theorems1.1 and 1.4 is a refinement of the method in [5]; it is based on a KAM scheme.

The central idea is to prove an inductive lemma where one conjugates a system which is close to a reducible one to another system which is even closer to something reducible. Iterating this lemma arbitrarily many times, one would then be able to conjugate the initial system to something which is arbitrarily close to a reducible one. An estimate on the reducing transformation would then imply almost reducibility. Now consider a system close to a reducible one;

if it is close to a system which can be reduced to a constant part satisfying some non-resonant conditions, then there exists a conjugation which is close to the identity in a good topology taking the first system to something closer to a reducible system. But the constant part might well be too resonant for such a conjugation to exist. In this case, it is possible to remove the resonances in the

constant part, but then the conjugation will not stay very close to the identity except if one accepts to give up a lot of regularity. Now we want to avoid this loss of regularity in order to obtain a strong version of almost reducibility. So we will have to improve the step of removing the resonances and use the following two facts: when resonances have been removed up to some order N, firstly, the eigenvalues will be so close together that resonances are in fact removed up to an order RN which is much greater than N; secondly, the eigenvalues are removed in a durable way, that is, one will not have to remove resonances again until a large number of conjugations is made that will take the cocycle to something much closer to a reducible one. The article is organized as follows:

Section2is dedicated to the proof of the theorem in the analytic case. Here are the main steps of the proof:

– Removing of the resonances by a map Φcalled a reduction of the eigen- values at orderR,N¯ (Proposition2.6) forR, N ∈N\ {0}.

In dimension 2 (i.e if n= 2),Φwill be such that for allH continuous onT^d,ΦHΦ⁻¹is continuous onT^d.

This step is crucial in the obtention of strong almost reducibility. The reduction of the eigenvalues is defined in a way similar to [5], however here it will remove resonances up to an orderRN¯ which is much greater than the value of the parameter N¯ appearing in the estimates. The pa- rameter R will be used to define a map of reduction of the eigenvalues at order R,N¯ where N¯ does not depend on the loss of analyticity. This way, the map of reduction of the eigenvalues will stay under control on a neighbourhood of the torus which will not have to fade totally.

– Resolution of the homological (also called cohomological) equation (Proposition 2.8): if A˜ has a spectrum fulfilling some non-resonance conditions and F˜ is a function with nice periodicity properties with respect to A, then there exists a solution˜ X˜ of equation

∂ωX˜ = [ ˜A,X] + ˜˜ F^RN¯; Xˆ˜(0) = 0

having the same periodicity properties as F; it takes its values in the˜ same Lie algebra as doesF˜. Moreover, it can be well controlled by losing some analyticity.

– Inductive lemma (Proposition2.14): IfF˜ ∈C_r^ω(2T^d,

G

⁾has some period- icity properties (with respect toA), if˜

∂ωΨ = ¯AΨ−Ψ ˜A

andF¯= Ψ ˜FΨ⁻¹, then there existsZ∈C_r^ω0(2T^d, G)such that (2) ∂_ωZ= ( ¯A+ ¯F)Z−Z( ¯A⁰+ ¯F⁰)

with A¯⁰ reducible,F¯⁰ is much smaller thanF,¯ Z is close to the identity and Ψ⁰⁻¹F¯⁰Ψ⁰ has periodicity properties with respect to A⁰ which are similar to the properties of F.˜

The estimate of F¯⁰ depends on F˜ −F˜^RN¯, on the reduction of the eigenvalues Φ, and on the solution X˜ of the homological equation.

– Iteration of the inductive lemma (Theorem 2.16): We shall iterate Lemma 2.14 so as to obtain estimates of analytic functions on a se- quence of neighbourhoods of the torus not shrinking to 0, by means of a numerical lemma (Lemma2.15), to reduce the perturbation arbitrarily.

In Section 3, some lemmas are given (3.1) which show that it is possible to adapt the proof to the Gevrey case; namely, the estimates will be analogous to those that are obtained in the analytic case and so, by slightly modifying the parameters, the argument works in the same way: one obtains analogous reduction of the eigenvalues (3.2), homological equation (3.3) and inductive lemmas (3.4).

Notations, further definitions and a general assumption. — For a function f ∈ C¹(2T^d, gl(n,C)), for allθ∈2T^d we will denote by

(3) ∂ωf(θ) = d

dtf(θ+tω)_|t=0

the derivative off in the directionω. Denote byh·,·ithe complex euclidian scalar product, taking it antilinear in the second variable. For a linear oper- ator M, we shall call M^∗ its adjoint, M^∗ = ^tM¯, which is identical to the transpose of M if M is real. Also denote by M_N the nilpotent part of M, as follows: let M = P AP⁻¹ with A in Jordan normal form, let AD be the diagonal part of A, then M_N = P(A−AD)P⁻¹. To simplify the notation, if A : 2T^d → GL(n,C), we will denote by A⁻¹ the map θ7→A(θ)⁻¹. For all m = (m1, . . . , md) ∈ ¹₂Z^d, we shall denote | m |=| m1 | +· · ·+ | md |. The letterJ will stand for the matrixJ = 0 −Id

Id 0

! .

Definition 1. — A function f is analytic on an r-neighbourhood of the torus (resp. double torus) if f is holomorphic on {x = (x1, . . . , xd) ∈ C^d,sup_j|Imxj| < r} and 1-periodic (resp. 2-periodic) in Rexj for all 1≤j≤d.

For all subsetEofgl(n,C), denote byC_r^ω(T^d, E)the set of functions which are analytic on an r-neighbourhood of the torus and whose restriction to R^d takes its values inE; letC_r^ω(2T^d, E)be the set of functions which are analytic

on an r-neighbourhood of the double torus and whose restriction to2T^d takes its values inE. For allf ∈C_r^ω(2T^d, E), denote

(4) |f|r= sup

|Imx|<r

||f(x)||

where ||.||stands for the operator norm.

Definition 2. — A functionf is Gevreyβ with parameterr if it satisfies X

α∈N^d

r^β|α|

α!^β sup

θ

||∂^αF(θ)||<+∞.

Let C_r^G,β be the class of Gevrey β functions with parameter r. Denote by

||.||β,r the norm

||F ||β,r= X

α∈N^d

r^β|α|

α!^β sup

θ

||∂^αF(θ)||.

To formalize the notion of reducibility, we shall introduce an equivalence relation on cocycles.

Definition 3. — LetGbe a Lie group and

G

the Lie algebra associated toG.

Let r, r⁰ > 0 and A, B ∈ C_r^ω(2T^d,

G

). We say that A and B are conjugate in C_r^ω0(2T^d, G)if there existsZ ∈C_r^ω0(2T^d, G)such that for all θ∈2T^d,

∂ωZ(θ) =A(θ)Z(θ)−Z(θ)B(θ)

where ∂ω means the derivative in the directionω. If B is constant inθ, we say thatA is reducible inC_r^ω0(2T^d, G), or reducible byZ toB.

We will use an analogous definition withC^G,β instead ofC^ω.

Note that if X is the quasi-periodic cocycle associated toA, then the map A is reducible byZ toB if and only if

(5) ∀(t, θ), X^t(θ) =Z(θ+tω)⁻¹e^tBZ(θ)

Reducibility is also equivalent to the fact that the map from 2T^d×Rⁿ to itself:

(6) θ

v

!

7→ θ+ω X¹(θ)v

!

is conjugate to a mapχsuch that

(7) dχ

dθ θ v

!

≡ ¯1 0

! .

Assumption:The frequencyω is in the diophantine classDC(κ, τ), i.e

(8) ∀m∈Z^d\ {0}, |hm, ωi| ≥ κ

|m|^τ

where κ, τ are fixed throughout the paper and0< κ <1, τ ≥max(1, d−1).

2. Strong almost reducibility for analytic quasi-periodic cocycles

2.1. Nice periodicity properties. — A few definitions will first be given. The no- tion of “triviality with respect to a decomposition” will make the construction of the map of reduction of the eigenvalues easier; the “nice periodicity prop- erties” have been introduced in [5] and are used in the real case to make sure that only one period doubling will be needed in iterating the inductive lemma.

2.1.1. Invariant decompositions

Definitions 1. — – The set

L

^={L1, . . . , L_R} is called a decomposition of Cⁿ if

Cⁿ=M

j

L_j.

– If

L

^,

L

⁰ are decompositions of Cⁿ, then

L

is said to be finer than

L

^{0 if}

for all L∈

L

^{, there isL0 ∈}

L

^{0 such that L⊂L0;}

–

L

is said strictly finer than

L

^{0 if}

L

is finer than

L

^{0 and}

L

⁶⁼

L

^0.

Let A ∈ gl(n,C); then

L

^{= {L}1, . . . , Ls} is an A-decomposition, or else A-invariant decomposition, if it is a decomposition ofCⁿ and for alli,AL_i⊂ Li. Subsets Li are called subspaces of

L

^.

A Jordan decomposition for Ais an A-decomposition which is minimal (i.e no finer decomposition is anA-decomposition).

Remark:

– A matrix might have many Jordan decompositions. For instance, the identity has infinitely many Jordan decompositions.

– A decomposition is an A-decomposition if and only if it is less fine than some Jordan decomposition forA. Therefore, if operatorsAandA⁰ have a common Jordan decomposition, then anA-decomposition which is less fine than this common Jordan decomposition is an A⁰-decomposition.

Notation: Let

L

^{be an} A-decomposition. For all L ∈

L

, denote by σ(A_|L) the spectrum of the restriction of Ato subspaceL.

Definition 4. — Letκ⁰≥0. Let

L

A,κ⁰ be the uniqueA-decomposition

L

^such

that for allL6=L⁰∈

L

^,α∈σ(A|L)andβ ∈σ(A_|L0)⇒ |α−β|> κ⁰ and such that noA-decomposition strictly finer than

L

has this property.

Remark:Forκ⁰≥0, any Jordan decomposition is finer than

L

A,κ⁰.

Definition 5. — Let

L

be a decomposition of Cⁿ. For all u ∈ Cⁿ, there is a unique decomposition u =P

L∈LuL such that uL ∈ L for all L ∈

L

^{. For}

all L ∈

L

, the projection on L with respect to

L

, denoted by P_L^L, is the map defined byP_L^Lu=uL.

Remark:LetA ∈gl(n,C) andκ⁰ >0. If

L

^{is an} A-decomposition which is less fine than

L

A,κ⁰, then one has the following lemma, which can be found in [5], appendix, Lemma A⁽²⁾:

Lemma 2.1. — There is a constantC₀≥1 depending only onnsuch that for all subspace L∈

L

^,

(9) ||P_L^L||≤C0

Å1+||A_N ||

κ⁰

ãn(n+1)

.

In what follows, C0 will always stand for this constant fixed in Lemma2.1.

Definition 6. — An (A, κ⁰, γ)-decomposition is an A-decomposition

L

^such

that for all L∈

L

, the projection on Lwith respect to

L

^satisfies

(10) ||P_L^L ||≤C0

Å1+||A_N ||

κ⁰ ãγ

.

Remark:ForA∈gl(n,C), one always hasA=P

L,L⁰∈LP_L^LAP_L^L0. In partic- ular, if

L

^{is an}A-decomposition, thenA=P

L∈LP_L^LAP_L^L. Definitions 2. — Let

L

be a decomposition. We say that

–

L

is a real decomposition if for allL∈

L

^,L¯∈

L

^;

–

L

is a symplectic decomposition if it is a decomposition of Cⁿ with even n and for allL∈

L

, there is a unique L⁰∈

L

^{such that hL, J L0i 6= 0;}

–

L

is a unitary decomposition if for allL6=L⁰∈

L

^{,hL, L0i= 0.}

Remark:

– IfAis a real matrix, then for allκ⁰≥0,

L

A,κ⁰ is a real decomposition.

– For allL, there is at least oneL⁰ such thathL, J L⁰i 6= 0. This comes from the fact that the symplectic form h., J.iis non-degenerate.

– IfA∈sp(n,R), then anyA-decomposition

L

which is less fine than

L

A,0

is a real and symplectic decomposition. To see this, let L, L⁰ ∈

L

^such

that hL, J L⁰i 6= 0; let v ∈ L, v⁰ ∈ L⁰ be eigenvectors of A such that hv, J v⁰i 6= 0and λ, λ⁰ their associated eigenvalues. Then

λhv, J v⁰i=hAv, J v⁰i=hv, A^∗J v⁰i=−hv, J Av⁰i=−¯λ⁰hv, J v⁰i and sincehv, J v⁰i 6= 0, thenλ=−¯λ⁰.

(2)Lemma A from [5] gives in fact an estimate which depends on||A||, but the proof shows clearly that the estimate in fact only depends onA_N.

– If A ∈ U(n), then any decomposition which is less fine than

L

A,0 is unitary.

– If

L

is unitary, then for everyL∈

L

^{, P}L^L is an orthogonal projection so

||P_L^L||≤1.

2.1.2. Triviality and nice periodicity properties with respect to a decomposition Definitions 3. — Let

L

be a decomposition ofCⁿ. We say a mapΨis trivial with respect to

L

if there exist{m_L, L∈

L

^{} ⊂ 1}₂Z^d such that for all θ∈2T^d,

(11) Ψ(θ) = X

L∈L

e^2iπhmL,θiP_L^L.

We say that the functionΨis trivial if there exists a decomposition

L

^such

that Ψis trivial with respect to

L

^.

Remark:

– IfΨis trivial with respect to

L

^and

L

⁰ is finer than

L

^{, thenΨis trivial}

with respect to

L

^0.

– IfΦ,Ψ : 2T^d→GL(n,C)are trivial with respect to

L

, then the product ΦΨ is trivial with respect to

L

^.

– IfΦis trivial with respect to anA-decomposition

L

, then for allθ∈2T^d, [A,Φ(θ)] = 0.

Lemma 2.2. — Let

L

be a real decomposition ofCⁿ,{mL, L∈

L

^{} ⊂ 1}₂Z^dand Ψdefined by

(12) Ψ(θ) = X

L∈L

e^2iπhmL,θiP_L^L.

ThenΨ is real if and only if for allL,m_L =−mL¯. Moreover, ifΨ is real, then Ψtakes its values inSL(n,R).

Proof. — Assume that for allL∈

L

^,mL=−mL¯. Letu∈Rⁿ. Then Ψ(θ)u= X

L∈L

e^{2iπh−mL,θi}P_L^Lu= X

L∈L

e^{2iπhmL¯,θi}P_L¯^Lu= Ψ(θ)u

so Ψ(θ)is real.

Now suppose thatΨis real. Then for allθ, X

L∈L

e^2iπhmL,θiP_L^L =X

L∈L

e^{2iπh−mL,θi}P_L^L= X

L∈L

e^{2iπh−mL,θi}P_L¯^L

so mL=−mL¯.

SupposeΨis real; then for allL, mL=−mL¯ so Ψ(θ)is the exponential of a trace-zero matrix, so it has determinant 1.

Remark:Any map which is trivial with respect to a unitary decomposition is unitary: let

L

be a unitary decomposition, letΦbe trivial with respect to

L

and letL, L⁰∈

L

. Then for allu∈

L

^{, v∈}

L

^0,

hΦ(θ)u,Φ(θ)vi=he^2iπhmL,θiu, e^2iπhmL0,θivi=hu, vi.

Lemma 2.3. — Let

L

be a real and symplectic decomposition and{mL, L∈

L

^}

be a family of elements of ¹₂Z^d. Let Ψ =P

L∈Le^2iπhmL,.iP_L^L. ThenΨtakes its values inSp(n,R)if and only if

– for all L,mL=−mL¯

– and if hL, J L⁰i 6= 0, thenmL=mL⁰.

Proof. — By Lemma2.2,Ψis real if and only if for allL,m_L=−mL¯. Assume nowΨis real.

We show first that if for allL, L⁰ ∈

L

^{, hL, J L0i 6= 0⇒m}L =m_L⁰, thenΨ takes its values inSp(n,R). Letu, v∈Rⁿ. Then

hu,Ψ(θ)^∗JΨ(θ)vi=hΨ(θ)u, JΨ(θ)vi=X

L

e^{2iπhmL−mM(L),θi}hP_L^Lu, J P_M(L)^L vi whereM(L)stands for the unique subspace such thathL, J M(L)i 6= 0. Assume that ifhL, J L⁰i 6= 0, thenmL=mL⁰. This implies that

hu,Ψ(θ)^∗JΨ(θ)vi=X

L

hP_L^Lu, J P_M(L)^L vi=hu, J vi so Ψ(θ)∈Sp(n,R).

Now we will show that if Ψ(θ)∈Sp(n,R)and if hL, J L⁰i 6= 0, then mL = m_L⁰. SupposeΨ(θ)∈Sp(n,R). For any two vectorsu, v,

hu, J vi=hu,Ψ(θ)^∗JΨ(θ)vi=hΨ(θ)u, JΨ(θ)vi.

Ifu∈Landv∈m(L)satisfyhu, J vi 6= 0, then

hu, J vi=hΨ(θ)u, JΨ(θ)vi=e^{2iπhmL−mM(L),θi}hu, J vi so mL=mM(L).

We will now define the periodicity properties.

Definition:Let

L

be a decomposition ofCⁿ. We say thatF ∈C⁰(2T^d, gl(n,R)) has nice periodicity properties with respect to

L

if there exists a mapΦwhich is trivial with respect to

L

and such thatΦ⁻¹FΦis continuous onT^d.

To make the family (mL)explicit, we say thatF has nice periodicity prop- erties with respect to

L

^and(mL).

Remark:

– If F ∈C⁰(2T^d, gl(n,R)) has nice periodicity properties with respect to a decomposition

L

^andΦ is trivial with respect to

L

^{, thenΦFΦ−1 has}

nice periodicity properties with respect to

L

^.

– If

L

⁰ is a decomposition ofCⁿwhich is finer than

L

^andF has nice peri- odicity properties with respect to

L

^,thenFhas nice periodicity properties with respect to

L

^0.

– Let

L

be a decomposition of Cⁿ and (mL)_L∈L be a family of elements of ¹₂Z^d. IfF1, F2∈C⁰(2T^d, gl(n,R))have nice periodicity properties with respect to

L

^and(mL), then the productF₁F₂has nice periodicity prop- erties with respect to

L

^and(mL).

2.2. Removing the resonances. — In the following we will have to solve a homo- logical equation and estimate the solution on a neighbourhood of the torus; in order to have a sufficient estimate on the solution of the homological equation, one will assume that the coefficients of the equation satisfy some diophantine conditions:

LetA∈gl(n,R)and0< κ⁰<1. LetN ∈N.

Definition:Letz∈C, ν ∈ {1,2}. We say thatz isdiophantine moduloν with respect to ω, with constant κ⁰, exponentτ and orderN if for every m∈ _ν¹Z^d such that0<|m| ≤N,

(13) |z−2iπhm, ωi| ≥ κ⁰

|m|^τ. This property will be denoted by

(14) z∈DC_ω,ν^N (κ⁰, τ)

Note that

(15) DC_ω,2^N (κ⁰, τ)⊂DC_ω,1^N (κ⁰, τ)

and that every real number zis in DC_ω,2^N (₂^κτ, τ)since for allm∈ ¹₂Z^d, (16) |z−2iπhm, ωi|= |z|²+ (2π|hm, ωi|)²¹2 ≥ πκ

|2m|^τ ≥ κ

|2m|^τ.

Remark:In the definition above, the condition is required only for non van- ishing m, so (13) has a meaning.

Definition:Ais said to haveDC_ω^N(κ⁰, τ)spectrum if (17)

( ∀α, β∈σ(A), α−β ∈DC_ω,1^N (κ⁰, τ)

∀α, β∈σ(A), α6= ¯β⇒α−β ∈DC_ω,2^N (κ⁰, τ).

LetN ∈N. LetA in a Lie algebra

G

. The aim is to show that there exists κ⁰>0,A˜∈

G

^{such thatA˜hasDC}ω^N(κ⁰, τ)spectrum andAandA˜are conjugate

(in the sense of cocycles, following the definition given in the introduction). To achieve this, one has to find a family (m1, . . . , mn)satisfying

(18)

( ∀ α_j, α_k∈σ(A), α_j−α_k+ 2iπhm_j−m_k, ωi ∈DC_ω,1^N (κ⁰, τ)

∀ αj, αk∈σ(A), αj 6= ¯αk ⇒αj−αk+ 2iπhmj−mk, ωi ∈DC_ω,2^N (κ⁰, τ) We shall construct the so-called map of reduction of the eigenvaluesΦcon- jugating (in the sense of cocycles) A to the matrix obtained from A by sub- stituting an eigenvalue αj by αj + 2iπhmj, ωi, then we will prove that Φ is G-valued.

2.2.1. Diophantine conditions. —

Lemma 2.4. — Let {α1, . . . , αn} ⊂ C. Let N˜ ∈ N and κ⁰ ≤ ^κ

n(8 ˜N)^τ. There existsm1, . . . , mn ∈¹₂Z^d such that sup_j|mj| ≤N, and such that letting for all˜ j,α˜_j=α_j−2iπhmj, ωi, then

(19) {α1, . . . , αn}={α1, . . . , αn} ⇒ ∀j, k, αj= ¯αk ⇒mj =−mk, (20) n= 2, α₂=−α1⇒m₁=−m2,

(21) ∀j, k, αj=−¯αk ⇒mj =mk, (22) ∀j, k, |α_j−α_k| ≤κ⁰ ⇒m_j=m_k, (23) ∀j, |Im ˜αj| ≤ |Imαj|,

(24) ∀j, k, α_j= ¯α_k ⇒α˜_j−α˜_k∈DC_ω,1^N˜ (κ⁰, τ) and

(25) ∀j, k, αj6= ¯α_k ⇒α˜_j−α˜_k∈DC_ω,2^N˜ (κ⁰, τ) and such that if not all m_j vanish, then there existj, k such that (26) |αj−αk| ≥κ⁰, |α˜j−α˜k|< κ⁰.

Moreover, there exist m1, . . . mn ∈ Z^d, with |mj| ≤ N˜ for all j, fulfilling conditions (21),(22),(23), such that

(27) ∀j, k, α˜j−α˜k∈DC_ω,1^N˜ (κ⁰, τ)

and such that if not all mj vanish, then there existj, k such that (26) holds.

Proof. — We shall proceed in two steps. The first step consists in removing resonances which might occur between two eigenvalues whose imaginary parts are nearly opposite to each other. Once this first lot of resonances is removed, the second step consists in removing the resonances which might occur between two eigenvalues whose imaginary parts are far from opposite.

• Let1≤j≤n. Suppose that there is anm∈Z^d,0<|m|≤N˜ such that

|2 Imα_j−2πhm, ωi |< κ⁰

|m|^τ

then letα⁰_j =αj−2iπh^m₂, ωi. Otherwise, letα⁰_j =αj. Note that if|αj−αk| ≤κ⁰ and if there exist m_j6=m_k such that

|2 Imαj−2πhmj, ωi |< κ⁰

|mj |^τ; |2 Imαk−2πhmk, ωi |< κ⁰

|mk|^τ then

|2iπhm_j−m_k, ωi | ≤ κ

|mj−mk|^τ

which is impossible sinceωis diophantine. Therefore conditions (19) to (24) hold with α⁰_j = ˜αj andmj such thatαj−α⁰_j = 2iπhmj, ωi.

• LetI_−r, . . . , I_rbe the finest partition of {1, . . . , n}such that

|Im(α⁰_j−α⁰_k)|≤κ⁰⇒ ∃ −r≤r⁰ ≤r|j, k∈Ir⁰

and choose the indices in such a way that

r⁰ < r⁰⁰⇒ ∀j∈Ir⁰,∀k∈Ir⁰⁰,Imα⁰_j≤Imα⁰_k.

Note that I0 might be empty. We will proceed by induction on r⁰ to prove the following property

P

^(r0):

. — There are m⁰₁, m⁰₋₁, . . . , m⁰_r0, m⁰_−r0 ∈Z^d withsup_|j|≤r0|m⁰_j| ≤N˜ such that properties (19)to (25) hold for all −r⁰ ≤r₁, r₂ ≤r⁰, j ∈I_r1, k ∈ I_r2 with m⁰_j instead of mj andα⁰_j instead ofαj.

• Case r⁰ = 0: if I0 is empty, then

P

⁽⁰⁾ trivially holds. Assume I0 is non empty. Then for all j, k∈I0 and allm∈ ¹₂Z^d such that0<|m|≤N,˜

|α⁰_j−α⁰_k−2iπhm, ωi |≥|Im(α⁰_j−α⁰_k)−2πhm, ωi |≥ κ

|m|^τ −nκ⁰≥κ⁰ so α⁰_j−α⁰_k ∈DC_ω,2^N˜ (κ⁰, τ)and

P

⁽⁰⁾holds true.

• Letr⁰ ≤r−1. Assume

P

^(r0)holds. ConsiderIr⁰+1 andI−r⁰−1. There are two possible cases.

– There exist −r⁰ ≤ r⁰⁰ ≤ r⁰, j ∈ Ir⁰⁰, k ∈ Ir⁰+1 and m ∈ Z^d such that

|m|≤N˜ and

|α⁰_j−α⁰_k−2iπhmr⁰⁰+m, ωi |< κ⁰

|m|^τ – The case above does not hold.

In the first case, letm⁰_r0+1=m=−m⁰_−r0−1. In the second case, letm⁰_r0+1= m⁰_−r0−1= 0.

Nowm⁰_r0+1andm⁰_−r0−1are independent fromj, k. To see this, suppose there arej₁, j₂∈I_r1, k₁, k₂∈I_r2, m₁6=m₂∈Z^d such that forl= 1,2,

|α⁰_jl−α⁰_kl−2iπhml, ωi |< κ⁰

|ml|^τ. Then

|2πhm1−m2, ωi |≤ κ

|m1−m2|^τ which is impossible. Therefore

P

^{(r0+ 1)} holds true.

• Oncem⁰₁, . . . , m⁰_r, m⁰₋₁, . . . , m⁰_−r∈Z^d are defined, conditions (19) to (25) hold with, for all j ∈Ir⁰,α˜j =α⁰_j−2iπhm⁰_r0, ωiand mj such thatαj−α˜j = 2iπhmj, ωi. Condition (26) is obvious by construction.

• By proceeding only with the second step, one gets m₁, . . . m_n ∈Z^d, with

|m_j| ≤N˜ for allj, satisfying conditions (21), (22), (23), such that

∀j, k, α˜_j−α˜_k∈DC_ω,1^N˜ (κ⁰, τ)

and such that if not all mj vanish, then there are j, k such that (26) holds true.

Lemma 2.5. — Let {α1, . . . , αn} ⊂ C. For every R, N ∈ N, N ≥ 2, R ≥ 1, there existsN¯ ∈[N, R^12n(n−1)N]andm₁, . . . , m_n∈ ¹₂Z^d with

(28) sup

j

|mj| ≤2 ¯N such that letting α˜j =αj−2iπhmj, ωiand

(29) κ⁰⁰= κ

n(8R^12n(n−1)+1N)^τ

conditions (19)to (23)of Lemma2.4hold forκ⁰=κ⁰⁰, and such that (30) ∀j, k, α˜_j−α˜_k ∈DC_ω,1^RN¯(κ⁰⁰, τ)

and

(31) ∀j, k, αj 6= ¯αk⇒α˜j−α˜k ∈DC_ω,2^RN¯(κ⁰⁰, τ).

Moreover, there exist m1, . . . mn ∈ Z^d with |mj| ≤ N¯ for all j such that conditions (21),(22),(23)and (30)hold true.

Proof. — Ifαj satisfy for allj, k (32)

(α_j= ¯α_k⇒α_j−α_k∈DC_ω,1^RN(κ⁰⁰, τ) αj6= ¯αk⇒αj−αk∈DC_ω,2^RN(κ⁰⁰, τ) then we are done withN¯ =N andm₁=· · ·=m_n = 0.

Suppose (32) does not hold. Then apply Lemma2.4withN˜ =RN, κ⁰ =κ⁰⁰ to getm_1,1, . . . , m_n,1 such that

(33)















∀j, k, αj= ¯αk ⇒mj,1=−mk,1

∀j, k, α_j=−¯α_k ⇒m_j,1=m_k,1

∀j, k, |αj−αk| ≤κ⁰⁰⇒mj,1=mk,1

∀j, |Imαj−2iπhmj,1, ωi| ≤ |Imαj| and

(34)

(αj= ¯αk ⇒αj−αk−2iπhmj,1−mk,1, ωi ∈DC_ω,1^RN(κ⁰⁰, τ) α_j6= ¯α_k ⇒α_j−α_k−2iπhm_j,1−m_k,1, ωi ∈DC_ω,2^RN(κ⁰⁰, τ)

and such that there exist j1, k1 satisfying | Im(αj₁ − αk₁)− 2iπhmj₁,1 − m_k1,1, ωi |< κ⁰⁰.

Assume there arem1,s, . . . , mn,ssuch thatsup|mj,s| ≤(R+R²+· · ·+R^s)N and that for allj, k,

(35)















∀j, k, αj= ¯α_k⇒m_j,s=−mk,s

∀j, k, αj=−¯αk ⇒mj,s=mk,s

∀j, k, |α_j−α_k| ≤κ⁰⁰⇒m_j,s=m_k,s

∀j, |Imαj−2iπhmj,s, ωi| ≤ |Imαj| and

(36)

(α_j = ¯α_k⇒α_j−α_k−2iπhmj,s−m_k,s, ωi ∈DC_ω,1^RsN(κ⁰⁰, τ) αj 6= ¯αk⇒αj−αk−2iπhmj,s−mk,s, ωi ∈DC_ω,2^RsN(κ⁰⁰, τ) and suppose there exist distinct(j₁, k₁), . . . ,(j_s, k_s)such that for alll≤s, (37) |Imαj_l−Imαk_l−2iπhmj_l,s−mk_l,s, ωi |< κ⁰⁰.

If moreover one has for allj, k (38)

(α_j= ¯α_k⇒α_j−α_k−2iπhmj,s−m_k,s, ωi ∈DC_ω,1^Rs+1N(κ⁰⁰, τ) αj6= ¯αk⇒αj−αk−2iπhmj,s−mk,s, ωi ∈DC_ω,2^Rs+1N(κ⁰⁰, τ) then the process ends and one may take N¯ =R^sN andm_j =m_j,s since it is true that

(39) |mj,s|≤(R+R²+· · ·+R^s)N ≤R^sN1−_R¹s

1−_R¹ ≤2R^sN.

Otherwise, iterate once more Lemma 2.4 with N˜ = R^s+1N and αj − 2iπhmj,s, ωi in place of αj to get m1,s+1, . . . , mn,s+1 such that sup|m^s+1_j | ≤ (R+R²+· · ·+R^s+1)N and for allj, k,

(40)















∀j, k, αj= ¯αk⇒mj,s+1=−mk,s+1

∀j, k, α_j=−¯α_k ⇒m_j,s+1=m_k,s+1

∀j, k, |αj−αk| ≤κ⁰⁰⇒mj,s+1=mk,s+1

∀j, |Imα_j−2iπhm_j,s+1, ωi| ≤ |Imα_j| and

(41)

(αj= ¯αk⇒αj−αk−2iπhmj,s+1−mk,s+1, ωi ∈DC_ω,1^Rs+1N(κ⁰⁰, τ) αj6= ¯αk⇒αj−αk−2iπhmj,s+1−mk,s+1, ωi ∈DC_ω,2^Rs+1N(κ⁰⁰, τ) and that there exist distinct (j1, k1), . . . ,(js+1, ks+1) such that for all l ≤ s+ 1,

(42) |Imαj_l−Imαk_l−2iπhmj_l,s+1−mk_l,s+1, ωi |< κ⁰⁰. Therefore, for all1≤l≤s+ 1,

(43) |αj_l−αk_l−2iπhmj_l,s+1−mk_l,s+1, ωi| < κ⁰⁰.

This implies that for all m ∈ ¹₂Z^d such that 0 < |m| ≤ RN¯ and for all l,1≤l≤s+ 1,

(44) |αj_l−αk_l−2iπhmj_l,s+1−mk_l,s+1, ωi−2iπhm, ωi| ≥ κ

2^τ+1(RN)¯ ^τ−κ⁰⁰≥κ⁰⁰ so for alll≤s+ 1,

(45) αj_l−αk_l−2iπhmj_l,s+1−mk_l,s+1, ωi ∈DC_ω,2^RN¯(κ⁰⁰, τ).

Therefore, after s¯≤ ⁿ⁽ⁿ⁻¹⁾₂ steps, one gets conditions (30) and (31) with mj =mj,¯s and α˜j =αj−2iπhmj, ωi and|Imαj−2iπhmj, ωi| ≤ |Imαj|. It is true that | m_j,¯s |≤ 2 ¯N and conditions (19) to (23) of Lemma 2.4 are also satisfied.

Lemma 2.4 implies that if conditions (19) and (31) are not required, then one can getm₁, . . . m_n∈Z^d.

2.2.2. Reduction of the eigenvalue. — Now the preceding lemmas will be used to define the map of reduction of the eigenvalues Φwhich will conjugateAto a matrix withDC_ω^RN(κ⁰⁰, τ)spectrum for someκ⁰⁰, withR, N arbitrarily large andΦbounded independently ofR.

In all that follows,Gwill be a Lie group among

GL(n,C), GL(n,R), Sp(n,R), SL(2,C), SL(n,R), O(n), U(n) and

G

will be the Lie algebra associated toG.

Proposition 2.6. — Let A ∈

G

^{, R ≥ 1 and N ∈} N. There exists N¯ ∈ [N, R^12n(n−1)N]such that if

(46) κ⁰⁰= κ

n(8R^12n(n−1)+1N)^τ

then there exists a map Φ which is trivial with respect to

L

A,κ⁰⁰ andG-valued and such that

1. for all r⁰≥0, (47)

|Φ|r⁰ ≤nC0

Å1 +||A_N||

κ⁰⁰

ãn(n+1)

e^{4πN r¯ 0}, |Φ⁻¹|r⁰≤nC0

Å1 +||A_N||

κ⁰⁰

ãn(n+1)

e^{4πN r¯ 0}.

2. Let A˜ be such that

(48) ∀θ∈2T^d, ∂ωΦ(θ) =AΦ(θ)−Φ(θ) ˜A then

(49) ||A˜−A|| ≤4πN¯ andA˜ has DC_ω^RN¯(κ⁰⁰, τ)spectrum.

3. If

G

^=gl(n,C)or u(n),Φis defined on T^d. 4. If

G

^{=o(n)oru(n), then}

(50) |Φ|r⁰ ≤ne^{4πN r¯ 0}, |Φ⁻¹|r⁰ ≤ne^{4πN r¯ 0}.

5. If

G

^=sl(2,C)or sl(2,R), then either Φis the identity or||A˜||≤κ⁰⁰. Proof. — Let{α1, . . . , αn}=σ(A). Two cases must be considered:

– If

G

^=gl(n,C)oru(n), Lemma2.5givesN¯ andmj∈Z^d forj= 1, . . . , n such that

N ≤N¯ ≤R^12n(n−1)N; sup

j

|mj| ≤2 ¯N

and such that conditions (21) to (23) of Lemma2.4 hold withκ⁰ =κ⁰⁰, as well as conditions (30).

– If

G

^{= gl(n,}R), sp(n,R), sl(n,R), sl(2,C) or o(n), Lemma 2.5 gives N¯ andmj∈ ¹₂Z^d forj = 1, . . . , nsuch that

N ≤N¯ ≤R^12n(n−1)N; sup

j

|mj| ≤2 ¯N

and such that conditions (19) to (23) of Lemma2.4 hold withκ⁰ =κ⁰⁰, as well as conditions (30) and (31).