PRÉPUBLICATIONS DU LABORATOIRE DE PROBABILITÉS & MODÈLES ALÉATOIRES

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1 Universités de Paris 6 & Paris 7 - CNRS UMR 7599 PRÉPUBLICATIONS DU LABORATOIRE DE PROBABILITÉS & MODÈLES ALÉATOIRES 4, place Jussieu - Case Paris cedex 05

2 Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles A. AVILA & R. KRIKORIAN MARS 2004 Prépublication n o 891 Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599, Université Paris VI & Université Paris VII, 4, place Jussieu, Case 188, F Paris Cedex 05.

3 REDUCIBILITY OR NON-UNIFORM HYPERBOLICITY FOR QUASIPERIODIC SCHRÖDINGER COCYCLES ARTUR AVILA AND RAPHAËL KRIKORIAN Abstract. We show that for almost every frequency α R\Q, for every C ω potential v : R/Z R, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or non-uniformly hyperbolic. This result gives a very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrödinger operator, and allows us to complete the proof of the Aubry-André conjecture on the measure of the spectrum of the Almost Mathieu Operator. 1. Introduction A one-dimensional quasiperiodic C r -cocycle in SL2, R briefly, a C r -cocycle is a pair α, A R C r R/Z, SL2, R, viewed as a linear skew-product: 1.1 α, A : R/Z R 2 R/Z R 2 x, w x + α, Ax w. For n Z, we let A n C r R/Z, SL2, R be defined by the rule α, A n = nα, A n we will keep the dependence of A n on α implicit. Thus A 0 x = id, A n x = Ax + jα = Ax + n 1α Ax, for n 1, j=n 1 and A n x = A n x nα 1. The Lyapunov exponent of α, A is defined as Lα, A = lim ln A n x dx 0. n n R/Z We say that α, A is uniformly hyperbolic if there exists a continuous splitting E s x E u x = R 2, and C > 0, 0 < λ < 1 such that 1.4 A n x w Cλ n w, A n x w Cλ n w, w E s x, w E u x. Such splitting is automatically unique and thus invariant, that is AxE s x = E s x + α and AxE u x = E u x + α. The set of uniformly hyperbolic cocycles is open in the C 0 -topology one allows perturbations both in α and in A. Uniformly hyperbolic cocycles have a positive Lyapunov exponent. If α, A has positive Lyapunov exponent but is not uniformly hyperbolic then it will be called non-uniformly hyperbolic. We say that a C r -cocycle α, A is C r -reducible if there exists B C r R/2Z, SL2, R and A SL2, R such that 1.5 Bx + αaxbx 1 = A, x R. Date: April 18,

4 2 ARTUR AVILA AND RAPHAËL KRIKORIAN We say that α, A is C r -reducible modulo Z if one can take B C r R/Z, SL2, R. 1 We say that α R \ Q satisfies a Diophantine condition DCκ, τ, κ > 0, τ > 0 if 1.6 qα p > κ q τ, p, q Z 2, q 0. Let DC = κ>0,τ>0 DCκ, τ. It is well known that κ>0 DCκ, τ has full Lebesgue measure if τ > 1. We say that α R \ Q satisfies a recurrent Diophantine condition RDCκ, τ if there are infinitely many n > 0 such that G n {α} DCκ, τ, where {α} is the fractionary part of α and G : 0, 1 [0, 1 is the Gauss map Gx = {x 1 }. We let RDC = κ>0,τ>0 RDCκ, τ. Notice that RDCκ, τ has full Lebesgue measure as long as DCκ, τ has positive Lebesgue measure since the Gauss map dx 1+x ln 2 is ergodic with respect to the probability measure. It is possible to show that R \ RDC has Hausdorff dimension 1/2. Given v C r R/Z, R, let us consider the Schrödinger cocycle E vx S v,e x = C r R/Z, SL2, R 1 0 v is called the potential and E is called the energy. There is a fairly good comprehension about the dynamics of Schrödinger cocycles in the case of either small or large potentials: Proposition 1.1 Sorets-Spencer [SS]. Let v C ω R/Z, R be a non-constant potential, and let α R. There exists λ 0 = λ 0 v > 0 such that if λ > λ 0 then for every E R we have Lα, S λv,e > 0. Proposition 1.2 Eliasson [E] 2. Let v C ω R/Z, R, and let α DC. There exists λ 0 = λ 0 v, α such that if λ < λ 0 then for almost every E R the cocycle α, S λv,e is C ω -reducible. Remark 1.1. Sorets-Spencer s result is non-perturbative: the largeness condition λ 0 does not depend on α. On the other hand, the proof of Eliasson s result is perturbative: the smallness condition λ 0 depends in principle on α in the full measure set DC R. We will come back to this issue cf. Theorem 1.4. Remark 1.2. In general, one can not replace almost every by every in Eliasson s result above. Indeed, in [E] it is also shown that the set of energies for which α, S λv,e is not even C 0 reducible is non-empty for a generic in an appropriate topology choice of λ, v satisfying λ < λ 0 v. Those exceptional energies do have zero Lyapunov exponent. Remark 1.3. Let α DC and A C r R/Z, SL2, R, r =, ω. In this case, α, A is uniformly hyperbolic if and only if it is C r -reducible and has a positive Lyapunov exponent. Thus, there are lots of simple cocycles for which one has positive Lyapunov exponent, resp. reducibility, and indeed both at the same time: this is the case in particular for E large in the Schrödinger case. Those examples are also stable here we fix α DC and stability is with respect to perturbations of A. However, cocycles with a positive Lyapunov exponent, resp. reducible, but which are not uniformly hyperbolic do happen for a positive measure set of energies for many choices of the potential, and in particular in the situations described by the results of Sorets-Spencer this follows from [B], Theorem 12.14, resp. Eliasson. Our main result for Schrödinger cocycles aims to close the gap and describe the situation for almost every energy without largeness/smallness assumption on the potential: 1 Obviously, reducibility modulo Z is a stronger notion than plain reducibility, but in some situations one can show that both definitions are equivalent see Remark 1.5. The advantage of defining reducibility modulo 2Z is to include some special situations notably certain uniformly hyperbolic cocycles. 2 This result was originally stated for the continuous time case, but the proof also works for the discrete time case.

5 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES 3 Theorem A. Let α RDC and let v : R/Z R be a C ω potential. Then, for Lebesgue almost every E, the cocycle α, S v,e is either non-uniformly hyperbolic or C ω -reducible. For θ R, let cos 2πθ sin 2πθ 1.8 R θ =. sin 2πθ cos 2πθ Given a C r -cocycle α, A, we associate a canonical one-parameter family of C r -cocycles θ α, R θ A. Our proof of Theorem A goes through for the more general context of cocycles homotopic to the identity, with the role of the energy parameter replaced by the θ parameter. Theorem A. Let α RDC, and let A : R/Z SL2, R be C ω and homotopic to the identity 3. Then for Lebesgue almost every θ R/Z, the cocycle α, R θ A is either non-uniformly hyperbolic or C ω -reducible. Remark 1.4. Theorems A and A also hold in the smooth setting. The only modification in the proof is in the use of a KAM theoretical result of Eliasson see Theorem 2.7, which must be replaced by a smooth version. They also generalize to the case of continuous time differential equations: in this case the adaptation is straightforward. See [AK3] for a discussion of those generalizations. Remark 1.5. One can distinguish two distinct behaviors among the reducible cocycles α, A given by Theorems A and A. The first is uniformly hyperbolic behavior, see Remark 1.3. The second is totally elliptic behavior, corresponding projectively to an irrational rotation of T 2 R/Z P 1. More precisely, we call a cocycle totally elliptic if it is C r -reducible and the constant matrix A in 1.5 can be chosen to be a rotation R ρ, where 1, α, ρ are linearly independent over Q. In this case it is easy to see that the cocycle α, A is automatically C r -reducible modulo Z possibly replacing ρ by ρ + α 2. To see that almost every reducible cocycle is either uniformly hyperbolic or totally elliptic, it is enough to use Theorems 2.3 and 2.4 which are due to Johnson-Moser and Deift-Simon. Theorems A and A give a nice global picture for the theory of quasiperiodic cocycles, extending known results for cocycles taking values on certain compact groups see [K1] for the case of SU2. They fit with the Palis conjecture for general dynamical systems [Pa], and have a strong analogy with the work of Lyubich in the quadratic family [Ly], generalized in [ALM]. More importantly, reducible and non-uniformly hyperbolic systems can be efficiently described through a wide variety of methods, especially in the analytic case. With respect to reducible systems, the dynamics of the cocycle itself is of course very simple, and the use of KAM theoretical methods [DiS], [E] allowed also a good comprehension of their perturbations. With respect to non-uniformly hyperbolic systems, there has been recently lots of success in the application of subtle properties of subharmonic functions [BG], [GS], [BJ1] to obtain large deviation estimates with important consequences such as regularity properties of the Lyapunov exponent Application to Schrödinger operators. We now discuss the application of the previous results to the quasiperiodic Schrödinger operator 1.9 H v,α,x un = un un 1 + vx + αnun, u l 2 Z, where α R \ Q, x R and v : R/Z R is C ω. The properties of H v,α,x are closely connected to the properties of the family of cocycles α, S v,e, E R. Notice for instance that if u n n Z is a solution of H v,α,x u = Eu then 1.10 E vx + nα 1 un = 1 0 un 1 3 For the case of cocycles non-homotopic to the identity, see [AK1]. un + 1. un

6 4 ARTUR AVILA AND RAPHAËL KRIKORIAN Let Σ be the spectrum of H v,α,x. It is well known see [JM] that 1.11 Σ = {E R, α, S v,e is not uniformly hyperbolic}, so Σ = Σv, α does not depend on x. Let Σ sc = Σ sc α, v, x respectively, Σ ac, Σ pp be the support of the singular continuous respectively, absolutely continuous, pure point part of the spectrum of H v,α,x. It has been shown by Last-Simon [LS], Theorem 1.5 that Σ ac does not depend on x for α R \ Q there are no hypothesis on the smoothness of v beyond continuity. It is known that Σ sc and Σ pp do depend on x in general. We will also introduce some decompositions of Σ that only depend on the cocycle, and hence are independent of x. We split Σ = Σ 0 Σ + in the parts corresponding to zero Lyapunov exponent and positive Lyapunov exponent for the cocycle α, S v,e. By [BJ1], Σ 0 is closed. Let Σ r be the set of E Σ such that α, S v,e is C ω -reducible. It is easy to see that Σ r Σ 0. Notice that by the Ishii-Pastur Theorem see [I] and [P], we have Σ ac Σ 0. By Theorem A, Σ 0 \ Σ r has zero Lebesgue measure if α RDC and v C ω. One way to interpret Σ 0 \ Σ r = 0 using the Ishii-Pastur Theorem is that generalized eigenfunctions in the essential support of the absolutely continuous spectrum are very regular Bloch waves. This already gives in the particular cases under consideration strong versions of some conjectures in the literature see for instance the discussion after Theorem 7.1 in [DeS]. Analogous statements hold in the continuous time case. Another immediate application of Theorem A is a non-perturbative version of Eliasson s result stated in Proposition 1.2. It is based on the following non-perturbative result: Proposition 1.3 Bourgain-Jitomirskaya. Let α DC, v C ω. There exists λ 0 = λ 0 v > 0 only depending on the bounds of v, but not on α such that if λ < λ 0, then the spectrum of H λv,α,x is purely absolutely continuous for almost every x. Theorem 1.4. Let α RDC, v C ω. There exists λ 0 > 0 which may be taken the same as in the previous proposition such that if λ < λ 0, then α, S λv,e is reducible for almost every E. Proof. By the previous proposition, Σ ac = Σ, so Σ + =. There are several other interesting results which can be concluded easily from Theorem A and current results and techniques: 1 Zero Lebesgue measure of Σ sc for almost every frequency, 2 Persistence of absolutely continuous spectrum under perturbations of the potential, 3 Continuity of the Lebesgue measure of Σ under perturbations of the potential. Although the key ideas behind those results are quite transparent given the appropriate background, a proper treatment would take as too far from the proof of Theorem A, which is the main goal of this paper. We will thus concentrate on a particular case which provides one of the most striking applications of Theorem A. For the applications mentioned above and others, see [AK2] and [AK3] Almost Mathieu. Certainly the most studied family of potentials in the literature is vθ = λ cos 2πθ, λ > 0. In this case, H v,α,x is called the Almost Mathieu Operator. The Aubry-André conjecture on the measure of the spectrum of the Almost Mathieu Operator states that the measure of the spectrum of H λ cos 2πθ,α,x is 4 2λ for every α R \ Q, x R see [AA]. 4 There is a long story of developments around this problem, which led to several partial results [HS], [AMS], [L], [JK]. In particular, it has already been proved for every λ 2 see [JK], and for 4 The critical case λ = 2 can be traced even further back to Hofstadter [H].

7 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES 5 every α not of constant type 5 [L]. However, for α, say, the golden mean, and λ = 2, where one should prove zero Lebesgue measure of the spectrum, it was still unknown even whether the spectrum has empty interior. Using Theorem A, we can deal with the last cases which are also Problem 5 of [Si2]. Theorem 1.5. The spectrum of H λ cos 2πθ,α,x has Lebesgue measure 4 2λ for every α R \ Q. Proof. As stated above, it is enough to consider λ = 2 and α of constant type, in particular α RDC. Let Σ be the spectrum of H 2 cos 2πθ,α,x. By Corollary 2 of [BJ1], Σ + =. By Theorem A, for almost every E Σ 0, α, S 2 cos 2πθ,E is C ω -reducible. Thus, it is enough to show that α, S 2 cos 2πθ,E is not C ω -reducible for every E Σ. Assume this is not the case, that is, α, S 2 cos 2πθ,E is reducible for some E Σ. To reach a contradiction, we will approximate the potential 2 cos 2πθ by λ cos 2πθ with λ > 2 close to 2. Then, by Theorem A of [E], if λ, E is sufficiently close to 2, E, either α, S λ cos 2πθ,E is uniformly hyperbolic or Lα, S λ cos 2πθ,E = 0. In particular since the spectrum depends continuously on the potential, there exists E R such that Lα, S λ cos 2πθ,E = 0. But it is well known, see [H], that the Lyapunov exponent of S λ cos 2πθ,E is bounded from below by max{ln λ 2, 0} > 0 and the result follows. Remark 1.6. Barry Simon has pointed out to us an alternative argument based on duality that shows that if α R \ Q and if E Σ = Σ2 cos 2πθ, α then the cocycle α, S 2 cos 2πθ,E is not C ω -reducible. Indeed, if α, S v,e is C ω -reducible and E Σ, then by duality there exists x R such that E is an eigenvalue for H 2 cos 2πθ,α,x, and the corresponding eigenvector decays exponentially, hence Lα, S v,e > 0 which gives a contradiction. This argument actually can be used to show that α, S v,e is not C 1 -reducible. By [GJLS], we get: Corollary 1.6. The spectrum of H 2 cos 2πθ,α,x is purely singular continuous for every α R \ Q, and for almost every x R/Z. Theorem A also gives a fairly precise dynamical picture for λ < 2 completing the spectral picture obtained by Jitomirskaya in [J]: Theorem 1.7. Let λ < 2, α RDC. For almost every E R, α, S λ cos 2πθ,E is reducible. Proof. By Corollary 2 of [BJ1], the Lyapunov exponent is zero on the spectrum. The result is now a consequence of Theorem A Outline of the proof of Theorem A. The proof has some distinct steps, and is based on a renormalization scheme. This point of view, which has already been used in the study of reducibility properties of quasiperiodic cocycles with values in SU2 and SL2, R, has proved to be very useful in the non-perturbative case see [K1], [K2]. However, the scheme we present in this paper is somehow simpler and fits better at least in the SL2, R case with the general renormalization philosophy see [S] for a very nice description of this point of view on renormalization: 1 The starting point is the theory of Kotani 6. For almost every energy E, if the Lyapunov exponent of α, S v,e is zero, then the cocycle is L 2 -conjugate to a cocycle in SO2, R. Moreover, the fibered rotation number of the cocycle is Diophantine with respect to α. The set of those energies will be precisely the set of energies for which we will be able to conclude reducibility. 5 A number α R is said to be of constant type if the coefficients of its continued fraction expansion are bounded. It follows that α is of constant type if and only if α κ>0 DCκ, 1 if and only if α κ>0 RDCκ, 1. 6 This step holds in much greater generality, namely for cocycles over ergodic transformation.

8 6 ARTUR AVILA AND RAPHAËL KRIKORIAN 2 We now consider a smooth cocycle α, A which is L 2 -conjugate to rotations. An explicit estimate allows us to control the derivatives of iterates of the cocycle restricted to certain small intervals. 3 After introducing the notion of renormalization of cocycles, we interpret item 2 as a priori bounds or precompactness for a sequence of renormalizations α nk, A nk. 4 The recurrent Diophantine condition for α allows us to take α nk uniformly Diophantine, so the limits of renormalization are cocycles ˆα, Â where ˆα satisfies a Diophantine condition. Those limits are essentially that is, modulo a constant conjugate to cocycles in SO2, R, and are trivial to analyze: they are always reducible. 5 Since limα nk, A nk is reducible, Eliasson s Theorem [E] allows us to conclude that some renormalization α nk, A nk must be reducible, provided the fibered rotation number of α nk, A nk is Diophantine with respect to α nk. 6 This last condition is actually equivalent to the fibered rotation number of α, A being Diophantine with respect to α. It is easy to see that reducibility is invariant under renormalization, so α, A is itself reducible. We conclude that for almost every E R such that Lα, S v,e = 0, the cocycle α, S v,e is reducible, which is equivalent to Theorem A by Remark 1.3. The above strategy uses α RDC in order to take good limits of renormalization. It would be interesting to try to obtain results under the weaker condition α DC by working directly with deep renormalizations without considering limits. 2. Parameter exclusion 2.1. L 2 -estimates. We say that α, A is L 2 -conjugated to a cocycle of rotations if there exists a measurable B : R/Z SL2, R such that B L 2 and 2.1 Bx + αaxbx 1 SO2, R. Theorem 2.1. Let v : R/Z R be continuous. Then for almost every E, either Lα, S v,e > 0 or S v,e is L 2 -conjugated to a cocycle of rotations. Proof. Looking at the projectivized action of α, S v,e on the upper half-plane H, one sees that the existence of a L 2 conjugacy to rotations is equivalent to the existence of a measurable invariant section 7 m +, E : R/Z H satisfying 1 R/Z Im +x,e dx <. This holds for almost every E such that Lα, S v,e = 0 by Kotani Theory, as described in [Si1] 8. It turns out that this result generalizes to the setting of Theorem A : Theorem 2.2. Let A : R/Z SL2, R be continuous. Then for almost every θ R, either Lα, R θ A > 0 or α, R θ A is L 2 -conjugated to a cocycle of rotations. The proof of this generalization is essentially the same as in the Schrödinger case. We point the reader to [AK1] for a discussion of this and further generalizations. Remark 2.1. Both theorems above are valid in a much more general setting, namely for cocycles over transformations preserving a probability measure. The requirement on the cocycle is the minimal to speak of Lyapunov exponents and Oseledets theory, namely integrability of the logarithm of the norm. 7 That is Sv,E x m + x, E = m + x + α, E. 8 This reference was pointed out to us by Hakan Eliasson.

9 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES Fibered rotation number. Besides the Lyapunov exponent, there is one important invariant associated to continuous cocycles which are homotopic to the identity. This invariant, called the fibered rotation number will be denoted by ρα, A R/Z, and was introduced in [H], [JM]. The fibered rotation number is a continuous function of α, A, where α, A varies in the space of continuous cocycles which are homotopic to the identity. Another important elementary fact is that both E ρα, S v,e and θ ρα, R θ A have non-decreasing lifts R R, and in particular, those functions have non-negative derivatives almost everywhere. The following result was proved in [JM], in the continuous time case, and in [DeS], in the discrete time case used here and where an optimal estimate is given. Theorem 2.3. Let v C 0 R/Z, R. Then for almost every E such that Lα, S v,e = 0, we have d de ρα, S v,e > 0. This result and proof also generalize to the setting of Theorem A see [AK1] for further generalizations: Theorem 2.4. Let A C 0 R/Z, SL2, R be continuous and homotopic to the identity. Then for almost every E such that Lα, R θ A = 0, we have d dθ ρα, R θa > 0. Remark 2.2. In the Schrödinger case, it is possible to show that the fibered rotation number is a surjective function of E onto [0, 1/2]. In [AS] it is also shown that NE = 1 2ρα, S v,e can be interpreted as the integrated density of states. The arithmetic properties of the fibered rotation number are also important for the analysis of cocycles α, A. Fix α R. Let us say that β R/Z is Diophantine with respect to α if there exists κ > 0, τ > 0 such that 2.2 2β kα l κ1 + k + l τ, k, l Z 2. Notice that for every α DC, the set of β R which are Diophantine with respect to α has full Lebesgue measure. By Theorems 2.3 and 2.4 we conclude: Corollary 2.5. Let α DC, v C 0 R/Z, R. Then for almost every E R such that Lα, S v,e = 0, we have that ρα, S v,e is Diophantine with respect to α. Corollary 2.6. Let α DC, A C 0 R/Z, SL2, R. Then for almost every θ R such that Lα, R θ A = 0, we have that ρα, R θ A is Diophantine with respect to α. The fibered rotation number and its arithmetic properties play a role in the following result of Eliasson [E]: Theorem 2.7. Let α, A R C ω R/Z, SL2, R. Assume that: 1 α DCκ, τ for some κ > 0, τ > 0, 2 ρα, A is Diophantine with respect to α, 3 A admits a holomorphic extension to some strip R/Z ɛ, ɛ, 4 A is sufficiently close to a constant  SL2, R: 2.3 sup Az  < δ = δκ, τ, ɛ, Â. z R/Z ɛ,ɛ Then α, A is reducible. This theorem was originally proved in the case of differential equations, but the adaptation to our setting is immediate. For further generalizations, see [AK3].

10 8 ARTUR AVILA AND RAPHAËL KRIKORIAN 3. Estimates for derivatives In this section, we will assume that α, A is L 2 -conjugated to a cocycle of rotations: there exist measurable B : R/Z SL2, R and R : R/Z SO2, R such that 3.1 x R/Z Ax = Bx + αrxbx 1 and φxdx < where we set φx = Bx 2 = Bx 1 2 here and in what follows, R 2 is supplied with the Euclidean norm and the space of real 2 2 matrices M2, R is supplied with the operator norm. We introduce the maximal function S of φ: n Sx = sup φx + kα. n 1 n k=0 Since the dynamics of x x + α is ergodic on R/Z endowed with Lebesgue measure, the Maximal Ergodic Theorem gives us the weak-type inequality 3.3 M > 0, Leb{x R/Z, Sx > M} 1 φxdx, M and for a.e x 0 R/Z the quantity Sx 0 is finite. If X GL2, R, we let AdX be the linear operator in M2, R which is given by AdX Y = X Y X 1. Notice that the operator norm of AdX satisfies the bound AdX X X 1. Lemma 3.1. Assume that A is Lipschitz with constant LipA. Then for every x 0, x R/Z such that Sx 0 <, we have 3.4 A n x 0 1 A n x A n x 0 e n x x0 A C 0 LipAφx0Sx0 1, and in particular 3.5 A n x e n x x0 A C 0 LipASx0φx0 φx 0 φx 0 + nα 1/2. Proof. We compute I n x 0, x := A n x 0 1 A n x A n x 0 : 0 I n x 0, x = A n x 0 1 Ax0 + kα + Ax + kα Ax 0 + kα 3.6 A n x 0 where we have set = n k=n 1 r=1 0 i r<...<i 1 n 1 j=1 R/Z R/Z r AdA ij x 0 1 H ij x 0, x 3.7 H i x 0, x = Ax 0 + iα 1 Ax + iα Ax 0 + iα, so that 3.8 H i x 0, x A C 0LipA x x 0. The assumptions we made give 3.9 A i x 0 = A i x 0 1 Bx 0 + iα 1 Bx 0, that is 3.10 AdA i x 0 1 Bx 0 + iα 1 Bx 0 2 = φx 0 φx 0 + iα.

11 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES 9 Thus we have 3.11 I n x 0, x hence for every x R/Z, n r r=1 0 i r<...<i 1 n 1 j=1 A C 0LipA x x 0 φx 0 φx 0 + i j α n 1 = A C 0LipA x x 0 φx 0 φx 0 + kα 1 + exp k=0 n 1 k=0 A C 0LipA x x 0 φx 0 φx 0 + kα, 3.12 A n x 0 1 A n x A n x 0 e n x x0 A C 0 LipAφx0Sx0 1, which implies 3.13 A n x e n x x0 A C 0 LipAφx0Sx0 A n x 0 e n x x0 A C 0 LipAφx0Sx0 φx 0 φx 0 + nα 1/2. We now give estimates for the derivatives. Lemma 3.2. Assume that A : R/Z SL2, R is of class C k 1 k. Then for every 0 r k, and any x 0, x R/Z such that Sx 0 <, we have 3.14 r A n x C r n r φx 0 + nα 1/2 c 1 x 0 e nc2x0 x x0 r+ 1 2 r A C 0 where C is an absolute constant and c 1 x 0 = φx 0 Sx 0 A 2 C 0, 3.15 c 2 x 0 = 2Sx 0 φx 0 A C0 A C0. Proof. We compute 3.16 r A n x = r 0 k=n 1 A + kα which by Leibniz formula is a sum of n r terms of the form s r i1+1 i I i x = Ax + lα m1 Ax + i 1 α l=n 1 m2 Ax + i 2 α ms Ax + i s α x l=i 1 1 i3+1 l=i l=i s 1 Ax + lα Ax + lα Ax + lα

12 10 ARTUR AVILA AND RAPHAËL KRIKORIAN where i runs through I = {0,..., n 1} {1,...,r} and where {i 1,..., i s } = i {1,..., r} satisfy n 1 i 1 > i 2 > i s 0 and m l = #i 1 i l notice that m m s = r. Each term I i can be written 3.18 I i x = 0 l=n 1 From the previous lemma, where Ad 0 Ax + lα Ad 0 l=i p 1 0 l=i p 1 l=i Ad l=i Ad l=i s 1 Ax + lα Ax + lα Ax + lα Ax + lα Kφx 0 φx 0 + i p α 1/2, Ax + lα 0 l=i p 1 Ax + lα 3.21 K = e 2n x x0 φx0sx0 A C 0 A C 0, p=1 2 Ax + i 1 α 1 m1 Ax + i 1 α Ax + i 2 α 1 m2 Ax + i 2 α Ax + i s α 1 ms Ax + i s α. Kφx 0 φx 0 + i p α and hence we get the following bound 1/2 s 3.22 I i x Kφx 0 φx 0 + nα Kφx 0 φx 0 + i p α A C 0 mp A C 0. From this and the convexity Hadamard-Kolmogorov inequalities [Ko] 3.23 m A C 0 C A 1 m/r 0 r A m r C 0, 0 m r, we deduce using s p=1 m p = r 1/2 s 3.24 I i x Kφx 0 φx 0 + nα K s φx 0 s A s C A so that 3.25 C 0 p=1 C s K s+ 1 2 φx0 s+ 1 2 φx0 + nα 1/2 A 2s 1 C 0 r A C 0 C r K A 2 C 0φx 0 r+ 1 2 φx 0 + nα 1/2 r A C 0 r A n x I i x i I mp 1 r C 0 s φx 0 + i p α p=1 s φx 0 + i p α, p=1 r mp r A C φx i p α C r K A 2 C 0φx 0 r+ 1 2 φx 0 + nα 1/2 r A C 0 φx 0 + i 1 α φx 0 + i s α. i I

13 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES 11 But the last sum in this estimate satisfies the inequality r 3.26 φx 0 + i 1 α φx 0 + i s α φx φx 0 + n 1α n r Sx 0 r i I recall that φ 1 which implies the result. We can now conclude easily: Lemma 3.3. Assume that A : R/Z SL2, R is C k 1 k. For almost every x R/Z, there exists K > 0, such that for every d > 0 and for every n > n 0 d, if αn R/Z d n, then 3.27 r A n x K r+1 n r A C r, x x d n. Proof. Let X R/Z be the set of all x such that Sx < and which are measurable continuity points of S and φ. This means that for every ɛ > 0, x is a density point of 3.28 Y x, ɛ = S 1 Sx ɛ, Sx + ɛ φ 1 φx ɛ, φx + ɛ. It is a classical fact that X has full Lebesgue Measure. Fix x X, d > 0 and ɛ > 0. If n is sufficiently big then [ 3.29 Y x, ɛ x 2d n, x + 2d ] n If αn R/Z < d n, this implies 3.30 Y x, ɛ αn Y x, ɛ 4 ɛd. n [ x d n, x + d ] n 2 2ɛd, n and in particular, each point x [ x d n, x + n] d is at distance at most 2ɛd n of a point x 0 such that x 0 Y x, ɛ and x 0 + αn Y x, ɛ. In particular, for every δ > 0, if ɛ > 0 is sufficiently small then c 1 x 0 c 1 x + δ, c 2 x 0 c 2 x + δ where c 1 and c 2 are as in the previous lemma. The previous lemma implies that 3.31 r A n x C r n r φx 0 + nα 1/2 c 1 x 0 e c2x0n x x0 r+ 1 2 r A C 0 C r n r φx + ɛ 1/2 c 1 x + δe 2ɛdc2x +δ r+ 1 2 r A C 0. It immediately follows that for every ɛ > 0, for every n sufficiently big such that αn R/Z < d n, we have r r A n x n Cc r 1 x + ɛ A C r, x x d n. Lemma 3.4. Assume that A : R/Z SL2, R is Lipschitz. For almost every x R/Z, for every d > 0, for every ɛ > 0, if n > n 0 d, ɛ and αn R/Z d n, then the matrix Bx A n xbx 1 is ɛ close to SO2, R provided that x x d n. Proof. Let x be a measurable continuity point of S and B. By the same argument of the previous lemma, for n big enough, if αn R/Z < d n, then every x such that x x < d n is at distance at most ɛ n from some x 0 such that Sx 0 Sx < ɛ, Bx 0 Bx < ɛ and Bx 0 + nα Bx < ɛ. By 3.4, we have 3.33 A n x 0 1 A n x A n x 0 e n x x0 A C 0 LipAφx0Sx0 1 Kɛ

14 12 ARTUR AVILA AND RAPHAËL KRIKORIAN so it is enough to show that Bx A n x 0 Bx 1 is close to SO2, R. But this is clear since Bx 0 + αna n x 0 Bx 0 1 SO2, R and Bx 0, Bx 0 + nα are close to Bx. 4. Renormalization Let Ω r = R C r R, SL2, R. We will view Ω r as a subgroup of Diff r R R 2 : 4.1 α, A x, w = x + α, Ax w. A C r fibered Z 2 -action is a homomorphism Φ : Z 2 Ω r that is, Φn, m Φn, m = Φn + n, m + m. We let Λ r denote the space of C r fibered Z 2 -actions. We endow Λ r with the pointwise topology. This topology is induced from the embedding Λ r Ω r Ω r, Φ Φ1, 0, Φ0, 1. 9 Let Π 1 : R C r R, SL2, R R, Π 2 : R C r R, SL2, R C r R, SL2, R be the coordinate projections. Let also γ Φ n,m = Π 1 Φn, m R and A Φ n,m = Π 2 Φn, m C r R, SL2, R. The action Φ will be called non-degenerate if Π 1 Φ : Z 2 R is injective. Let Γ r be the set of non-degenerate actions. We let Λ r 0 be the set of Φ Λ r such that γ Φ 1,0 = 1 and γ Φ 0,1 [0, 1]. For Φ Λ r 0, we let α Φ = γ Φ 0,1. We let Γ r 0 = Γ r Λ r 0 = {Φ Λ r 0, α Φ R \ Q} Some operations. Let λ 0. Define M λ : Λ r Λ r by 4.2 M λ Φn, m = λ 1 γ Φ n,m, x A Φ n,mλx. Let x R. Define T x : Λ r Λ r by 4.3 T x Φn, m = γ Φ n,m, x A Φ n,mx + x. Let U GL2, Z. Define N U : Λ r Λ r by 4.4 N U Φn, m = Φn, m, n = U 1 m n. m The operations M, T, and N will be called rescaling, translation, and base change. Notice that M λ M λ = M λλ, T x T x = T x +x, and N U N U = N UU that is, M, T, and N are left actions of R, R and GL2, Z on Λ r. Moreover, base changes commute with translations and rescalings. Notice that C r R, SL2, R acts on Ω r by Ad B α, A = α, B + αa B 1. This action extends to an action still denoted Ad B on Λ r. We will say that Φ and Ad B Φ are C r -conjugate via B Continued fraction expansion. Let 0 < α < 1 be irrational. We will discuss some elementary facts and fix notation regarding the continued fraction expansion 4.5 α = a a Here and in what follows, spaces of C r functions such as C r R, SL2, R are always endowed with the weak topology of uniform C r -convergence on compacts. In the C ω case which is the most important for us, this means that a sequence A n converges to A if and only if for every compact K there exists a complex neighborhood V K such that the holomorphic extensions of A n are defined and converge to A uniformly on V. We recall that the weak topology is metrizable for r ω, but not even separable for r = ω.

15 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES 13 Define α n = G n α where G is the Gauss map Gx = {x 1 } { } denotes fractionary part. The coefficients a n in 4.5 are given by a n = [αn 1 1 ], where [ ] denotes integer part. We also set a 0 = 0 for convenience. Then 4.6 α n = a n+1 + Let β n = n j=0 α j. Define q0 p 4.7 Q 0 = 0 = q n 1 q 1 qn p 4.8 Q n = n = that is, p n 1 p a n+2 + an , Q n = Uα n 1 Uα 0, where 4.10 Ux = Then we have 4.11 β n = 1 n q n α p n = [x 1 ] qn 1 p n 1 q n 2 p n 2 1 q n+1 + α n+1 q n, < β n < 1. q n+1 + q n q n Renormalization. We define the renormalization operator around 0, R R 0 : Γ r 0 Γ r 0, by RΦ = M α N Uα Φ where α = α Φ and U is given by The renormalization operator around x R, R x : Γ r 0 Γ r 0 is defined by R x = Tx 1 R T x. Notice that if Φ Γ r 0 and αφ = α then α RΦ = Gα and so 4.13 R n Φ = M αn 1 N Uαn 1 M α0 N Uα0Φ = M βn 1 N 1n Q n Φ Normalized actions, relation to cocycles. An action Φ Λ r 0 will be called normalized if Φ1, 0 = 1, id. If Φ is normalized then Φ0, 1 = α, A can be viewed as a C r -cocycle, since A is automatically defined modulo Z. 10 Inversely, given a C r -cocycle α, A, α [0, 1], we associate a normalized action Φ α,a by setting 4.14 Φ α,a 1, 0 = 1, id, Φ α,a 0, 1 = α, A. Lemma 4.1. Any Φ Γ r 0 is Cr -conjugate to a normalized action. Moreover, if Φ n 1, 0 Γ r 0 converges to 1, id in Γ r 0 then one can choose a sequence of conjugacies converging to id in the C r topology Since the commutativity relation 1, id α, A = α, A 1, id is equivalent to Ax = Ax The reason we refer to sequences instead of speaking of closeness is because the C ω topology is not separable.

16 14 ARTUR AVILA AND RAPHAËL KRIKORIAN Proof. We first assume that r ω. Let Φ1, 0 = 1, A. Let B C r [0, 3/2], SL2, R be such that Bx = id, x [0, 1/2], Bx = Ax 1, x [1, 3/2]. Let us extend B to R forcing Ad B 1, id = 1, A B is still smooth after the modification. If A is C r close to id, we can select B : [0, 3/2] SL2, R to be C r close to id, and in this case B : R SL2, R is also C r close to id. Let us now assume that r = ω. Let us first deal with the case where the holomorphic extension of A is close to the identity in a definite neighborhood of R. Extend A to a real-symmetric C function A : C SL2, C which is C close to the identity and which is holomorphic on a definite neighborhood V of R. We will assume that V satisfies after shrinking 4.15 z V = z + 1 V, Rz 0, 4.16 z V = z 1 V, Rz 1, 4.17 [0, 1] [ ɛ, ɛ] V. Let B C C, SL2, C be C close to the identity, real-symmetric, and satisfying Az = Bz + 1Bz 1, z C B is obtained as in the previous case. Notice that Bz + 1 = AzBz + Az Bz, so for z V we have Bz Bz + 1 = Bz Az Bz = Bz 1 Bz. Moreover, 4.18 Bz 1 Bz < δ, z [0, 1] [ ɛ, ɛ] for some small δ. Given C : R/Z [ 1, 1] SL2, C, we let D = BC 1 and we obviously have Az = Dz + 1Dz 1. We want to choose C so that 4.19 Cz 1 Cz = Bz 1 Bz, z [0, 1] [ ɛ, ɛ], for this will assure us that 4.20 Bz 1 DzCz = Bz 1 Bz + Cz 1 Cz vanishes for z [0, 1] [ ɛ, ɛ] and also in V R [ ɛ, ɛ] this guarantees that D is holomorphic in a definite neighborhood of R, and we also want to impose that C and hence D is C 0 close to the identity. Here the smoothness requirement on C is for it to be of class W 1,1, that is, it should be continuous and have distributional derivatives in L 1. Equation 4.19 is equivalent to 4.21 Cz 1 Cz = Bz 1 Bz. To conclude, we use the following proposition: Proposition 4.2. There exists κ > 0 with the following property. Let η L R/Z [ 1, 1], sl2, R and assume that η L < κ. Then there exists C : R/Z [ 1, 1] SL2, R of class W 1,1 such that Cz 1 Cz = η and C id C 0 κ 1 η L close to the identity for z R/Z [ 1, 1]. Moreover, C is real-symmetric provided η is real-symmetric. Proof. Let W 1,1 R/Z [ 1, 1], sl2, R be the space of continuous maps a : R/Z [ 1, 1] sl2, R with integrable distributional derivatives, endowed with the natural norm. We can obtain a bounded linear map P : L R/Z [ 1, 1], sl2, C W 1,1 R/Z [ 1, 1], sl2, C which is real-symmetric and solves P = id. Indeed P can be given explicitly in terms of the Cauchy transform 4.22 P αz = 1 π R [ 1,1] αζ 1 dζ dζ = lim z ζ t π [ t,t] [ 1,1] αζ dζ dζ. z ζ Define an analytic map T : L R/Z [ 1, 1] L R/Z [ 1, 1] by T = e P e P. Then T 0 = 0, DT 0 = id. It follows that T is a diffeomorphism in a neighborhood of η = 0, so we may

17 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES 15 solve e P α e P α = η with α K η L the conclusion of the proposition. provided η is close to 0. It follows that C = e P α satisfies We may now obtain C with the required properties by taking η = B 1 B in [0, 1] [ ɛ, ɛ] and η = 0 otherwise and applying the previous proposition. This concludes the second part of the lemma in the case r = ω. This argument also works if we only assume that A is close to the identity in the C topology indeed the C 1 topology is enough, as this is all that we need to get 4.18, and gives the first part of the lemma also in this case but we obviously do not get that the holomorphic extension of the normalizing matrix is close to the identity. In order to treat the global case, we first consider B C R, SL2, R with Ax = Bx + 1Bx 1, and then approximate B in the C topology by B C ω R, SL2, R. Then B x AxB x is C close to the identity and we can apply the previous case Degree and rotation number. Let us recall two basic notions for actions which were defined in [K2]. Let Φ be a Z 2 -action. If w is a point of the usual euclidean circle S 1 R 2 C we set 4.23 f Φ n,mx, w = AΦ n,mx w A Φ n,mx w, and we define 4.24 F Φ n,m : R S 1 R S 1 x, w x + γ Φ n,m, f Φ n,mx, w If π : R S 1 is the projection πy = exp2πiy we can find a continuous lift d Φ n,m : R R R of f Φ n,mx, ww 1, that is 4.25 πy + d Φ n,mx, y = f Φ n,mx, πy. Observe that such a lift is not uniquely defined, every other lift being of the form d Φ n,mx, y + k n,m, where k n,m is a constant integer. Also, for any x, y R R we have d Φ n,mx, y + 1 = d Φ n,mx, y and thus d Φ n,mx, w can be defined for any x R, w S 1. Let e 1, e 2 be a directed basis of the Z-module Z 2 that is if e 1 = n 1, m 1, e 2 = n 2, m 2 then we assume that n 1 m 2 n 2 m 1 = 1. Then it is easy to see that the quantity 4.26 d Φ e 1 F Φ e 2 x, w + d Φ e 2 x, w d Φ e 2 F Φ e 1 x, w + d Φ e 1 x, w is independent of the choices made for the lifts, does not depend on x, w and is a constant integer. Moreover it is shown in [K2] that this integer does not depend on the chosen directed basis e 1, e 2, while the effect of a non-oriented base change is a change of sign. This is what we call the degree of the action Φ and denote it by deg Φ. Also, this integer is invariant by the operation of rescaling, translation and conjugacies that is degm λ Φ = degt x Φ = degad B Φ = degφ, and is equal, when the action is normalized, to the usual degree of the map A : R/Z SL2, R defined by Φ0, 1 = α, A. Assume now that the action Φ has degree zero. Let us denote by M the set of measures on R S 1 that project on the first factor to Lebesgue measure on R. It is not difficult to see that one can find a measure µ in M that is invariant by F Φ n,m for any n, m Z 2. Take as before e 1, e 2 a directed basis of Z 2 and define the quantity: 4.27 II = I0, γ Φ e 2 ; d Φ e 1 I0, γ Φ e 1 ; d Φ e 2,

18 16 ARTUR AVILA AND RAPHAËL KRIKORIAN where we have defined for any function h : R S 1 R and a, b R 2 the quantity 4.28 Ia, b; h = sgnb a hx, vdµx, v. [a,b] S 1 If we make other choices for the lifts of F Φ, the number we obtain just differ by the addition of an element of the module of frequency of Φ, that is the Z-module Γ Φ generated by γ Φ e 1 and γ Φ e 2 where e 1, e 2 is any basis of Z 2 the module of frequency of Φ is independent of this basis. Moreover the class of II modulo Γ Φ is invariant by conjugacy and does not depend on µ see [K2]. We shall call the element of R/Γ Φ thus obtained the fibered rotation number of the action Φ and denote it by rotφ. As defined, rotφ is only invariant under oriented base changes: the effect of a non-oriented base change is a change of sign. If σ λ : R/Γ Φ R/Γ Mλ Φ is the isomorphism of modules induced by x λ 1 x λ 0 we also have rotm λ Φ = σ λ rotφ. We shall say that an element in R/Γ Φ is Diophantine if for some representative β and some κ > 0, τ > 0 one has β kγ e1 lγ e2 κ1 + k + l τ, k, l Z 2. This definition is clearly independent of the choice of the representative and of the chosen basis κ then has to be changed. Finally, we say that the action Φ is fiberwise Diophantine if rotφ is Diophantine. This notion is stable under conjugation, translation, rescaling, and base change, so it is also stable under renormalization. The following result follows immediately from the definition of the fibered rotation number of a cocycle in [H], [JM]. Lemma 4.3. If Φ is a normalized action of degree 0 which is associated to the cocycle α, A then the fibered rotation number ρα, A is a representative of rotφ. In particular, Φ is Diophantine if and only if ρα, A is Diophantine with respect to α Reducibility. An action Φ is called constant if for every n, m Z 2, x A Φ n,mx is constant. We will say that an action Φ Λ r 0 is C r -reducible if it is C r -conjugate to a constant action. It immediately follows that reducibility is invariant under conjugation, translation, rescaling and base change. Thus reducibility is also invariant under renormalization: an action Φ Γ r 0 is C r -reducible if and only if its renormalization RΦ is C r -reducible. Moreover, reducibility of a non-degenerate normalized action Φ α,a can be interpreted in familiar terms: Lemma 4.4. Let α, A R \ Q C r R/Z, SL2, R. Then Φ α,a is C r -reducible if and only if α, A is C r -reducible. Proof. Assume that Φ α,a is reducible. Then there exists B C r R, SL2, R such that Bx + 1Bx 1 = U, Bx + αaxbx 1 = V, where U, V SL2, R commute. Write U = εe u, where u sl2, R commutes with V, and ε {1, 1}. Let B x = e xu Bx. Then B x+1b x 1 = ε id, and so B x + 2 = B x. Moreover, B x + αaxb x 1 = e αu V is a constant. Thus α, A is reducible. Assume that α, A is reducible. Thus there exists B C r R/2Z, SL2, R such that Bx + αaxbx 1 = C for some C SL2, R. Let Dx = Bx + 1Bx 1, so Dx + 2 = Dx. Then CDxC 1 = Dx + α. Assume that C is not conjugate to a rotation of angle θ = kα/2 for any k Z \ {0}. Write in Fourier series 4.30 Dx = k Z ˆDke πikx, ˆDk M2, C.

19 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES 17 Then 4.31 ˆDke πikα 1 = C ˆDkC If ˆDk 0 for some k 0 then e πikα is an eigenvalue of AdC : M2, C M2, C. This implies that C is conjugate to R θ where θ = ± kα 2, contradicting our assumption. Thus Dx = ˆD0 is a constant, and it follows that Ad B Φ α,a is a constant action. Assume that C is conjugate to a rotation of angle θ = kα/2 for some k Z \ {0}: C = UR θ U 1, U SL2, R. Let B x = UR θ/αx U 1 Bx. Then B x+2 = B x and B x+αaxb x 1 = UR θ/αx+α U 1 CUR θ/αx U 1 = id. Thus, up to changing B to B we may assume that C = id, and we can apply the previous case. We will need the following version of a well-known reducibility result: Lemma 4.5. Let Φ Γ r 0, r = ω, be C r -conjugate to a SO2, R action of degree 0. If α Φ DC then Φ is C r -conjugate to a normalized constant action. In particular, Φ is C r -reducible. Proof. We may assume that Φ is normalized, since we can always conjugate Φ1, 0 to 1, id via C r R, SO2, R: this can be done in the same way as in Lemma 4.1 it is indeed easier to proceed for the SO2, R case. Let α, A = Φ0, 1, and let φ : R R satisfy Ax = R φx. Since Φ is normalized, A is defined modulo Z, and since Φ is of degree 0, this implies that φ is defined modulo Z as well. Consider the Fourier series 4.32 φθ = ˆφke 2kπiθ, k Z and let 4.33 ψθ = where k Z\{0} ˆψke 2kπiθ, 4.34 ˆψk = ˆφk 1 e 2kπiα, k 0 so that 4.35 φx ˆφ0 = ψx ψx + α. The fact that α DC implies that 1 e 2kπiα > κk τ for some κ > 0, τ > 0. In particular ψ C r R/Z, R. Let Bx = R ψx. Then B C r R/Z, SO2, R, and we have Bx + 1Bx 1 = id, Bx + αaxbx 1 = R ˆφ0. This implies that Ad B Φ is a normalized constant action. The following is a restatement of Theorem 2.7 in the language of actions. Lemma 4.6. Let Ψ Λ ω 0 be C ω -conjugate to a normalized constant action, and let κ > 0, τ > 0 be fixed. Let Ψ n be a sequence of Diophantine actions converging to Ψ in Λ ω 0 and satisfying α n α Ψn DCκ, τ. Then Ψ n is C ω -reducible for n large enough. Proof. After performing a conjugation, we may assume that Ψ1, 0 = 1, id and Ψ0, 1 = ˆα,  where  SL2, R is a constant. By Lemma 4.1, there exists a sequence B n C ω R, SL2, R converging to id which conjugates Ψ n to a normalized cocycle Ψ n = Ad B nψ n. It follows that α n, A n Ψ n0, 1 converges to ˆα,  in the Cω -topology, so Theorem 2.7 applies and α n, A n is C ω -reducible for n large enough. This implies that Ψ n and Ψ n are C ω -reducible as well.

20 18 ARTUR AVILA AND RAPHAËL KRIKORIAN 5. A priori bounds and limits of renormalization The language of renormalization allows us to restate Lemma 3.3 as a precompactness result: Theorem 5.1 A priori bounds. Let Φ Γ r 0, r 1, be a normalized action, and assume that the cocycle α, A = Φ0, 1 is L 2 -conjugated to a cocycle of rotations. Then for almost every x R, there exists K > 0 such that for every d > 0 and for every n > n 0 d, 5.1 k A Rn x Φ 1,0 x K k+1 A C k, 0 k r, x x < d. 5.2 k A Rn x Φ 0,1 x K k+1 A C k, 0 k r, x x < d. In particular, if r = ω, then {R n x Φ} n is precompact in Λ r 0. Proof. Apply Lemma 3.3 to both α, A and to α, A 1, obtaining a full measure set of good points x. Notice that 5.3 A Rn x Φ 1,0 x = A 1 n 1 q n 1 x + β n 1 x x, 5.4 A Rn x Φ 0,1 x = A 1n q n x + β n 1 x x. Fix d we may assume d > 1. Since β n 1 < 1 q n < 1 q n 1, the estimates of Lemma 3.3 imply that for 0 k r and for x x < d, k A Rn x Φ 1,0 x β k 5.5 n 1 k A 1 n 1 q n 1 x + β n 1 x x 5.6 β n 1 q n 1 k K k+1 A C k K k+1 A C k, k A Rn x Φ 0,1 x β k n 1 k A 1n q n x + β n 1 x x β n 1 q n k K k+1 A C k K k+1 A C k notice that A C k = A 1 C k. The precompactness statement is then obvious. This result allows us to consider limits of renormalization. Those are easy to analyze due to the following simple corollary of Lemma 3.4: Theorem 5.2 Limits. Let Φ Γ Lip 0 be a normalized action, and assume that the cocycle α, A = Φ0, 1 is L 2 -conjugated to a cocycle of rotations. Then for almost every x R, any limit of R n x Φ is conjugate to an action of rotations, via a constant B SL2, R. We can now prove the following rigidity result. Theorem 5.3 Rigidity. Let α RDC, and let A : R/Z SL2, R be C ω and homotopic to the identity. If α, A is L 2 -conjugated to a cocycle of rotations, and the fibered rotation number of α, A is Diophantine with respect to α, then α, A is C ω -reducible. Proof. Let α RDCκ, τ and let n k be such that α nk DCκ, τ. Consider the renormalizations Ψ k = R n k x Φ α,a, where x is as in Theorems 5.1 and 5.2. Notice that for every k, α Ψ k DCκ, τ and Ψ k is a Diophantine action. Passing to a subsequence, we may assume that Ψ k Ψ in the C ω topology. Since DCκ, τ is compact, α Ψ = lim α nk DCκ, τ. By Theorem 5.2, Ψ is C ω -conjugate to a SO2, R action, so by Lemma 4.5, Ψ is C ω -conjugate to a normalized constant action. Thus Lemma 4.6 applies and we conclude that Ψ k is C ω -reducible for k large enough. It follows that Φ α,a is reducible, so α, A is reducible as well.

21 REDUCIBLE OR NON-UNIFORMLY HYPERBOLIC SCHRÖDINGER COCYCLES 19 Proof of Theorems A and A. We can now easily prove Theorem A. Let α RDC, v C ω R/Z, R, and let be the set of E R such that α, S v,e is L 2 -conjugated to a cocycle of rotations and the fibered rotation number of α, S v,e is Diophantine with respect to α. By Theorem 2.1 and Corollary 2.5, {E R, Lα, S v,e > 0} has full Lebesgue measure in R, and Theorem 5.3 implies that α, S v,e is C ω -reducible for all E. This shows that α, S v,e is C ω -reducible for almost every E R such that Lα, S v,e = 0. By Remark 1.3, if E R is such that Lα, S v,e > 0 then α, S v,e is either non-uniformly hyperbolic or C ω -reducible, and the result follows. This argument also works for Theorem A, using Theorem 2.2 and Corollary 2.6 instead of Theorem 2.1 and Corollary 2.5. Acknowledgements: We would like to thank Hakan Eliasson, Svetlana Jitomirskaya, Barry Simon, and Jean-Christophe Yoccoz for several discussions and suggestions. References [AA] Aubry S.; Andre, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc , [AK1] Avila, A.; Krikorian, R. Quasiperiodic SL2, R cocycles. In preparation. [AK2] Avila, A.; Krikorian, R. Some continuity properties of the measure of the spectrum of one-dimensional quasiperiodic Schrödinger operators. In preparation. [AK3] Avila, A.; Krikorian, R. Some remarks on local and semi-local results for Schrödinger cocycles. In preparation. [ALM] Avila, A.; Lyubich, M.; de Melo, W. Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math , no. 3, [AMS] Avron, J.; van Mouche, P. H. M.; Simon, B. On the measure of the spectrum for the almost Mathieu operator. Comm. Math. Phys , no. 1, [AS] Avron, Joseph; Simon, Barry Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J , no. 1, [B] Bourgain, Jean Green s function estimates for lattice Schrdinger operators and applications. pp , Ann. of Math. Stud., to appear. [BG] Bourgain, J.; Goldstein, M. On nonperturbative localization with quasi-periodic potential. Ann. of Math , no. 3, [BJ1] Bourgain, J.; Jitomirskaya, S. Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Statist. Phys , no. 5-6, [BJ2] Bourgain, J.; Jitomirskaya, S. Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math , no. 3, [DeS] Deift, P.; Simon, B. Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension. Comm. Math. Phys , no. 3, [DiS] Dinaburg, E. I.; Sinai, Ja. G. The one-dimensional Schrödinger equation with quasiperiodic potential. Funkcional. Anal. i Prilozen , no. 4, [E] Eliasson, L. H. Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Comm. Math. Phys , no. 3, [GS] Goldstein, M.; Schlag, W. Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math , no. 1, [GJLS] Gordon, A. Y.; Jitomirskaya, S.; Last, Y.; Simon, B. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math , no. 2, [HS] Helffer, B.; Sjöstrand, J. Semiclassical analysis for Harper s equation. III. Cantor structure of the spectrum. Mém. Soc. Math. France N.S. No , [H] Herman, Michael-R. Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d un théorème d Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv , no. 3, [Ho] Hofstadter D.R. Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field. Phys. Rev. B , [I] Ishii, K. Localization of eigenstates and transport phenomena in one-dimensional disordered systems. Suppl. Prog. Theor. Phys , [J] Jitomirskaya, Svetlana Ya. Metal-insulator transition for the almost Mathieu operator. Ann. of Math , no. 3,