Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles


Code(s) de Classification MSC:

Résumé: We show that for almost every frequency $\alpha \in \R \setminus \Q$, for every $C^\omega$ potential $v:\R/\Z \to \R$, and for almost every energy $E$ the corresponding quasiperiodic Schr\"odinger 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 Schr\"odinger operator, and allows us to complete the proof of the Aubry-Andr\'e conjecture on the measure of the spectrum of the Almost Mathieu Operator.

Mots Clés: Quasi-periodic Schrödinger operator ; spectrum ; Lyapunov exponents ; Floquet reducibility ; renormalization

Date: 2004-03-10

Prépublication numéro: PMA-891