Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles

Artur Ávila¹ ; Raphaël Krikorian²

¹ CNRS UMR 7599, Université Pierre et Marie Curie, 75252 Paris, France
² Université Pierre et Marie Curie, 75252 Paris, France

Annals of mathematics, Tome 164 (2006) no. 3, pp. 911-940.

Résumé

We show that for almost every frequency $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, for every $C^\omega$ potential $v:\mathbb{R}/\mathbb{Z} \to \mathbb{R}$, and for almost every energy $E$ the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic. This result gives 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.

Détail

DOI : 10.4007/annals.2006.164.911