Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles

Abstract

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.

Authors

Artur Ávila

CNRS UMR 7599
Université Pierre et Marie Curie
75252 Paris
France

Raphaël Krikorian

Université Pierre et Marie Curie
75252 Paris
France