Reducibility or non-uniform hyperbolicity for quasiperiodic Schrodinger cocycles

Artur Avila, Raphael Krikorian

Published 2003-06-26, updated 2007-04-26Version 3

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 Schrodinger 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 Schrodinger operator, and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Operator.