On the Almost Reducibility Conjecture

Abstract

Avila’s Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one-frequency \(SL(2,{\mathbb{R}})\) cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schrödinger operators, with many striking consequences, allowing to give a detailed characterization of the subcritical region. Here we give a proof, completely different from Avila’s, for the important case of Schrödinger cocycles with trigonometric polynomial potentials and non-exponentially approximated frequencies, allowing, in particular, to obtain all the desired spectral consequences in this case.