On the almost reducibility conjecture

TL;DR

This paper proves Avila's Almost Reducibility Conjecture for Schrödinger cocycles with non-exponentially approximated frequencies, providing a new approach that yields significant spectral theory insights.

Contribution

It offers a novel proof of the Almost Reducibility Conjecture for a key case, expanding understanding of spectral properties of Schrödinger operators.

Findings

Confirmed the conjecture for non-exponentially approximated frequencies

Derived spectral consequences for Schrödinger operators

Provided a new proof method distinct from Avila's approach

Abstract

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