Almost reducibility and absolute continuity I

TL;DR

This paper proves the Almost Reducibility Conjecture for one-frequency analytic SL(2,R) cocycles with exponentially Liouville frequencies, showing that near-constant cocycles are almost reducible regardless of frequency.

Contribution

It establishes the Almost Reducibility Conjecture for exponentially Liouville frequencies, extending previous results to a broader class of cocycles.

Findings

Almost reducibility holds for cocycles with exponentially Liouville frequencies.

All cocycles close to constant are almost reducible regardless of frequency.

Implications for the spectral analysis of one-frequency Schrödinger operators.

Abstract

We consider one-frequency analytic SL(2,R) cocycles. Our main result establishes the Almost Reducibility Conjecture in the case of exponentially Liouville frequencies. Together with our earlier work, this implies that all cocycles close to constant are almost reducible, independent of the frequency. In our forthcoming work, we discuss applications to the analysis of the absolutely continuous spectrum of one-frequency Schrodinger operators.