Reducibility of ultra-differentiable quasi-periodic cocycles under an adapted arithmetic condition

TL;DR

This paper proves a reducibility result for quasi-periodic cocycles in ultra-differentiable classes under an extended arithmetic condition, advancing understanding of their behavior near elliptic matrices.

Contribution

It extends reducibility results to ultra-differentiable classes using an adapted arithmetic condition, generalizing the Brjuno-Rüssmann condition from the analytic case.

Findings

Proves reducibility under an extended arithmetic condition

Shows necessity of a weaker arithmetic condition for reducibility

Uses fibered rotation number and weighted Fourier norms in proof

Abstract

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-R{\"u}ssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition.