Almost reducibility of quasiperiodic sl(2, R)-cocycles in ultradifferentiable classes

TL;DR

This paper proves that quasiperiodic sl(2, R)-cocycles near a constant are almost reducible within ultradifferentiable classes, under specific arithmetic conditions, and discusses the regularity of the Lyapunov exponent.

Contribution

It establishes almost reducibility results for quasiperiodic cocycles in ultradifferentiable classes with new arithmetic conditions.

Findings

Almost reducibility in ultradifferentiable classes

Arithmetic conditions on frequency vectors

Hölder regularity of the Lyapunov exponent

Abstract

Given a quasiperiodic cocycle in sl(2, R) sufficiently close to a constant, we prove that it is almost-reducible in ultradifferentiable class under an adapted arithmetic condition on the frequency vector. We also give a corollary on the H{\"o}lder regularity of the Lyapunov exponent.