Reducibility of cocycles under a Brjuno-R\"ussmann arithmetical condition

TL;DR

This paper extends the reducibility results of analytic quasi-periodic cocycles to cases with Brjuno-Rüssmann arithmetical conditions on frequency and rotation number, using a KAM method adapted from vector field linearization.

Contribution

It generalizes Eliasson's work from Diophantine to Brjuno-Rüssmann conditions, applying a KAM approach to cocycle reducibility.

Findings

Achieved reducibility under Brjuno-Rüssmann conditions

Extended KAM techniques to cocycle problems

Provided new insights into arithmetical conditions in dynamical systems

Abstract

The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasi-periodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H.Eliasson which deal with the diophantine case so as to implement a Brjuno-Russmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Poschel-Russmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.