Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

TL;DR

This paper establishes a strong form of almost reducibility for analytic and Gevrey quasi-periodic cocycles with Diophantine frequencies, ensuring conjugacies in the same regularity class independent of initial closeness to constant.

Contribution

It generalizes previous results by proving a uniform regularity conjugacy for cocycles close to constant, enhancing the understanding of their reducibility properties.

Findings

Almost reducibility holds with conjugacies in the same regularity class.

Density of reducible cocycles near a constant is established.

Algebraic structures can be preserved, possibly with period doubling.

Abstract

This paper is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H.Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density, of reducible cocycles near a constant. Some algebraic structure can also be preserved, by doubling the period if needed.