Rigidity of the Reducibility of Gevrey Quasi-periodic Cocycles on U(n)

TL;DR

This paper investigates the conditions under which Gevrey class cocycles on tori can be conjugated to constant cocycles, establishing the rigidity of reducibility in the Gevrey setting and extending results to the global case for one-dimensional tori.

Contribution

It proves that measurable conjugacies to constant cocycles in Gevrey classes can be upgraded to Gevrey smooth conjugacies for almost all constants, and addresses the global reducibility problem on the circle.

Findings

Measurable conjugacy implies Gevrey smooth conjugacy for almost all constants.

Near-constant cocycles can be conjugated to constant cocycles within the same Gevrey class.

Results extend to the global reducibility problem when the base dimension is one.

Abstract

We consider the reducibility problem of cocycles $(\alpha,A)$ on $\T^d\times U(n)$ in Gevrey classes, where $ \alpha$ is a Diophantine vector. We prove that, if a Gevrey cocycle is conjugated to a constant cocycle $(\alpha,C)$ by a suitable measurable conjugacy $(0,B)$, then for almost all $C$ it can be conjugated to $(\alpha,C)$ in the same Gevrey class, provided that $A$ is sufficiently close to a constant. If $B$ is continuous we obtain it is Gevrey smooth. We consider as well the global problem of reducibility in Gevrey classes when $d=1$.