Optimal linearization of vector fields on the torus in non-analytic Gevrey classes

TL;DR

This paper demonstrates that the optimal arithmetic condition for solving linear small divisors problems also applies to non-linear problems in Gevrey classes, using a linearization of vector fields on the torus.

Contribution

It proves that the Bruno arithmetic condition is optimal for non-linear small divisors problems in Gevrey classes, extending known results from analytic to non-analytic regularity.

Findings

Bruno condition characterizes the solvability of linear small divisors problems.

The optimality of the Bruno condition extends to non-linear problems in Gevrey classes.

The proof employs Moser's approximation method and builds on works by Popov, Rüssmann, and Pöschel.

Abstract

We study linear and non-linear small divisors problems in analytic and non-analytic regularity. We observe that the Bruno arithmetic condition, which is usually attached to non-linear analytic problems, can also be characterized as the optimal condition to solve the linear problem in some fixed non quasi-analytic class. Based on this observation, it is natural to conjecture that the optimal arithmetic condition for the linear problem is also optimal for non-linear small divisors problems in any reasonable non quasi-analytic classes. Our main result proves this conjecture in a representative non-linear problem, which is the linearization of vector fields on the torus, in the most representative non quasi-analytic class, which is the Gevrey class. The proof follows Moser's argument of approximation by analytic functions, and uses in an essential way works of Popov, R\"{u}ssmann and P\"{o}schel.