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

2020/06/10 by Abed Bounemoura, Bounemoura, Abed · 1 citation
Mathematics · #Holomorphic and Operator Theory #Mathematical functions and polynomials #Analytic and geometric function theory

paper · doi:10.48550/arxiv.2006.05673

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, Rüssmann and Pöschel.

Cited by

• Brjuno condition through best approximations and the linearization problem

Related

• Linearization of analytic and non--analytic germs of diffeomorphisms of (\mathbb C,0)
• Renormalization of Gevrey vector fields with a Brjuno type arithmetical condition
• Brjuno condition through best approximations and the linearization problem
• Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part
• Towards sharp regularity: Full-dimensional tori in C^∞ vector fields over \mathbbT^∞