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Article Dans Une Revue Annales de l'Institut Henri Poincaré C, Analyse non linéaire Année : 2022

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

Résumé

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.

Dates et versions

hal-03008322 , version 1 (16-11-2020)

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Citer

Abed Bounemoura. Optimal linearization of vector fields on the torus in non-analytic Gevrey classes. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2022, 39, pp.501-528. ⟨hal-03008322⟩
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