Optimal linearization of vector fields on the torus in non-analytic Gevrey classes
Bounemoura, Abed (2022), Optimal linearization of vector fields on the torus in non-analytic Gevrey classes, Annales de l'Institut Henri Poincaré (C) Analyse non linéaire, 39, 3, p. 501–528. 10.4171/AIHPC/12
TypeArticle accepté pour publication ou publié
Journal nameAnnales de l'Institut Henri Poincaré (C) Analyse non linéaire
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)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..
Subjects / KeywordsGevrey classes
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