Optimal linearization of vector fields on the torus in non-analytic Gevrey classes
Bounemoura, Abed (2020), Optimal linearization of vector fields on the torus in non-analytic Gevrey classes. https://basepub.dauphine.psl.eu/handle/123456789/22156
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-03008322
Series titleCahier de recherche du CEREMADE
MetadataShow full item record
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..
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