Geodesic
Optimal linearization of vector fields on the torus in non-analytic Gevrey classes
Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 3, pp. 501-528 Cet article a éte moissonné depuis la source Numdam

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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 natural non-quasi-analytic class, which is the Gevrey class. The proof follows Moser’s argument of approximation by analytic functions, and uses works of Popov, Rüssmann and Pöschel in an essential way.

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DOI : 10.4171/aihpc/12
Classification : 37J25, 37J40
Keywords: Gevrey classes 
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     author = {Bounemoura, Abed},
     title = {Optimal linearization of vector fields on the torus in non-analytic {Gevrey} classes},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {501--528},
     year = {2022},
     volume = {39},
     number = {3},
     doi = {10.4171/aihpc/12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/12/}
}
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Bounemoura, Abed. Optimal linearization of vector fields on the torus in non-analytic Gevrey classes. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 3, pp. 501-528. doi: 10.4171/aihpc/12

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