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.
Cite this article
Abed Bounemoura, Optimal linearization of vector fields on the torus in non-analytic Gevrey classes. Ann. Inst. H. Poincaré Anal. Non Linéaire 39 (2022), no. 3, pp. 501–528DOI 10.4171/AIHPC/12