Hamiltonian Perturbation Theory for Ultra-Differentiable Functions

Hamiltonian Perturbation Theory for Ultra-Differentiable Functions Synopsis

"Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M bounding the growth of derivatives. In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BRM, and which generalizes the Bruno-Russmann condition; and Nekhoroshev's theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M