Hamiltonian perturbation theory for ultra-differentiable functions

Some scales of spaces ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They well-suited for solving non-linear functional equations by means hard implicit function theorems. comprise Gevrey thus, as a limiting case, analytic functions. Using majorizing series, we manage characterize them in terms real sequence M M bounding the growth derivatives. In this setting, prove two fundamental results Hamiltonian perturbation theory: invariant torus theorem, where remains under assumption that its frequency satisfies some arithmetic condition which call BR_M , generalizes Bruno-Rüssmann condition; Nekhoroshev’s time depends on class pertubation, through same . Our proof uses periodic averaging, while substitute analyticity width allows us bypass smoothing. We also converse statements destruction tori existence diffusing orbits perturbations, respectively mimicking construction Bessi (in category) Marco-Sauzin non-analytic category). When space additional (we then it matching), narrow gap between hypotheses (e.g. condition) instability hypotheses, thus circumbscribing threshold. The formulas relating one hand, arithmetics robust frequencies or other bring light competition nearly integrable systems distance integrability. Due our method using regularity regularizing parameter, these closer optimal tends analyticity.