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For perturbations of integrable Hamiltonians systems, the Nekhoroshev theoremshows that all solutions are stable for an exponentially long interval of time,provided the integrable part satisfies a steepness condition and the system isanalytic. This fundamental result has been extended in two distinct directions.The first one is due to Niederman, who showed that under the analyticityassumption, the result holds true for a prevalent class of integrable systemswhich is much wider than the steep systems. The second one is due toMarco-Sauzin but it is limited to quasi-convex integrable systems, for whichthey showed exponential stability if the system is assumed to be only Gevreyregular. If the system is finitely differentiable, the author showed polynomialstability, still in the quasi-convex case. The goal of this work is togeneralize all these results in a unified way, by proving exponential orpolynomial stability for Gevrey or finitely differentiable perturbations ofprevalent integrable Hamiltonian systems.

Effective stability for Gevrey and finitely differentiable prevalent Hamiltonians