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