Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians

A major result about perturbations of integrable Hamiltonian systems is the Nekhoroshev theorem, which gives exponential stability for all solutions provided the system is analytic and the integrable Hamiltonian is generic. In the particular but important case where the latter is quasi-convex, these exponential estimates have been generalized by Marco and Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by Lochak in the analytic case. In this paper, using the same approach, we investigate the situation where the Hamiltonian is assumed to be only finitely differentiable, for which it is known that exponential stability does not hold but nevertheless we prove estimates of polynomial stability.