In this article, we present a new approach of Nekhoroshev theory for a
generic unperturbed Hamiltonian which completely avoids small divisors
problems. The proof is an extension of a method introduced by P. Lochak which
combines averaging along periodic orbits with simultaneous Diophantine
approximation and uses geometric arguments designed by the second author to
handle generic integrable Hamiltonians. This method allows to deal with generic
non-analytic Hamiltonians and to obtain new results of generic stability around
linearly stable tori
