In this article, we consider solutions starting close to some linearly stable
invariant tori in an analytic Hamiltonian system and we prove results of
stability for a super-exponentially long interval of time, under generic
conditions. The proof combines classical Birkhoff normal forms and a new method
to obtain generic Nekhoroshev estimates developed by the author and L.
Niederman in another paper. We will mainly focus on the neighbourhood of
elliptic fixed points, the other cases being completely similar
