  We consider linearly stable elliptic fixed points for a symplectic vector
field and prove generic results of super-exponential stability for nearby
solutions. Morbidelli and Giorgilli have proved a theorem of stability over
super-exponentially long times if one consider an analytic lagrangian torus,
invariant for an analytic hamiltonian system, with a diophantine translation
vector which admit a sign definite torsion. Then, the solutions of the system
move very little over times which are super-exponentially long with respect to
the inverse of the distance to the invariant torus. The proof is in two steps:
first the construction of a Birkhoff normal form at a high order, then the
application of Nekhoroshev theory. Bounemoura has shown that the second step of
this construction remains valid if the Birkhoff normal form linked to an
elliptic fixed point belongs to a generic set among the formal series. To
obtain a complete result, we also have to establish that most strongly non
resonant elliptic fixed point in a Hamiltonian system admit a Birkhoff normal
form in the set introduced by Bounemoura. We show here that this property is
satisfied generically in the sense of the measure (prevalence) through
infinite-dimensional probe spaces (that is an infinite number of parameter
chosen at random) with methods similar to those developed in a paper of
Gorodetski, Kaloshin and Hunt.
