Generic super-exponential stability of elliptic equilibrium positions for symplectic vector fields

Abstract

In this article, we consider linearly stable elliptic fixed points (equilibrium) for a symplectic vector field and we prove generic results of super-exponential stability for nearby solutions. We will focus on the neighbourhood of elliptic fixed points but the case of linearly stable isotropic reducible invariant tori in an Hamiltonian system should be similar.More specifically, Morbidelli and Giorgilli have proved a result 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 the invariant torus or the elliptic fixed point belongs to a generic set among the formal series. This is not sufficient to prove this kind of super-exponential stability results in a general setting. We should also establish that most strongly non resonant elliptic fixed point or invariant torus 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 in another setting.