Effective generic super-exponential stability of elliptic fixed points for symplectic vector fields

TL;DR

This paper proves that most elliptic fixed points in symplectic vector fields are generically super-exponentially stable over long times, using Birkhoff normal form and Nekhoroshev theory, with a measure-theoretic approach.

Contribution

It establishes that generic elliptic fixed points admit Birkhoff normal forms leading to super-exponential stability, extending previous results to a measure-theoretic generic setting.

Findings

Most strongly non-resonant elliptic fixed points admit Birkhoff normal forms.

Super-exponential stability holds for generic elliptic fixed points.

The results are valid in a measure-theoretic (prevalence) sense.

Abstract

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.