Generic Nekhoroshev theory without small divisors

Abstract

In this article, we present a new approach of Nekhoroshev's theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak, it 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.

Super-exponential stability for generic real-analytic elliptic equilibrium points

“…Even though the condition that h 2 is positive definite is open, it is far from being generic in any sense and recently some efforts have been made to improve this result, especially in [Bou11b] and [Nie13]. In [Bou11b], using results from [Nie07] and [BN12], it was proved that under a certain condition on the formal Birkhoff series h ∞ , the double exponential stability holds true. This condition, which includes the condition that h 2 is positive definite as a particular case, was proved to be prevalent (a possible generalization of "full measure" in infinite dimensional spaces) in the space of all formal series.…”

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

“…In [BN12], Nekhoroshev estimates are proved in a generic setting by means of successive periodic averagings and this method can be implemented in cartesian coordinates. These observations allowed Bounemoura [Bou11a] to prove that a Morse Diophantine condition is satisfied at all order on a full Lebesgue measure set of Birkhoff invariants which is strong enough to prove superexponential results of stability over times of order exp(exp(C/r a )).…”

“…Note also that h does not need to be polynomial, we only require that the Hamiltonian is smooth enough. Then, in order to obtain superexponential estimates of stability, we should be able to apply Nekhoroshev theory (with the proof given in [BN12]) on the Birkhoff normal forms (h m ) m∈N * at all order m with a perturbation f m which decrease geometrically.…”

Sharp Nekhoroshev estimates for the three-body problem around periodic orbits

“…A global result of stability is obtained once one covers the entire phase space with such neighborhoods with the help of Dirichlet's approximation theorem. Improvements in this second approach can be found in [5] and [25]. A brief overview on both proofs can be found in [20] and [31].…”