On the stability of the set of hyperbolic closed orbits of a Hamiltonian

Abstract

Abstract Let H be a Hamiltonian, e ∈ H ( M ) ⊂ ℝ and Ɛ H, e a connected component of H −1 ({ e }) without singularities. A Hamiltonian system, say a triple ( H , e , Ɛ H, e ), is Anosov if Ɛ H, e is uniformly hyperbolic. The Hamiltonian system ( H , e , Ɛ H, e ) is a Hamiltonian star system if all the closed orbits of Ɛ H, e are hyperbolic and the same holds for a connected component of −1 ({ẽ}), close to Ɛ H, e , for any Hamiltonian , in some C 2 -neighbourhood of H , and ẽ in some neighbourhood of e . In this paper we show that a Hamiltonian star system, defined on a four-dimensional symplectic manifold, is Anosov. We also prove the stability conjecture for Hamiltonian systems on a four-dimensional symplectic manifold. Moreover, we prove the openness and the structural stability of Anosov Hamiltonian systems defined on a 2 d -dimensional manifold, d ≥ 2.