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

Abstract

AbstractLetHbe a Hamiltonian,e∈H(M) ⊂ ℝ andƐH, ea connected component ofH−1({e}) without singularities. A Hamiltonian system, say a triple (H,e,ƐH, e), is Anosov ifƐH, eis uniformly hyperbolic. The Hamiltonian system (H,e,ƐH, e) is aHamiltonian star systemif all the closed orbits ofƐH, eare hyperbolic and the same holds for a connected component of−1({ẽ}), close toƐH, e, for any Hamiltonian, in someC2-neighbourhood ofH, and ẽ in some neighbourhood ofe.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 2d-dimensional manifold,d≥ 2.