Let H be a Hamiltonian, e ∈ H(M) ? ? and EH, e a connected component of H~1({e}) without singularities. A Hamiltonian system, say a triple (H, e, E_(H, e)), is Anosov if EH, e is uniformly hyperbolic. The Hamiltonian system (H, e, E_(H, e)) is a Hamiltonian star system if all the closed orbits of E_(H, e) are hyperbolic and the same holds for a connected component of H? ~1({?}), close to EH, e, for any Hamiltonian H?, in some C~2-neighbourhood of H, and ?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 2d-dimensional manifold, d ≥ 2.
展开▼
