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

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0305004110000253 How to cite this article: MÁRIO BESSA, CÉLIA FERREIRA and JORGE ROCHA (2010). On the stability of the set of hyperbolic closed orbits of a Hamiltonian. Abstract Let H be a Hamiltonian, e ∈ H (M) ⊂ R and E H,e a connected component of H −1 ({e}) without singularities. A Hamiltonian system, say a triple (H, e, E H,e ), is Anosov if E H,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 ({ẽ}), close to E H,e , for any HamiltonianH , in some C 2 -neighbourhood of H , andẽ 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.

Hyperbolicity in the volume-preserving scenario

“…We want to stress that we follow the same lines as Mañé's original argument, once we have shown that periodic points must have the same index. We observe also that Bessa and Rocha have established analogous results in [11] and, together with Ferreira, in [12] for the context of incompressible and Hamiltonian flows, but in lower dimensions (three and four, respectively). We shall discuss what kind of results our arguments could prove in the context of incompressible flows in any dimension.…”

Hyperbolicity and stability for Hamiltonian flows

“…Here, we generalize the results in [5] to higher dimensions and we prove that any Hamiltonian star system defined on 2d-dimensional compact symplectic manifold is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians.…”

Shades of hyperbolicity for Hamiltonians

“…The next result was proved in [10], for n = 2, and recently generalized by the authors in [16], for n ≥ 2. It is interesting to note that the specification property implies the topologically mixing property (see Lemma 6.2).…”