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


Let H be a Hamiltonian, eH(M) ⊂ ℝ and ƐH, e a connected component of H−1({e}) without singularities. A Hamiltonian system, say a triple (H, e, ƐH, e), is Anosov if ƐH, e is uniformly hyperbolic. The Hamiltonian system (H, e, ƐH, e) is a Hamiltonian star system if all the closed orbits of ƐH, e are hyperbolic and the same holds for a connected component of −1({ẽ}), close to ƐH, e, for any Hamiltonian , in some C2-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 2d-dimensional manifold, d ≥ 2.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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