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

Published online by Cambridge University Press:  03 June 2010

MÁRIO BESSA ,
CÉLIA FERREIRA  and
JORGE ROCHA
MÁRIO BESSA
Affiliation:
Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal. ESTGOH-Instituto Politécnico de Coimbra, Rua General Santos Costa, 3400-124 Oliveira do Hospital, Portugal. e-mail: bessa@fc.up.pt
CÉLIA FERREIRA
Affiliation:
Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal. e-mail: celiam@fc.up.pt, jrocha@fc.up.pt
JORGE ROCHA
Affiliation:
Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal. e-mail: celiam@fc.up.pt, jrocha@fc.up.pt
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Abstract

Let H be a Hamiltonian, e ∈ H(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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 373 - 383
Copyright
Copyright © Cambridge Philosophical Society 2010

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