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

• Type: Article in International Scientific Journal
• Date: 2010

Authors

Mario Bessa (Author) Other; Celia Ferreira (Author) Other; Jorge Rocha (Author) FCUP

Journal

Title: Mathematical Proceedings of the Cambridge Philosophical Society

Vol. 149

Pages: 373-383

ISSN: 0305-0041

Publisher: Cambridge University Press

Indexing

ISI Web of Knowledge - 9 Citations

Scopus - 9 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-003-3PG

DOI: 10.1017/s0305004110000253

Abstract (EN): Let H be a Hamiltonian, e is an element of H (M) subset of R and epsilon(H,e) a connected component of H(-1)({e}) without singularities. A Hamiltonian system, say a triple (H, e, epsilon(H,e)), is Anosov if epsilon(H,e) is uniformly hyperbolic. The Hamiltonian system (H, e, epsilon(H,e)) is a Hamiltonian star system if all the closed orbits of epsilon(H,e) are hyperbolic and the same holds for a connected component of (H) over tilde (-1)({(e) over tilde}), close to epsilon(H,e), for any Hamiltonian (H) over tilde, in some C(2)-neighbourhood of H, and (e) over tilde 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.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 11