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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hamiltonian elliptic dynamics on symplectic $4$-manifolds
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by Mário Bessa and João Lopes Dias PDF
Proc. Amer. Math. Soc. 137 (2009), 585-592 Request permission

Abstract:

We consider $C^2$-Hamiltonian functions on compact $4$-dimensional symplectic manifolds to study the elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set $U$ intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through $U$. Moreover, this implies that, for far from Anosov regular energy surfaces of a $C^2$-generic Hamiltonian, the elliptic closed orbits are generic.
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Additional Information
  • Mário Bessa
  • Affiliation: Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
  • MR Author ID: 804955
  • ORCID: 0000-0002-1758-2225
  • Email: bessa@fc.up.pt
  • João Lopes Dias
  • Affiliation: Departamento de Matemática, ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal
  • Email: jldias@iseg.utl.pt
  • Received by editor(s): January 23, 2008
  • Published electronically: August 20, 2008
  • Communicated by: Bryna Kra
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 585-592
  • MSC (2000): Primary 37J25, 37D30; Secondary 37C27
  • DOI: https://doi.org/10.1090/S0002-9939-08-09578-6
  • MathSciNet review: 2448579