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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hamiltonian elliptic dynamics on symplectic $ 4$-manifolds


Authors: Mário Bessa and João Lopes Dias
Journal: Proc. Amer. Math. Soc. 137 (2009), 585-592
MSC (2000): Primary 37J25, 37D30; Secondary 37C27
Published electronically: August 20, 2008
MathSciNet review: 2448579
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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
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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09578-6
PII: S 0002-9939(08)09578-6
Received by editor(s): January 23, 2008
Published electronically: August 20, 2008
Communicated by: Bryna Kra
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.