Hamiltonian Elliptic Dynamics on Symplectic 4-Manifolds

We consider C²-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 hav...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2009-02, Vol.137 (2), p.585-592
Main Authors: Bessa, Mário, Dias, João Lopes
Format: Article
Language:English
Subjects:
Differential geometry
Dynamical systems
Exact sciences and technology
General mathematics
General, history and biography
Geometry
Hyperbolic orbits
Manifolds and cell complexes
Mathematical analysis
Mathematical functions
Mathematical surfaces
Mathematical theorems
Mathematics
Ordinary differential equations
Sciences and techniques of general use
Topological theorems
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:We consider C²-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 C²-generic Hamiltonian, the elliptic closed orbits are generic.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-08-09578-6