Hamiltonian elliptic dynamics on symplectic 4-manifolds

Bessa, Mário; Dias, João

doi:10.1090/S0002-9939-08-09578-6

Hamiltonian elliptic dynamics on symplectic $4$-manifolds
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by Mário Bessa and João Lopes Dias

Proc. Amer. Math. Soc. 137 (2009), 585-592

DOI: https://doi.org/10.1090/S0002-9939-08-09578-6

Published electronically: August 20, 2008

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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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