Journal of Modern Dynamics (JMD)

Floer homology in disk bundles and symplectically twisted geodesic flows

Pages: 61 - 101, Issue 1, January 2009      doi:10.3934/jmd.2009.3.61

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Michael Usher - Department of Mathematics, University of Georgia, Athens, GA 30602, United States (email)

Abstract: We show that if $K: P\to\mathbb{R}$ is an autonomous Hamiltonian on a symplectic manifold $(P,\Omega)$ which attains a Morse-Bott nondegenerate minimum of 0 along a symplectic submanifold $M$ and if $c_1(TP)$↾M vanishes in real cohomology, then the Hamiltonian flow of $K$ has contractible periodic orbits with bounded period on all sufficiently small energy levels. As a special case, if the geodesic flow on T*M is twisted by a symplectic magnetic field form, then the resulting flow has contractible periodic orbits on all low energy levels. These results were proven by Ginzburg and Gürel when $\Omega$↾M is spherically rational, and our proof builds on their work; the argument involves constructing and carefully analyzing at the chain level a version of filtered Floer homology in the symplectic normal disk bundle to $M$.

Keywords:  Periodic orbits, Floer homology, symplectic submanifolds.
Mathematics Subject Classification:  53D40, 53D25, 37J10.

Received: September 2008;      Available Online: February 2009.