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Hamiltonian dynamics on convex symplectic manifolds

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Abstract

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results.

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Authors and Affiliations

  1. Mathematisches Institut der Universität München, Theresienstr. 39, 80333, München, Germany

    Urs Frauenfelder

  2. Département de Mathématiques, Université Libre de Bruxelles, CP 218, Boulevard du Triomphe, 1050, Bruxelles, Belgium

    Felix Schlenk

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  1. Urs Frauenfelder
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  2. Felix Schlenk
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Partially supported by the Swiss National Foundation.

Supported by the Swiss National Foundation and the von Roll Research Foundation.

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Frauenfelder, U., Schlenk, F. Hamiltonian dynamics on convex symplectic manifolds. Isr. J. Math. 159, 1–56 (2007). https://doi.org/10.1007/s11856-007-0037-3

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