15 May 2004 Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles
Viktor L. Ginzburg, Başak Z. Gürel
Duke Math. J. 123(1): 1-47 (15 May 2004). DOI: 10.1215/S0012-7094-04-12311-5


The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems.

We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions with a sequence of regular levels without periodic orbits, converging to an isolated, but very degenerate, minimum.

The proof of the relative almost existence theorem hinges on the notion of the relative Hofer-Zehnder capacity and on showing that this capacity of a small neighborhood of a symplectic submanifold is finite. The latter is carried out by proving that the flow of a Hamiltonian with sufficiently large variation has a nontrivial contractible one-periodic orbit when the Hamiltonian is constant and equal to its maximum near a symplectic submanifold and supported in a neighborhood of the submanifold.


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Viktor L. Ginzburg. Başak Z. Gürel. "Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles." Duke Math. J. 123 (1) 1 - 47, 15 May 2004. https://doi.org/10.1215/S0012-7094-04-12311-5


Published: 15 May 2004
First available in Project Euclid: 13 May 2004

zbMATH: 1066.53138
MathSciNet: MR2060021
Digital Object Identifier: 10.1215/S0012-7094-04-12311-5

Primary: 53D40 37J45

Rights: Copyright © 2004 Duke University Press


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Vol.123 • No. 1 • 15 May 2004
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