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Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds

Part of: Symplectic geometry, contact geometry Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  17 June 2016

Viktor L. Ginzburg  and
Başak Z. Gürel
Viktor L. Ginzburg
Affiliation:
Department of Mathematics, UC Santa Cruz, Santa Cruz, CA 95064, USA email ginzburg@ucsc.edu
Başak Z. Gürel
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA email basak.gurel@ucf.edu
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Abstract

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We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

Keywords

non-contractible periodic orbitsHamiltonian flowsFloer homologyaugmented action

MSC classification

Primary: 53D40: Floer homology and cohomology, symplectic aspects 37J10: Symplectic mappings, fixed points
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 9 , September 2016 , pp. 1777 - 1799
Copyright
© The Authors 2016 

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