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A symplectic proof of a theorem of Franks

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

Published online by Cambridge University Press:  11 October 2012

Brian Collier ,
Ely Kerman ,
Benjamin M. Reiniger ,
Bolor Turmunkh  and
Andrew Zimmer
Brian Collier
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: collier3@illinois.edu)
Ely Kerman
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: ekerman@illinois.edu)
Benjamin M. Reiniger
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: reinige1@illinois.edu)
Bolor Turmunkh
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: turmunk1@illinois.edu)
Andrew Zimmer
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (email: aazimmer@umich.edu)
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Abstract

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A celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area-preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this work we re-prove Franks’ theorem under the additional assumption that the map is smooth. Our proof uses only tools from symplectic topology and thus differs significantly from previous proofs. A crucial role is played by the results of Ginzburg and Kerman concerning resonance relations for Hamiltonian diffeomorphisms.

Keywords

periodic orbitsHamiltonian flowsFloer homology

MSC classification

Secondary: 53D40: Floer homology and cohomology, symplectic aspects 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 70H12: Periodic and almost periodic solutions
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1969 - 1984
Copyright
Copyright © Foundation Compositio Mathematica 2012

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