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Multiplicity of periodic orbits for dynamically convex contact forms

  • Miguel Abreu
  • Leonardo MacariniEmail author
Article

Abstract

We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on \(S^n\) carries at least two prime closed geodesics, multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically convex contact forms with finitely many geometrically distinct contractible closed orbits and precise estimates of the number of even periodic orbits of perfect contact forms. We also slightly relax the hypothesis of dynamical convexity. A fundamental ingredient in our proofs is the common index jump theorem due to Y. Long and C. Zhu.

Keywords

Reeb flows periodic orbits contact homology dynamical convexity 

Mathematics Subject Classification

53D42 37J45 37J55 53D25 

Notes

Acknowledgements

We thank IMPA and IST for the warm hospitality during the preparation of this work. We are grateful to Viktor Ginzburg, Basak Gurel and Jean Gutt for useful conversations regarding this paper. Part of these results were presented by the first author at the Workshop on Conservative Dynamics and Symplectic Geometry, IMPA, Rio de Janeiro, Brazil, August 3–7, 2015 and by the second author at the Contact and Symplectic Topology Session of the AMS-EMS-SPM Meeting, Porto, Portugal, June 10–13, 2015. They thank the organizers for the opportunity to participate in such wonderful events.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Center for Mathematical Analysis, Geometry and Dynamical Systems Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  2. 2.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroCentro de Tecnologia - Bloco C - Cidade Universitária - Ilha do FundãoBrazil

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