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Multiplicity of periodic orbits for dynamically convex contact forms

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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.

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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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Authors and Affiliations

  1. Center for Mathematical Analysis, Geometry and Dynamical Systems Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Miguel Abreu

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C - Cidade Universitária - Ilha do Fundão, CEP 21941-909 Rio de Janeiro, Brazil

    Leonardo Macarini

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  1. Miguel Abreu
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  2. Leonardo Macarini
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Correspondence to Leonardo Macarini.

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This paper is dedicated to Professor Paul H. Rabinowitz.

MA was partially funded by FCT/Portugal through UID/MAT/04459/2013 and Project EXCL/MAT-GEO/0222/2012 and by CNPq/Brazil through a visiting Grant. LM was partially supported by CNPq/Brazil and by FCT/Portugal through a visiting Grant. The present work is part of the authors’ activities within BREUDS, a research partnership between European and Brazilian Research Groups in dynamical systems, supported by an FP7 International Research Staff Exchange Scheme (IRSES) Grant of the European Union.

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Abreu, M., Macarini, L. Multiplicity of periodic orbits for dynamically convex contact forms. J. Fixed Point Theory Appl. 19, 175–204 (2017). https://doi.org/10.1007/s11784-016-0348-2

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