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Lusternik–Schnirelmann theory and closed Reeb orbits

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We develop a variant of Lusternik–Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. As an application, we prove new multiplicity results for simple closed Reeb orbits on the standard contact sphere, the unit cotangent bundle to the sphere and some other contact manifolds. We also show that the lower Conley–Zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of Long and Zhu. This is the second, combinatorial ingredient in the proof of the multiplicity results.

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Acknowledgements

The authors are grateful to Alberto Abbondandolo, Fréderic Bourgeois, River Chiang, Jean Gutt, Umberto Hryniewicz, Ely Kerman, Leonardo Macarini, Marco Mazzucchelli, Yaron Ostrover, Jeongmin Shon, Otto van Koert and the referee for useful comments, remarks and discussions. This project greatly benefited from the Workshop on Conservative Dynamics and Symplectic Geometry held at IMPA, Rio de Janeiro in August 2015. The authors would like to thank the organizers (Henrique Bursztyn, Leonardo Macarini, and Marcelo Viana) of the workshop. A part of this work was carried out while the first author was visiting the National Cheng Kung University (NCKU), Taiwan, and he would like to thank the NCKU for its warm hospitality and support.

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  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Başak Z. Gürel

  2. Department of Mathematics, UC Santa Cruz, Santa Cruz, CA, 95064, USA

    Viktor L. Ginzburg

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  1. Viktor L. Ginzburg
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Correspondence to Viktor L. Ginzburg.

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The work is partially supported by NSF CAREER award DMS-1454342 (Başak Z. Gürel) and NSF grants DMS-1414685 (Başak Z. Gürel) and DMS-1308501 (Viktor L. Ginzburg)

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Ginzburg, V.L., Gürel, B.Z. Lusternik–Schnirelmann theory and closed Reeb orbits. Math. Z. 295, 515–582 (2020). https://doi.org/10.1007/s00209-019-02361-2

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