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Mathematische Zeitschrift

, Volume 283, Issue 1–2, pp 197–242 | Cite as

On the existence of infinitely many invariant Reeb orbits

  • Will J. MerryEmail author
  • Kathrin Naef
Article

Abstract

In this article we extend results of Grove and Tanaka (Bull Am Math Soc 82:497–498, 1976, Acta Math 140:33–48, 1978) and Tanaka (J Differ Geom 17:171–184, 1982) on the existence of isometry-invariant geodesics to the setting of Reeb flows and strict contactomorphisms. Specifically, we prove that if M is a closed connected manifold with the property that the Betti numbers of the free loop space \( \Lambda (M)\) are asymptotically unbounded then for every fibrewise star-shaped hypersurface \(\Sigma \subset T^*M\) and every strict contactomorphism \( \varphi :\Sigma \rightarrow \Sigma \) which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits.

Notes

Acknowledgments

We are very grateful to Marco Mazzucchelli for explaining to us why studying strict contactomorphisms is interesting in this setting, and for numerous helpful discussions on generating functions, and to Viktor Ginzburg for his many detailed and useful comments, and in particular for suggesting Theorem 1.7 to us. The first author thanks Alberto Abbondandolo for many discussions about the \(L^{\infty }\)-estimates in Sect. 6.

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsETH ZürichZürichSwitzerland

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