Cylindrical contact homology and topological entropy

Alves, Marcelo

doi:10.2140/gt.2016.20.3519

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Cylindrical contact homology and topological entropy

Marcelo R R Alves

Geometry & Topology 20 (2016) 3519–3569

DOI: 10.2140/gt.2016.20.3519

Abstract

We establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We show that if a contact manifold $(M, ξ)$ admits a hypertight contact form $λ_{0}$ for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on $(M, ξ)$ has positive topological entropy. Using this result, we provide numerous new examples of contact $3$ –manifolds on which every Reeb flow has positive topological entropy.

Keywords

contact homology, Reeb flows, topological entropy, symplectic field theory

Mathematical Subject Classification 2010

Primary: 37B40, 53D35, 53D42, 37J05

References

Publication

Received: 18 August 2015

Revised: 14 November 2015

Accepted: 21 December 2015

Published: 21 December 2016

Proposed: Yasha Eliashberg

Seconded: Ciprian Manolescu, Leonid Polterovich

Authors

Marcelo R R Alves
Institut de Mathématiques Université de Neuchâtel Rue Emile-Argand 11 CH-2000 Neuchâtel Switzerland

Volume 20, issue 6 (2016)