Positive topological entropy of positive contactomorphisms

A positive contactomorphism of a contact manifold $M$ is the end point of a contact isotopy on $M$ that is always positively transverse to the contact structure. Assume that $M$ contains a Legendrian sphere $\Lambda$, and that $(M, \Lambda)$ is fillable by a Liouville domain $(W, \omega)$ with exact Lagrangian $L$. We show that if the exponential growth of the action filtered wrapped Floer homology of $(W, L)$ is positive, then every positive contactomorphism of $M$ has positive topological entropy. This result generalizes the result of Alves and Meiwes from Reeb flows to positive contactomorphisms, and it yields many examples of contact manifolds on which every positive contactomorphism has positive topological entropy, among them the exotic contact spheres found by Alves and Meiwes.

A main step in the proof is to show that wrapped Floer homology is isomorphic to the positive part of Lagrangian Rabinowitz–Floer homology.

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This work is supported by SNF grant 200021-163419/1 and SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’ funded by the DFG.

Received 1 October 2018

Accepted 16 July 2019

Published 30 July 2020