Positive topological entropy of positive contactomorphisms

Abstract

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 $L$ such that $\omega|_{\pi_2(W,L)}=0$. 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.