Positive topological entropy of positive contactomorphisms

TL;DR

This paper proves that positive contactomorphisms on certain contact manifolds exhibit positive topological entropy, extending previous results from Reeb flows and providing new examples of complex dynamical behavior.

Contribution

It generalizes the link between Floer homology growth and entropy from Reeb flows to positive contactomorphisms, introducing a new isomorphism with Lagrangian Rabinowitz-Floer homology.

Findings

Positive Floer homology growth implies positive topological entropy for contactomorphisms.

Established isomorphism between wrapped Floer homology and Lagrangian Rabinowitz-Floer homology.

Identified classes of contact manifolds with inherently chaotic dynamics.

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.