Cylindrical contact homology and topological entropy

TL;DR

This paper links the exponential growth of cylindrical contact homology in certain contact manifolds to the positivity of topological entropy in all Reeb flows on those manifolds, providing new examples of complex dynamical behavior.

Contribution

It establishes a direct relation between cylindrical contact homology growth and topological entropy, and demonstrates its implications for Reeb flows on contact 3-manifolds.

Findings

Exponential growth of cylindrical contact homology implies positive topological entropy for all Reeb flows.

Provides new examples of contact 3-manifolds with universally positive entropy Reeb flows.

Shows the relation holds under the existence of a hypertight contact form with exponential homotopical growth.

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 $(M,\xi)$ admits a hypertight contact form $\lambda_0$ for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on $(M,\xi)$ 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.