Cylindrical contact homology and topological entropy

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 $\left(M,\xi \right)$ 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 $\left(M,\xi \right)$ 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.