Legendrian contact homology and topological entropy

Abstract

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold $(M,\xi)$ the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on $(M,\xi)$ has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on $(M,\xi)$ the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.