Legendrian contact homology and topological entropy

TL;DR

This paper links the exponential growth of a specialized Legendrian contact homology to positive topological entropy in Reeb flows on 3-manifolds, revealing exponential orbit growth for all Reeb flows.

Contribution

It establishes a new connection between Legendrian contact homology growth and dynamical complexity of Reeb flows, extending to degenerate cases.

Findings

Exponential homotopical growth of Legendrian contact homology implies positive topological entropy.

All Reeb flows on certain contact manifolds have exponentially growing hyperbolic periodic orbits.

Existence of infinitely many contact structures with Legendrian knots exhibiting exponential homology growth.

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.