Legendrian contact homology and topological entropy

摘要

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.