Topological entropy and orbit growth in link complements

Abstract

In this article, we exhibit certain linking properties of periodic orbits of $C^{1+\alpha}$ flows with positive topological entropy on closed 3-manifolds M. It is shown that any such flow contains a link L of periodic orbits and a horseshoe K in M\L, such that all periodic orbits in K are unique in their homotopy class in M\L (among periodic orbits in M). Moreover, the entropy of the flow can be approximated by the entropies of such horseshoes K. A version of that result for chords is obtained. Our main motivation comes from Reeb dynamics, and as an application, we address a question by Alves-Pirnapasov, and obtain that the topological entropy of a 3-dimensional, $C^{\infty}$-generic Reeb flow can be approximated by the exponential homotopical growth rates of contact homology in link complements.