Topological entropy and orbit growth in link complements

TL;DR

This paper explores the relationship between topological entropy and orbit growth in 3-manifold flows, showing how entropy can be approximated by the growth rates of certain horseshoes and contact homology in link complements.

Contribution

It demonstrates linking properties of periodic orbits in flows with positive topological entropy and connects these properties to Reeb dynamics and contact homology growth rates.

Findings

Existence of a link of periodic orbits and a horseshoe with unique homotopy class orbits.

Flow entropy can be approximated by horseshoe entropies.

Topological entropy of generic Reeb flows relates to contact homology growth rates.

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.