Generic properties of $3$-dimensional Reeb flows: Birkhoff sections and entropy

Abstract

In this paper we use broken book decompositions to study Reeb flows on closed $3$-manifolds. We show that if the Liouville measure of a nondegenerate contact form can be approximated by periodic orbits, then there is a Birkhoff section for the associated Reeb flow. In view of Irie's equidistribution theorem, this is shown to imply that the set of contact forms whose Reeb flows have a Birkhoff section contains an open and dense set in the $C^{\infty }$-topology. We also show that the set of contact forms whose Reeb flows have positive topological entropy is open and dense in the $C^{\infty }$-topology.

39 pages, 3 figures; v2 contains several corrections and improvements pointed out by referees