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Generic properties of $3$-dimensional Reeb flows: Birkhoff sections and entropy (2202.01506v2)
Published 3 Feb 2022 in math.DS and math.SG
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.
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