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

arXiv (Cornell University)

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.

Authors 4

1. Vincent Colin

   Institut de Recherche Mathématique Avancée

   Affiliation as printed

   IRMA - Institut de Recherche Mathématique Avancée (7 rue René-Descartes, 67084 Strasbourg Cedex, France - France)

2. Pierre Dehornoy

   Laboratoire de Mathématiques Jean Leray

   Affiliation as printed

   LMJL - Laboratoire de Mathématiques Jean Leray (France)

3. Umberto L. Hryniewicz

   Institut Fourier

   Affiliation as printed

   IF - Institut Fourier (100, rue des maths 38610 Gières - France)

4. Ana Rechtman Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH - Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH Aachen Templergraben 55 52062 Aachen (Hausanschrift) 52056 Aachen (Postanschrift) - Germany)