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

Commentarii Mathematici Helvetici, vol. 99, pp. 557–611

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 non-degenerate 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

   Centre National de la Recherche Scientifique · Nantes Université · Laboratoire de Mathématiques Jean Leray

   Affiliation as printed

   Nantes Université, CNRS, LMJL, Nantes, France

2. Pierre Dehornoy

   Centre National de la Recherche Scientifique · Aix-Marseille Université · Institut de Mathématiques de Marseille

   Affiliation as printed

   Aix Marseille Université, CNRS, I2M, Marseille, France

3. Umberto L. Hryniewicz Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen, Aachen, Germany

4. Ana Rechtman

   Université Grenoble Alpes · Institut Universitaire de France

   Affiliation as printed

   Institut Universitaire de France (IUF), Paris, France

   Université Grenoble Alpes, Gières, France

   Institut Universitaire de France (IUF), Paris, France