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

Year: 2024
Type: article
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.
DOI: 10.4171/cmh/573
Institutions: Centre National de la Recherche Scientifique, Laboratoire de Mathématiques Jean Leray, Nantes Université, Aix-Marseille Université, Institut de Mathématiques de Marseille, RWTH Aachen University, Institut Universitaire de France, Université Grenoble Alpes
Language: English
Open Access status: diamond
Awards: Quantum topology and contact geometry, Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation, Conformally symplectic dynamics, beyond symplectic dynamics, New challenges in contact and symplectic topology, G470993797, TRR 191: Symplektische Strukturen in Geometrie, Algebra und Dynamik, Groups of homeomorphisms of manifolds
OpenAlex ID: W4221149919
FWCI: 5.395
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Cited by: 10
Related to: 10