Periodic orbits and Birkhoff sections of stable Hamiltonian structures

TL;DR

This paper investigates Reeb vector fields from stable Hamiltonian structures on 3-manifolds, classifying those with finitely many periodic orbits and establishing conditions for Birkhoff sections and broken book decompositions.

Contribution

It provides a classification of stable Hamiltonian structures with finitely many periodic orbits and offers criteria for the existence of Birkhoff sections and broken book decompositions.

Findings

Classified stable Hamiltonian structures with finitely many periodic orbits.

Established conditions for supporting broken book decompositions.

Provided criteria for the existence of Birkhoff sections.

Abstract

Stable Hamiltonian structures generalize contact forms and define a volume-preserving vector field known as the Reeb vector field. We study two aspects of Reeb vector fields defined by stable Hamiltonian structures on 3-manifolds: on one hand, we classify all the examples with finitely many periodic orbits under a non-degeneracy condition; on the other, we give sufficient conditions for the existence of a supporting broken book decomposition and for the existence of a Birkhoff section.