Birkhoff sections in 3-manifold with invariant toric foliation

TL;DR

This paper investigates the existence and construction of Birkhoff sections in 3-manifolds with invariant toric foliations, providing conditions for periodic orbits to form such sections and applying results to Hamiltonian systems.

Contribution

It establishes necessary and sufficient conditions for periodic orbits to serve as Birkhoff section boundaries in foliated 3-manifolds, with applications to toric domains and Hamiltonian energy surfaces.

Findings

Conditions for periodic orbits to be boundary orbits of Birkhoff sections

Construction methods for Birkhoff sections in invariant toric foliations

Applications to energy hypersurfaces of Hamiltonian systems

Abstract

In this paper, we study the Birkhoff sections in a 3-manifold foliated by invariant tori. We establish the necessary and sufficient conditions for various types of periodic orbits to serve as boundary orbits of a Birkhoff section. The construction relies on the dynamical behaviour of the flow combined with fundamental topological argument. As an application, we study the boundaries of toric domains and the energy hypersurfaces of separable Hamiltonian systems, providing conditions for the existence or non-existence of different types of Birkhoff sections. Additionally, we offer an alternative proof of part of the results presented in [23] and [14].