Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds

TL;DR

This paper proves that Kupka-Smale Reeb flows on closed 3-manifolds always have Birkhoff sections, with implications for generic contact forms and geodesic flows on surfaces, enhancing understanding of their dynamical structure.

Contribution

It establishes the existence of Birkhoff sections for Kupka-Smale Reeb flows on closed 3-manifolds, including generic cases for contact forms and geodesic flows.

Findings

Kupka-Smale Reeb flows admit Birkhoff sections

Generic contact forms have Birkhoff sections

Geodesic flows on surfaces have Birkhoff sections

Abstract

A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section.