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

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∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document}-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document}-generic Riemannian metric on a closed surface admits a Birkhoff section.