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

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.

25 pages, 9 figures, version 2: new title; the results are now extended beyond geodesic flows, to general Reeb flows satisfying the Kupka-Smale condition

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