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.
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Acknowledgements
We thank Vincent Colin, Pierre Dehornoy, Umberto Hryniewicz, and Ana Rechtman for a conversation concerning a subtle point in the proof of [CDR20, Corollary 3.2]. This work was completed while Marco Mazzucchelli was a visitor at the Fakultät für Mathematik of the Ruhr-Universität Bochum, Germany. He would like to thank Alberto Abbondandolo, Stefan Suhr, and Kai Zehmisch for their hospitality.
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Gonzalo Contreras is partially supported by CONACYT, Mexico, grant A1-S-10145. Marco Mazzucchelli is partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” funded by the Deutsche Forschungsgemeinschaft, and by the French Agence Nationale de la Recherche under grant ANR-21-CE40-0002 “New challenges in symplectic and contact topology”.
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Contreras, G., Mazzucchelli, M. Existence of Birkhoff sections for Kupka–Smale Reeb flows of closed contact 3-manifolds. Geom. Funct. Anal. 32, 951–979 (2022). https://doi.org/10.1007/s00039-022-00616-5
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DOI: https://doi.org/10.1007/s00039-022-00616-5