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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.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-03452529
Contributor : Marco Mazzucchelli Connect in order to contact the contributor
Submitted on : Friday, November 26, 2021 - 11:56:57 PM
Last modification on : Saturday, November 27, 2021 - 3:48:55 AM

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  • HAL Id : ensl-03452529, version 1
  • ARXIV : 2110.07491

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Marco Mazzucchelli, Gonzalo Contreras. Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds. 2021. ⟨ensl-03452529⟩

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