Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds - Archive ouverte HAL Access content directly
Journal Articles Geometric And Functional Analysis Year : 2022

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

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Marco Mazzucchelli
Gonzalo Contreras
• Function : Author

#### 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.

### Dates and versions

ensl-03452529 , version 1 (26-11-2021)

### Identifiers

• HAL Id : ensl-03452529 , version 1
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### Cite

Marco Mazzucchelli, Gonzalo Contreras. Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds. Geometric And Functional Analysis, 2022, 32 (5), pp.951-979. ⟨ensl-03452529⟩

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