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# Surfaces of section for geodesic flows of closed surfaces

Abstract : We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma$ that we construct are either Birkhoff sections, meaning that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partial\Sigma$ as limit sets of the orbits of the geodesic flow that do not return to $\Sigma$. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on Grayson's curve shortening flow.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal-ens-lyon.archives-ouvertes.fr/ensl-03816282
Contributor : Marco Mazzucchelli Connect in order to contact the contributor
Submitted on : Sunday, October 16, 2022 - 3:17:15 AM
Last modification on : Monday, October 17, 2022 - 3:09:33 AM

### Identifiers

• HAL Id : ensl-03816282, version 1
• ARXIV : 2204.11977

### Citation

Gonzalo Contreras, Gerhard Knieper, Marco Mazzucchelli, Benjamin H. Schulz. Surfaces of section for geodesic flows of closed surfaces. {date}. ⟨ensl-03816282⟩

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