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## Surfaces of section for geodesic flows of closed surfaces

Gonzalo Contreras
• Function : Author
Gerhard Knieper
• Function : Author
Marco Mazzucchelli
Benjamin H. Schulz
• Function : Author

#### 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.

### Dates and versions

ensl-03816282 , version 1 (16-10-2022)

### Identifiers

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

### Cite

Gonzalo Contreras, Gerhard Knieper, Marco Mazzucchelli, Benjamin H. Schulz. Surfaces of section for geodesic flows of closed surfaces. 2022. ⟨ensl-03816282⟩

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