Surfaces of section for geodesic flows of closed surfaces

Gonzalo Contreras , Gerhard Knieper , Marco Mazzucchelli (1) , Benjamin H. Schulz

1 CNRS - Centre National de la Recherche Scientifique

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.

Domaines: Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]

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