Seminar 1.31.19 Chen | Ergodic Theory at Ohio State

Title: $C^{r}$ closing lemma for geodesic flows on Finsler surfaces

Speaker: Dong Chen (Ohio State University)

Abstract: A Finsler metric on a smooth manifold is a smooth family of quadratically convex norms on each tangent space. The geodesic flow on a Finsler manifold is a 2-homogeneous Lagrangian flow. In this talk, I will give a proof of the $C^{r} (r \geq 2)$ closing lemma for geodesic flows on Finsler surfaces.

The $C^{r}$ closing lemma says that for any compact smooth Finsler surface and any vector $v$ in the unit tangent bundle, the Finsler metric can be perturbed in $C^{r}$ topology so that $v$ is tangent to a periodic geodesic in the resulting metric. This allows us to get the density of periodic geodesics in the tangent bundle of a $C^{r}$ generic Finsler surface.