**Title**: Cr 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 Cr(r≥2) closing lemma for geodesic flows on Finsler surfaces.

The Cr 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 Cr 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 Cr generic Finsler surface.