Existence of closed geodesics through a regular point on translation surfaces

Weixu Su (Fudan University)

We show that on any translation surface, every regular point is contained in either zero or infinitely many simple closed geodesics. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa's classifications of periodic points and of $\GL(2,\R)$ orbit closures in hyperelliptic components, as well as a recent result of Eskin-Filip-Wright. This work is jointed with Duc-Manh Nguyen and Huiping Pan.