Existence of closed geodesics through a regular point on translation surfaces

Abstract

We show that on any translation surface, if a regular point is contained in some simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in $${\mathbb {RP}}^1$$. 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 on some translation surfaces. 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 $$\text {GL}(2,{\mathbb {R}})$$ orbit closures in hyperelliptic components, as well as a recent result of Eskin–Filip–Wright.