Existence of closed geodesics through a regular point on translation surfaces

TL;DR

This paper proves that on any translation surface, points on simple closed geodesics are dense in directions, and the set of points not on any such geodesic is finite, with exceptions in hyperelliptic cases.

Contribution

It establishes the density of directions of simple closed geodesics through regular points and characterizes the finite exceptional set, using recent classifications and orbit closure results.

Findings

Points on simple closed geodesics have dense directions.

The set of points not on any simple closed geodesic is finite.

In hyperelliptic components, this exceptional set is empty.

Abstract

We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. 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.