Closed geodesics in dilation surfaces

TL;DR

This paper proves that in dilation surfaces, the directions of closed geodesics are densely distributed on the circle, using degeneration analysis of Delaunay triangulations under Teichmüller flow.

Contribution

It establishes the density of closed geodesic directions in dilation surfaces, linking geometric properties with Teichmüller dynamics.

Findings

Closed geodesic directions are dense in the circle for all dilation surfaces.

Degeneration of Delaunay triangulations under Teichmüller flow is key to the proof.

Provides new insights into the geometric structure of dilation surfaces.

Abstract

We prove that directions of closed geodesics in every dilation surface form a dense subset of the circle. The proof draws on a study of the degenerations of the Delaunay triangulation of dilation surfaces under the action of Teichm\"{u}ller flow in the moduli space.