The growth and distribution of large circles on translation surfaces

TL;DR

This paper studies how large circles and closed geodesics distribute on translation surfaces, showing that as their size grows, they become uniformly distributed according to the surface's area measure.

Contribution

It establishes the distribution of large circles and closed geodesics on translation surfaces, extending understanding of geometric and dynamical properties of these surfaces.

Findings

Large circles distribute according to the area measure as radius grows.

Analogous distribution results are obtained for closed geodesics.

The results connect geometric growth with measure-theoretic distribution.

Abstract

We consider circles on a translation surface $X$, consisting of points joined to a common center point by a geodesic of length $R$. We show that as $R \to \infty$ these circles distribute to a measure on $X$ which is equivalent to the area. In the last section we consider analogous results for closed geodesics.