# Equidistribution of saddle connections on translation surfaces

## TL;DR

This paper proves that saddle connections on translation surfaces become uniformly distributed as their length increases, and the directions of these connections are equidistributed in angle, using ergodic theory and measure convergence techniques.

## Contribution

It establishes the equidistribution of saddle connections and their angles on translation surfaces, extending understanding of their geometric and dynamical properties.

## Key findings

- Measures from saddle connections converge to Lebesgue measure
- Angles of saddle connections are equidistributed in the circle
- Results rely on unique ergodicity of directional flows

## Abstract

Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $R\to\infty$, the weak limit of these measures exists and is equal to the Lebesgue measure on $X$. We also show that any weak limit of a subsequence of the counting measures on $S^1$ given by the angles of all saddle connections of length at most $R_n$, as $R_n\to\infty$, is in the Lebesgue measure class. The proof of the first result uses the second result, together with the result of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.