Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits

TL;DR

This paper proves that for almost every flat surface within any given $ ext{SL}(2, ext{R})$ orbit, the lengths of saddle connections, when ordered, are uniformly distributed in the interval from 0 to 1.

Contribution

It establishes the uniform distribution of saddle connection lengths for almost all surfaces in any $ ext{SL}(2, ext{R})$ orbit, a new result in flat surface dynamics.

Findings

Almost every surface's saddle connection lengths are uniformly distributed.

The result applies to all $ ext{SL}(2, ext{R})$ orbits.

Provides a statistical property of saddle connections in flat surfaces.

Abstract

For every flat surface, almost every flat surface in its $\mathsf{SL}(2,\mathbb{R})$ orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval.