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

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.

23 pages, 2 figures