The distribution of gaps for saddle connections on the octagon

TL;DR

This paper computes the limiting gap distribution for saddle connection slopes on the octagon surface, revealing new insights into Veech surfaces with multiple cusps and their horocycle flows.

Contribution

It provides the first explicit gap distribution for a Veech surface with multiple cusps and introduces a parametrization of Poincaré sections for horocycle flows on such surfaces.

Findings

Explicit limiting gap distribution for the octagon surface.

First computation for Veech surfaces with multiple cusps.

Gap distribution is piecewise real analytic.

Abstract

We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincar\'e section for the horocycle flow on $SL(2,\mathbb{R})/SL(X,\omega)$ associated to an arbitrary Veech surface $(X, \omega)$. As a corollary, we show that the associated gap distribution is piecewise real analytic.