Shrinking rates of horizontal gaps for generic translation surfaces

TL;DR

This paper investigates how the angles between nearly horizontal saddle connections on translation surfaces decrease as their length increases, providing precise decay rates for their angular differences.

Contribution

It introduces new quantitative decay rates for the difference in angles between saddle connections on translation surfaces as their length tends to infinity.

Findings

Established precise decay rates for angular differences

Analyzed behavior of saddle connections on translation surfaces

Enhanced understanding of geodesic flow dynamics

Abstract

A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most $R$, we obtain precise decay rates as $R\to \infty$ for the difference in angle between two almost horizontal saddle connections.