Shrinking rates of horizontal gaps for generic translation surfaces

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.

References in corpus (7)

• Connected components of the moduli spaces of Abelian differentials with prescribed singularities
• The distribution of gaps for saddle connections on the octagon
• A short introduction to translation surfaces, Veech surfaces, and Teichmueller dynamics
• Equidistribution of saddle connections on translation surfaces
• Slope Gap Distribution of Saddle Connections on the 2n-gon
• Counting pairs of saddle connections
• Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals