Gaps of saddle connection directions for some branched covers of tori

Anthony Sanchez

doi:10.1017/etds.2021.81

Abstract

AbstractWe compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit computations of gap distributions for non-lattice translation surfaces. We show that this distribution has support at zero and quadratic tail decay. We also construct examples of translation surfaces in any genus$d>1$that have the same gap distribution as the gap distribution of two identical tori glued along a slit. The second class we consider are twice-marked tori and saddle connections between distinct marked points with a specific orientation. These results can be interpreted as the gap distribution of slopes of affine lattices. We obtain our results by translating the question of gap distributions to a dynamical question of return times to a transversal under the horocycle flow on an appropriate moduli space.

Highlights

We are interested in the distribution of directions of saddle connections on translation surfaces
We prove other results related to the gap distribution, but we wait until §4 Theorem 4.4 before more precisely stating them
We prove the following result for the space of affine lattices

Summary

Introduction

We are interested in the distribution of directions of saddle connections on translation surfaces. This paper makes use of unipotent flows on the space of affine lattices to prove a gap distribution result for doubled slit tori. This strategy involves turning the question of gap distributions of slopes of saddle connections to a dynamical question of return times to a transversal under horocycle flow on an appropriate moduli space.

Objectives

Results

Conclusion