Bounds on saddle connections for flat spheres

TL;DR

This paper derives explicit bounds on the number and length of saddle connections with limited self-intersections on flat spheres with conical singularities, impacting the study of polygonal billiards.

Contribution

It provides new explicit upper bounds on saddle connections' counts and lengths, linking geometric properties to billiard trajectory counting.

Findings

Bound on the number of saddle connections with at most k self-intersections

Upper bound on lengths of saddle connections for normalized area surfaces

Application of bounds to counting singular trajectories in irrational polygonal billiards

Abstract

We consider a flat metric with conical singularities on the sphere. Under the assumption that no partial sum of angle defects is equal to $2\pi$, we draw on the geometry of immersed disks to obtain an explicit upper bound on the number of saddle connections with at most $k$ self-intersections. Additionally, we establish an upper bound on their lengths for a surface with a normalized area. Finally, we apply these bounds to the counting of singular trajectories in irrational polygonal billiards.