Bounds on saddle connections for flat spheres

Kai Fu; Guillaume Tahar

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arXiv · 2308.08940

Bounds on saddle connections for flat spheres

Kai Fu·Guillaume Tahar

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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.

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Kai Fu, Guillaume Tahar. 2023-08-17. Bounds on saddle connections for flat spheres. https://arxiv.org/abs/2308.08940

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