On the Local Theory of Billiards in Polygons

TL;DR

This paper introduces a new proof technique using covering spaces to analyze the stability of periodic billiard trajectories in polygons, classifies trajectories near specific triangles, and explores the structure of orbit tiles.

Contribution

It provides a novel covering space approach to characterize stable trajectories and extends results on billiard dynamics in special triangles.

Findings

Stable trajectories correspond to null-homologous paths on punctured spheres.

No finite union of orbit tiles covers neighborhoods of certain triangles.

Certain Veech triangles admit no periodic trajectories for specific parameters.

Abstract

A periodic trajectory on a polygonal billiard table is stable if it persists under any sufficiently small perturbation of the table. It is a standard result that a periodic trajectory on an $n$-gon gives rise in a natural way to a closed path on an $n$-punctured sphere, and that the trajectory is stable iff this path is null-homologous. We present a novel proof of this result in the language of covering spaces, which generalizes to characterize the stable trajectories in neighborhoods of a polygon. Using this, we classify the stable periodic trajectories near the 30-60-90 triangle, giving a new proof of a result of Schwartz that no neighborhood of the triangle can be covered by a finite union of orbit tiles. We also extend a result of Hooper and Schwartz that the isosceles Veech triangles $V_n$ admit no periodic trajectories for $n=2^m,m\ge 2$, and examine their conjecture that no neighborhood of $V_n$ can be covered by finitely many orbit tiles.