On the Local Theory of Billiards in Polygons

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.