Billiards in regular polygons

Abstract

In this paper we study billiards in regular $n$-sided polygons and monodromy over Teichmüller curves from an arithmetic perspective. Our main results show that the combinatorics of billiards in a periodic direction $s$ is controlled by the projection of $s$ to a finite projective line.
This arithmetic control coexists, experimentally, with chaotic behavior that first emerges when $n=12$.