Complexity of Billiards in Polygons Associated to Hyperbolic $(p,q)$-Tilings

The complexity of the billiard language of regular polygons in the hyperbolic plane with $p$ sides and $2\pi/q$ internal angles is known to grow exponentially and the exponential growth rate is known to equal the topological entropy of the billiard system. In this paper we compute these exponential growth rates explicitly when $q$ is even and give bounds when $q$ is odd. Additionally, for the $q$ even case, we give complete grammar rules that establish when a word (finite, infinite or bi-infinite) in $p$ letters is realized by a billiard path. This latter result is roughly stated and not rigorously proved in a paper of Giannoni and Ullmo (1995). In this paper, we provide a precise statement and a complete proof using new methods relating to minimal tiling paths.