Complexity for billiards in regular N-gons

Abstract

We compute the complexity of the billiard language of the regular Euclidean $N$-gons (and other families of rational lattice polygons), answering a question posed by Cassaigne-Hubert-Troubetzkoy. Our key technical result is a counting result for saddle connections on lattice surfaces, when we count by combinatorial length.

42 pages, 12 figures, updated proof of Lemma 4.8 (added auxiliary lemma Lemma 4.9 and some new figures)