Looking for a Billiard Table which is not a Lattice Polygon but Satisfies Veech's Dichotomy

Abstract

Over the course of studying billiard dynamics, several questions were raised. One of the questions was, which surfaces satisfy the following property (which is called Veech's dichotomy): Any direction is either completely periodic or uniquely ergodic. In an important paper Veech gave a sufficient condition for this dichotomy. He showed that if the stabilizer of a translation surface is a lattice in $SL_2(\R)$, then the surface satisfies Veech's dichotomy. Later, Smillie and Weiss proved that this condition is not necessary. They constructed a translation surface which satisfies Veech's dichotomy but is not a lattice surface. Their construction was based on previous work of Hubert and Schmidt, by taking a branched cover over a lattice surface, where the branch locus is a single non-periodic connection point. In this work we tried to answer the following question: Is there a flat structure obtained from a billiard table that satisfies Veech's dichotomy, but its Veech group is not a lattice? We prove that in the entire list of possible candidates for such a construction, an example does not exist.