Let P be a polygon whose angles are rational multiples of pi. A direction is said to be minimal (resp. uniquely ergodic) if every billiard trajectory with the prescribed initial direction is dense (resp. uniformly distributed.)
Veech discovered a class of polygons, which includes all the regular n-gons, with the property that every minimal direction is uniquely ergodic. He also showed that there are polygons (2-by-1 rectangle with a barrier down the middle of irrational length) that do not have this property, i.e. they admit dense but not uniformly distributed trajectories.
In this talk, I will describe some joint work with Howard Masur and Pascal Hubert that allow us to give further examples of polygons that admits minimal non-uniquely ergodic directions.