Unipotent flows on the space of branched covers of Veech surfaces

There is a natural action of SL(2, R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup U = {((1)(*)(0)(1))}. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL(2, R)-orbit of the set of branched covers of a fixed Veech surface.) For the U-action on these submanifolds, this is an analogue of Ratner's theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-Veech rational triangles, namely, the isosceles triangles in which exactly one angle is 2 pi/n, with n >= 5 and it odd.