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 2π/n, with n ≥ 5 and n odd.