Unipotent flows on the space of branched covers of Veech surfaces

Details Unipotent flows on the space of branched covers of Veech surfaces Unipotent flows on the space of branched covers of Veech surfaces

2004-08-06

arxiv.org/abs/math/0408090v2

Mathematics

Dynamical Systems

Added a corollary regarding orbit closures. Greatly expanded the part involving the counting application, giving more detailed

Scientific paper

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 $U = {\begin{pmatrix} 1 & * 0 & 1 \end{pmatrix}}$. 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 $n \ge 5$ and $n$ odd.

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