{
  "id": "math/0408090",
  "version": "v2",
  "published": "2004-08-06T16:49:21.000Z",
  "updated": "2005-05-09T15:30:53.000Z",
  "title": "Unipotent flows on the space of branched covers of Veech surfaces",
  "authors": [
    "Alex Eskin",
    "Jens Marklof",
    "Dave Witte Morris"
  ],
  "comment": "Added a corollary regarding orbit closures. Greatly expanded the part involving the counting application, giving more detailed proofs and a summary of previous results used",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-05-09T15:30:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A99",
      "37E15",
      "37D40",
      "37D50"
    ],
    "keywords": [
      "unipotent flows",
      "branched covers",
      "moduli space",
      "non-veech rational triangles",
      "u-invariant ergodic measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......8090E"
    }
  }
}