{
  "id": "1705.10847",
  "version": "v1",
  "published": "2017-05-30T20:00:59.000Z",
  "updated": "2017-05-30T20:00:59.000Z",
  "title": "Equidistribution of saddle connections on translation surfaces",
  "authors": [
    "Benjamin Dozier"
  ],
  "comment": "25 pages, 4 figures. arXiv admin note: text overlap with arXiv:1701.00175",
  "categories": [
    "math.DS",
    "math.GT"
  ],
  "abstract": "Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $R\\to\\infty$, the weak limit of these measures exists and is equal to the Lebesgue measure on $X$. We also show that any weak limit of a subsequence of the counting measures on $S^1$ given by the angles of all saddle connections of length at most $R_n$, as $R_n\\to\\infty$, is in the Lebesgue measure class. The proof of the first result uses the second result, together with the result of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-30T20:00:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "saddle connections",
      "translation surface",
      "weak limit",
      "equidistribution",
      "lebesgue measure class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}