{
  "id": "1102.4719",
  "version": "v1",
  "published": "2011-02-23T12:04:26.000Z",
  "updated": "2011-02-23T12:04:26.000Z",
  "title": "Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges",
  "authors": [
    "Yair N. Minsky",
    "Barak Weiss"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "A translation surface on (S, \\Sigma) gives rise to two transverse measured foliations \\FF, \\GG on S with singularities in \\Sigma, and by integration, to a pair of cohomology classes [\\FF], \\, [\\GG] \\in H^1(S, \\Sigma; \\R). Given a measured foliation \\FF, we characterize the set of cohomology classes \\B for which there is a measured foliation \\GG as above with \\B = [\\GG]. This extends previous results of Thurston and Sullivan. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation \\sigma \\in \\mathcal{S}_d, the space \\R^d_+ parametrizes the interval exchanges on d intervals with permutation \\sigma. We describe lines \\ell in \\R^d_+ such that almost every point in \\ell is uniquely ergodic. We also show that for \\sigma(i) = d+1-i, for almost every s>0, the interval exchange transformation corresponding to \\sigma and (s, s^2, \\ldots, s^d) is uniquely ergodic. As another application we show that when k=|\\Sigma| \\geq 2, the operation of `moving the singularities horizontally' is globally well-defined. We prove that there is a well-defined action of the group B \\ltimes \\R^{k-1} on the set of translation surfaces of type (S, \\Sigma) without horizontal saddle connections. Here B \\subset \\SL(2,\\R) is the subgroup of upper triangular matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-23T12:04:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "measured foliation",
      "cohomology classes",
      "mahlers question",
      "translation surface",
      "upper triangular matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.4719M"
    }
  }
}