{
  "id": "1811.00184",
  "version": "v1",
  "published": "2018-11-01T01:58:50.000Z",
  "updated": "2018-11-01T01:58:50.000Z",
  "title": "Rigidity of a class of smooth singular flows on $\\mathbb T^2$",
  "authors": [
    "Changguang Dong",
    "Adam Kanigowski"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field $\\mathcal{X}$ on $\\mathbb T^2\\setminus \\{a\\}$, where $\\mathcal{X}$ is not defined at $a\\in \\mathbb T^2$. It follows that the phase space can be decomposed into a (topological disc) $D_\\mathcal{X}$ and an ergodic component $E_\\mathcal{X}=\\mathbb T^2\\setminus D_\\mathcal{X}$. Let $\\omega_\\mathcal{X}$ be the 1-form associated to $\\mathcal{X}$. We show that if $|\\int_{E_{\\mathcal{X}_1}}d\\omega_{\\mathcal{X}_1}|\\neq |\\int_{E_{\\mathcal{X}_2}}d\\omega_{\\mathcal{X}_2}|$, then the corresponding flows $(v_t^{\\mathcal{X}_1})$ and $(v_t^{\\mathcal{X}_2})$ are disjoint. It also follows that for every $\\mathcal{X}$ there is a uniquely associated frequency $\\alpha=\\alpha_{\\mathcal{X}}\\in \\mathbb T$. We show that for a full measure set of $\\alpha\\in \\mathbb T$ the class of smooth time changes of $(v_t^\\mathcal{X_\\alpha})$ is joining rigid, i.e. every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to a problem of Ratner (Problem 3 in \\cite{Rat4}) is positive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-01T01:58:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A35",
      "37A05"
    ],
    "keywords": [
      "smooth singular flows",
      "smooth time changes",
      "full measure set",
      "von neumann flows",
      "smooth vector field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}