{
  "id": "1102.5358",
  "version": "v1",
  "published": "2011-02-25T21:50:02.000Z",
  "updated": "2011-02-25T21:50:02.000Z",
  "title": "Ergodic properties of infinite extensions of area-preserving flows",
  "authors": [
    "Krzysztof Fraczek",
    "Corinna Ulcigrai"
  ],
  "comment": "57 pages, 4 pictures",
  "journal": "Math. Ann. 354 (2012), 1289-1367",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "We consider volume-preserving flows $(\\Phi^f_t)_{t\\in\\mathbb{R}}$ on $S\\times \\mathbb{R}$, where $S$ is a closed connected surface of genus $g\\geq 2$ and $(\\Phi^f_t)_{t\\in\\mathbb{R}}$ has the form $\\Phi^f_t(x,y)=(\\phi_tx,y+\\int_0^t f(\\phi_sx)ds)$, where $(\\phi_t)_{t\\in\\mathbb{R}}$ is a locally Hamiltonian flow of hyperbolic periodic type on $S$ and $f$ is a smooth real valued function on $S$. We investigate ergodic properties of these infinite measure-preserving flows and prove that if $f$ belongs to a space of finite codimension in $\\mathscr{C}^{2+\\epsilon}(S)$, then the following dynamical dichotomy holds: if there is a fixed point of $(\\phi_t)_{t\\in\\mathbb{R}}$ on which $f$ does not vanish, then $(\\Phi^f_t)_{t\\in\\mathbb{R}}$ is ergodic, otherwise, if $f$ vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension $(\\Phi^0_t)_{t\\in\\mathbb{R}}$. The proof of this result exploits the reduction of $(\\Phi^f_t)_{t\\in\\mathbb{R}}$ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of $(\\phi_t)_{t\\in\\mathbb{R}}$ on which $f$ does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-25T21:50:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C10",
      "37A10",
      "37A40"
    ],
    "keywords": [
      "ergodic properties",
      "infinite extensions",
      "area-preserving flows",
      "fixed point",
      "interval exchange transformation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.5358F"
    }
  }
}