{
  "id": "0709.2667",
  "version": "v2",
  "published": "2007-09-17T16:43:42.000Z",
  "updated": "2008-04-22T14:52:26.000Z",
  "title": "Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts",
  "authors": [
    "Artur Avila",
    "Jairo Bochi",
    "David Damanik"
  ],
  "comment": "Final version. To appear in Duke Mathematical Journal",
  "journal": "Duke Mathematical Journal, 146, no. 2 (2009), 253-280",
  "doi": "10.1215/00127094-2008-065",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We consider continuous $SL(2,\\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schr\\\"odinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schr\\\"odinger operator is a Cantor set.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-04-22T14:52:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized skew-shifts",
      "schrödinger operators",
      "cantor spectrum",
      "potentials arising",
      "uniform hyperbolicity"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.2667A"
    }
  }
}