{
  "id": "2207.12153",
  "version": "v1",
  "published": "2022-07-25T12:55:15.000Z",
  "updated": "2022-07-25T12:55:15.000Z",
  "title": "Uniformity Aspects of $\\mathrm{SL}(2,\\mathbb{R})$ Cocycles and Applications to Schrödinger Operators Defined Over Boshernitzan Subshifts",
  "authors": [
    "David Damanik",
    "Daniel Lenz"
  ],
  "comment": "24 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We consider continuous $\\mathrm{SL}(2,\\mathbb{R})$ valued cocycles over general dynamical systems and discuss a variety of uniformity notions. In particular, we provide a description of uniform one-parameter families of continuous $\\mathrm{SL}(2,\\mathbb{R})$ cocycles as $G_\\delta$-sets. These results are then applied to Schr\\\"odinger operators with dynamically defined potentials. In the case where the base dynamics is given by a subshift satisfying the Boshernitzan condition, we show that for a generic continuous sampling function, the associated Schr\\\"odinger cocycles are uniform for all energies and, in the aperiodic case, the spectrum is a Cantor set of zero Lebesgue measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-25T12:55:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniformity aspects",
      "schrödinger operators",
      "boshernitzan subshifts",
      "applications",
      "zero lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}