{
  "id": "1311.5394",
  "version": "v1",
  "published": "2013-11-21T13:16:03.000Z",
  "updated": "2013-11-21T13:16:03.000Z",
  "title": "The dynamics of a class of quasi-periodic Schrödinger cocycles",
  "authors": [
    "Kristian Bjerklöv"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $f:\\mathbb{T}\\to\\mathbb{R}$ be a Morse function of class $C^2$ with exactly two critical points, let $\\omega\\in\\mathbb{T}$ be Diophantine, and let $\\lambda>0$ be sufficiently large (depending on $f$ and $\\omega$). For any value of the parameter $E\\in \\mathbb{R}$ we make a careful analysis of the dynamics of the skew-product map $$\\Phi_E(\\theta,r)=\\left(\\theta+\\omega,\\lambda f(\\theta)-E-1/r\\right),$$ acting on the \"torus\" $\\mathbb{T}\\times\\widehat{\\mathbb{R}}$. The map $\\Phi_E$ is intimately related to the quasi-periodic Schr\\\"odinger cocycle $(\\omega,A_E): \\mathbb{T}\\times \\mathbb{R}^2 \\to \\mathbb{T}\\times \\mathbb{R}^2$, $(\\theta,x)\\mapsto (\\theta+\\omega, A_E(\\theta)\\cdot x)$, where $A_E:\\mathbb{T}\\to \\text{SL}(2,\\mathbb{R})$ is given by $A_{E}(\\theta)=\\left(\\begin{matrix}0 & 1 \\\\ -1 & \\lambda f(\\theta)-E \\end{matrix} \\right), \\quad E\\in \\mathbb{R}. $ More precisely, $(\\omega,A_E)$ naturally acts on the space $\\mathbb{T}\\times\\widehat{\\mathbb{R}}$, and $\\Phi_E$ is the map thus obtained. The analysis of $\\Phi_E$ allows us to derive three main results: (1) The (maximal) Lyapunov exponent of the Schr\\\"odinger cocycle $(\\omega,A_E)$ is $\\gtrsim \\log \\lambda$, uniformly in $E\\in \\mathbb{R}$. This implies that the map $\\Phi_E$ has exactly two ergodic probability measures for all $E\\in \\mathbb{R}$; (2) If $E$ is on the edge of an open gap in the spectrum $\\sigma(H)$ of the associated Schr\\\"odinger operator $H_\\theta$, then there exist a phase $\\theta\\in\\mathbb{T}$ and a vector $u\\in l^2(\\mathbb{Z})$, exponentially decaying at $\\pm\\infty$, such that $H_\\theta u=Eu$; (3) The map $\\Phi_E$ is minimal iff $E\\in \\sigma(H)\\setminus\\{\\text{edges of open gaps}\\}$. In particular, $\\Phi_E$ is minimal for all $E$ for which the fibered rotation number $\\alpha(E)$ associated to $(\\omega,A_E)$ is irrational with respect to $\\omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-21T13:16:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasi-periodic schrödinger cocycles",
      "ergodic probability measures",
      "skew-product map",
      "main results",
      "morse function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.5394B"
    }
  }
}