{
  "id": "2106.12087",
  "version": "v1",
  "published": "2021-06-22T22:34:01.000Z",
  "updated": "2021-06-22T22:34:01.000Z",
  "title": "Generalized eigenvalues of the Perron-Frobenius operators of symbolic dynamical systems",
  "authors": [
    "Hayato Chiba",
    "Masahiro Ikeda",
    "Isao Ishikawa"
  ],
  "categories": [
    "math.DS",
    "math.FA"
  ],
  "abstract": "The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace $X$ of $\\mathcal{H}$ and its dual space $X'$. The three topological spaces $X \\subset \\mathcal{H} \\subset X'$ is called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron-Frobenius operators of the one-sided and two-sided shifts of finite types (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ration on $[0,1]$ modulo $1$ is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron-Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-22T22:34:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symbolic dynamical systems",
      "perron-frobenius operators",
      "generalized eigenvalues",
      "generalized spectrum",
      "hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}