{
  "id": "1710.03841",
  "version": "v1",
  "published": "2017-10-10T22:29:12.000Z",
  "updated": "2017-10-10T22:29:12.000Z",
  "title": "The Double Transpose of the Ruelle Operator",
  "authors": [
    "L. Cioletti",
    "A. C. D. van Enter",
    "R. Ruviaro"
  ],
  "comment": "19 pages",
  "categories": [
    "math.DS",
    "math.FA"
  ],
  "abstract": "In this paper we study the double transpose extension of the Ruelle transfer operator $\\mathscr{L}_{f}$ associated to a general real continuous potential $f\\in C(\\Omega)$, where $\\Omega=E^{\\mathbb{N}}$ and $E$ is any compact metric space. For this extension, we prove the existence of non-negative eigenfunctions, in the Banach lattice sense, associated to $\\lambda_{f}$, the spectral radius of the Ruelle operator acting on $C(\\Omega)$. As an application, we show that the natural extension of the Ruelle operator to $L^1(\\Omega,\\mathscr{B}(\\Omega),\\nu)$ (for a suitable Borel probability measure $\\nu$) always has an eigenfunction associated to $\\lambda_{f}$. These eigenfunctions agree with the usual maximal eigenfunctions, when the potential $f$ is either in H\\\"older or Walters spaces. We also constructed solutions to the classical (finite-state spaces) and generalized (general compact metric spaces) variational problem avoiding the standard normalization technique.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-10T22:29:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28Dxx",
      "37D35"
    ],
    "keywords": [
      "ruelle operator",
      "double transpose",
      "general compact metric spaces",
      "banach lattice sense",
      "general real continuous potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}