{
  "id": "1910.04902",
  "version": "v1",
  "published": "2019-10-10T22:54:45.000Z",
  "updated": "2019-10-10T22:54:45.000Z",
  "title": "Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism",
  "authors": [
    "Artur O. Lopes",
    "Ali Messaoudi",
    "Victor Vargas"
  ],
  "categories": [
    "math.DS",
    "math.FA",
    "math.PR"
  ],
  "abstract": "We denote by $X$ a real vector space that can be either $ l^p(\\mathbb{R}),\\; 1 \\leq p < \\infty,$ or $c_0(\\mathbb{R})$. We consider a certain class of linear maps $L : X \\to X$ (called weighted shift operators). We fix a suitable {\\it a priori} probability measure $m$ on the kernel of $L$ in order to introduce the Ruelle operator $\\mathcal{L}_A$ associated to a bounded H\\\"older continuous potential $A: X \\to \\mathbb{R}$. We adapt the Thermodynamic Formalism for generalized $XY$ models (in the case the alphabet is non-compact) to the present setting. We are able to show the existence of eigenvalues, eigenfunctions and eigenprobabilities for $\\mathcal{L}_A$. Moreover, we show the existence of $L$-invariant ergodic probabilities with full support in a similar manner as in Classical Thermodynamic Formalism. Finally, these results are extended to a large class of linear operators defined on Banach spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-10T22:54:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35"
    ],
    "keywords": [
      "discrete time linear dynamics",
      "invariant probabilities",
      "probability",
      "real vector space",
      "invariant ergodic probabilities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}