{
  "id": "2205.09044",
  "version": "v1",
  "published": "2022-05-17T14:26:00.000Z",
  "updated": "2022-05-17T14:26:00.000Z",
  "title": "Normalized image of a vector by an infinite product of nonnegative matrices",
  "authors": [
    "Alain Thomas"
  ],
  "comment": "23 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\nu$ be a measure on the set $\\{0,1,\\dots,b-1\\}^\\mathbb N$, linearly representable by means of a finite set $\\mathcal M$ of square nonnegative matrices. To prove that $\\nu$ has the weak Gibbs property, one generally use the uniform convergence on $\\mathcal M^\\mathbb N$ of the sequence of vectors $n\\mapsto\\frac{A_1\\cdots A_nc}{\\Vert A_1\\cdots A_nc\\Vert}$, where $c$ is a positive column-vector. We give a sufficient condition for this sequence to converge, that we apply to some measures defined by Bernoulli convolution. Secondarily we prove that the sequence of matrices $n\\mapsto\\frac{A_1\\cdots A_n}{\\Vert A_1\\cdots A_n\\Vert}$ in general diverges.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-17T14:26:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B48",
      "28A12"
    ],
    "keywords": [
      "infinite product",
      "normalized image",
      "weak gibbs property",
      "square nonnegative matrices",
      "uniform convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}