{
  "id": "1808.09803",
  "version": "v1",
  "published": "2018-08-22T17:29:49.000Z",
  "updated": "2018-08-22T17:29:49.000Z",
  "title": "Normalized image of a vector by an infinite product of nonnegative matrices",
  "authors": [
    "Alain Thomas"
  ],
  "comment": "32 pages",
  "categories": [
    "math.FA",
    "math.DS"
  ],
  "abstract": "Let $P_n=A_1\\cdots A_n$~, where $\\mathcal A=(A_n)_{n\\in\\mathbb N}$ is a sequence of $d\\times d$ matrices, and let $V$ be a $d$-dimensional column-vector. We call \"normalized image of~$V$ by the infinite product of the matrices $A_n$\", the vector $\\lim_{n\\to\\infty}P_nV/{\\Vert P_nV\\Vert}$ if exists. In general the sequence of matrices $n\\mapsto P_n/{\\Vert P_n\\Vert}$ does not converge but, under some sufficient conditions specified in Theorem~A the sequence of vectors $n\\mapsto P_nV/{\\Vert P_nV\\Vert}$ converges. We use Theorem~A to prove that certain sofic (i.e. linearly representable) measures satisfy the multifractal formalism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-22T17:29:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B48",
      "28A12"
    ],
    "keywords": [
      "infinite product",
      "normalized image",
      "nonnegative matrices",
      "multifractal formalism",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}