{
  "id": "1308.2996",
  "version": "v1",
  "published": "2013-08-13T23:59:55.000Z",
  "updated": "2013-08-13T23:59:55.000Z",
  "title": "The natural measure of a symbolic dynamical system",
  "authors": [
    "Wen-Guei Hu",
    "Song-Sun Lin"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "This study investigates the natural or intrinsic measure of a symbolic dynamical system $\\Sigma$. The measure $\\mu([i_{1},i_{2},...,i_{n}])$ of a pattern $[i_{1},i_{2},...,i_{n}]$ in $\\Sigma$ is an asymptotic ratio of $[i_{1},i_{2},...,i_{n}]$, which arises in all patterns of length $n$ within very long patterns, such that in a typical long pattern, the pattern $[i_{1},i_{2},...,i_{n}]$ appears with frequency $\\mu([i_{1},i_{2},...,i_{n}])$. When $\\Sigma=\\Sigma(A)$ is a shift of finite type and $A$ is an irreducible $N\\times N$ non-negative matrix, the measure $\\mu$ is the Parry measure. $\\mu$ is ergodic with maximum entropy. The result holds for sofic shift $\\mathcal{G}=(G,\\mathcal{L})$, which is irreducible. The result can be extended to $\\Sigma(A)$, where $A$ is a countably infinite matrix that is irreducible, aperiodic and positive recurrent. By using the Krieger cover, the natural measure of a general shift space is studied in the way of a countably infinite state of sofic shift, including context free shift. The Perron-Frobenius Theorem for non-negative matrices plays an essential role in this study.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-13T23:59:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symbolic dynamical system",
      "natural measure",
      "sofic shift",
      "context free shift",
      "countably infinite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.2996H"
    }
  }
}