{
  "id": "1702.07698",
  "version": "v1",
  "published": "2017-02-24T18:33:50.000Z",
  "updated": "2017-02-24T18:33:50.000Z",
  "title": "Complexity and fractal dimensions for infinite sequences with positive entropy",
  "authors": [
    "Carlos Gustavo Moreira",
    "Christian Mauduit"
  ],
  "comment": "31 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. One of the goals of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a complexity function bounded by a given function $f$ with exponential growth. We describe the combinatorial structure of such sets of infinite words and we introduce a real parameter, the $word\\,\\, entropy$ $E_W(f)$ associated to a given function $f$. We give estimates on the word entropy $E_W(f)$ in terms of the exponential rate of growth of $f$ and we determine fractal dimensions of sets of infinite sequences with complexity function bounded by $f$ in terms of its word entropy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-24T18:33:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68R15",
      "37B10",
      "37B40",
      "28A78"
    ],
    "keywords": [
      "infinite sequences",
      "infinite word",
      "positive entropy",
      "complexity function",
      "word entropy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}