{
  "id": "math/0609015",
  "version": "v1",
  "published": "2006-09-01T05:46:04.000Z",
  "updated": "2006-09-01T05:46:04.000Z",
  "title": "The Measure-Theoretical Entropy of a Linear Cellular Automata with respect to a Markov Measure",
  "authors": [
    "Hasan Akin"
  ],
  "comment": "7 pages",
  "categories": [
    "math.DS",
    "math.FA"
  ],
  "abstract": "In this paper we study the measure-theoretical entropy of the one-dimensional linear cellular automata (CA hereafter) $T_{f[-l,r]}$, generated by local rule $f(x_{-l},...,x_{r})= \\sum\\limits_{i=-l}^{r}\\lambda_{i}x_{i}(\\text{mod}\\ m)$, where $l$ and $r$ are positive integers, acting on the space of all doubly infinite sequences with values in a finite ring $\\mathbb{Z}_{m}$, $m \\geq 2$, with respect to a Markov measure. We prove that if the local rule $f$ is bipermutative, then the measure-theoretical entropy of linear CA $T_{f[-l,r]}$ with respect to a Markov measure $\\mu_{\\pi P}$ is $ h_{\\mu_{\\pi P}}(T_{f[-l,r]})=-(l+r)\\sum\\limits_{i,j=0}^{m-1}p_ip_{ij}\\text{log} p_{ij}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-01T05:46:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28D15",
      "37A15"
    ],
    "keywords": [
      "markov measure",
      "measure-theoretical entropy",
      "local rule",
      "one-dimensional linear cellular automata",
      "linear ca"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9015A"
    }
  }
}