{
  "id": "1402.7229",
  "version": "v3",
  "published": "2014-02-28T12:52:59.000Z",
  "updated": "2014-05-28T07:54:27.000Z",
  "title": "On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure",
  "authors": [
    "Simon Baker"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\lambda_{1},\\ldots,\\lambda_{n}$ be real numbers in $(0,1)$ and $p_{1},\\ldots,p_{n}$ be points in $\\mathbb{R}^{d}$. Consider the collection of maps $f_{j}:\\mathbb{R}^{d}\\to\\mathbb{R}^{d} $ given by $$f_{j}(x)=\\lambda_{j} x +(1-\\lambda_{j})p_{j}.$$ It is a well known result that there exists a unique compact set $\\Lambda\\subset \\mathbb{R}^{d}$ satisfying $\\Lambda=\\cup_{j=1}^{n} f_{j}(\\Lambda).$ Each $x\\in \\Lambda$ has at least one coding, that is a sequence $(\\epsilon_{i})_{i=1}^{\\infty}\\in \\{1,\\ldots,n\\}^{\\mathbb{N}}$ that satisfies $\\lim_{N\\to\\infty}f_{\\epsilon_{1}}\\cdots f_{\\epsilon_{N}} (0)=x.$ We study the size and complexity of the set of codings of a generic $x\\in \\Lambda$ when $\\Lambda$ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every $x\\in\\Lambda$ has a continuum of codings. We also show that almost every $x\\in\\Lambda$ has a universal coding. Our work makes no assumptions on the existence of holes in $\\Lambda$ and improves upon existing results when it is assumed $\\Lambda$ contains no holes.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-28T07:54:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37A45"
    ],
    "keywords": [
      "positive lebesgue measure",
      "self-similar sets",
      "complexity",
      "cardinality",
      "unique compact set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.7229B"
    }
  }
}