{
  "id": "1408.1663",
  "version": "v1",
  "published": "2014-08-07T17:33:17.000Z",
  "updated": "2014-08-07T17:33:17.000Z",
  "title": "Piecewise contractions defined by iterated function systems",
  "authors": [
    "Arnaldo Nogueira",
    "Benito Pires",
    "Rafael A. Rosales"
  ],
  "comment": "16 pages, two figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\phi_1,\\ldots,\\phi_n:[0,1]\\to (0,1)$ be Lipschitz contractions. Let $I=[0,1)$, $x_0=0$ and $x_n=1$. We prove that for Lebesgue almost every $(x_1,...,x_{n-1})$ satisfying $0<x_1<\\cdots <x_{n-1}<1$, the piecewise contraction $f:I\\to I$ defined by $x\\in [x_{i-1},x_i)\\mapsto \\phi_i(x)$ is asymptotically periodic. More precisely, $f$ has at least one and at most $n$ periodic orbits and the $\\omega$-limit set $\\omega_f(x)$ is a periodic orbit of $f$ for every $x\\in I$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-07T17:33:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "37C20",
      "37E15"
    ],
    "keywords": [
      "iterated function systems",
      "piecewise contractions",
      "periodic orbit",
      "limit set",
      "lipschitz contractions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.1663N"
    }
  }
}