{
  "id": "1503.07129",
  "version": "v1",
  "published": "2015-03-24T18:22:48.000Z",
  "updated": "2015-03-24T18:22:48.000Z",
  "title": "The dynamical system generated by the floor function $\\lfloorλ x\\rfloor$",
  "authors": [
    "U. A. Rozikov",
    "I. A. Sattarov",
    "J. B. Usmonov"
  ],
  "comment": "8 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate the dynamical system generated by the function $\\lfloor\\lambda x\\rfloor$ defined on $\\mathbb R$ and with a parameter $\\lambda\\in \\mathbb R$. For each given $m\\in \\mathbb N$ we show that there exists a region of values of $\\lambda$, where the function has exactly $m$ fixed points (which are non-negative integers), also there is another region for $\\lambda$, where there are exactly $m+1$ fixed points (which are non-positive integers). Moreover the full set $\\mathbb Z$ of integer numbers is the set of fixed points iff $\\lambda=1$. We show that depending on $\\lambda$ and on the initial point $x$ the limit of the forward orbit of the dynamical system may be one of the following possibilities: (i) a fixed point, (ii) a two-periodic orbit or (iii) $\\pm\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-24T18:22:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05"
    ],
    "keywords": [
      "dynamical system",
      "floor function",
      "fixed point",
      "two-periodic orbit",
      "initial point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}