{
  "id": "math/0612407",
  "version": "v1",
  "published": "2006-12-14T17:46:23.000Z",
  "updated": "2006-12-14T17:46:23.000Z",
  "title": "Dynamics of the third order Lyness' difference equation",
  "authors": [
    "Anna Cima",
    "Armengol Gasull",
    "Victor Manosa"
  ],
  "comment": "46 pages. 5 figures",
  "journal": "J. Difference Equations and Applications 13 (10) (2007), 855--884",
  "doi": "10.1080/10236190701264735",
  "categories": [
    "math.DS"
  ],
  "abstract": "This paper studies the iterates of the third order Lyness' recurrence $x_{k+3}=(a+x_{k+1}+x_{k+2})/x_k,$ with positive initial conditions, being $a$ also a positive parameter. It is known that for $a=1$ all the sequences generated by this recurrence are 8-periodic. We prove that for each $a\\ne1$ there are infinitely many initial conditions giving rise to periodic sequences which have almost all the even periods and that for a full measure set of initial conditions the sequences generated by the recurrence are dense in either one or two disjoint bounded intervals of $\\R.$ Finally we show that the set of initial conditions giving rise to periodic sequences of odd period is contained in a codimension one algebraic variety (so it has zero measure) and that for an open set of values of $a$ it also contains all the odd numbers, except finitely many of them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-12-14T17:46:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39A11",
      "39A20"
    ],
    "keywords": [
      "third order lyness",
      "difference equation",
      "initial conditions giving rise",
      "periodic sequences",
      "recurrence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12407C"
    }
  }
}