{
  "id": "1210.7715",
  "version": "v1",
  "published": "2012-10-29T16:28:14.000Z",
  "updated": "2012-10-29T16:28:14.000Z",
  "title": "Preperiodic points for families of rational map",
  "authors": [
    "Dragos Ghioca",
    "Liang-Chung Hsia",
    "Thomas J. Tucker"
  ],
  "comment": "30 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and b are preperiodic for f_l. In particular we show that if P,Q are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and there exists l such that a is periodic for P(x)/Q(x) + l, but b is not preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such that both a and b are preperiodic for P(x)/Q(x)+l. We also prove a similar result for certain two-dimensional families of endomorphisms of P^2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-29T16:28:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P05",
      "37P10"
    ],
    "keywords": [
      "preperiodic points",
      "algebraic numbers",
      "similar result",
      "two-dimensional families",
      "rational maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.7715G"
    }
  }
}