{
  "id": "1407.1573",
  "version": "v1",
  "published": "2014-07-07T03:54:13.000Z",
  "updated": "2014-07-07T03:54:13.000Z",
  "title": "Portraits of preperiodic points for rational maps",
  "authors": [
    "Dragos Ghioca",
    "Khoa Nguyen",
    "Thomas J. Tucker"
  ],
  "categories": [
    "math.NT",
    "math.AG",
    "math.DS"
  ],
  "abstract": "Let $K$ be a function field over an algebraically closed field $k$ of characteristic $0$, let $\\varphi\\in K(z)$ be a rational function of degree at least equal to $2$ for which there is no point at which $\\varphi$ is totally ramified, and let $\\alpha\\in K$. We show that for all but finitely many pairs $(m,n)\\in \\mathbb{Z}_{\\ge 0}\\times \\mathbb{N}$ there exists a place $\\mathfrak{p}$ of $K$ such that the point $\\alpha$ has preperiod $m$ and minimum period $n$ under the action of $\\varphi$. This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under suitable modification, also when $\\varphi$ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple $(c_1,\\dots , c_{d-1})\\in k^{n-1}$ and for almost all pairs $(m_i,n_i)\\in \\mathbb{Z}_{\\ge 0}\\times \\mathbb{N}$ for $i=1,\\dots, d-1$, there exists a polynomial $f\\in k[z]$ of degree $d$ in normal form such that for each $i=1,\\dots, d-1$, the point $c_i$ has preperiod $m_i$ and minimum period $n_i$ under the action of $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-07T03:54:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "preperiodic points",
      "rational maps",
      "minimum period",
      "normal form",
      "similar result"
    ],
    "publication": {
      "doi": "10.1017/S0305004115000274",
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2015,
      "month": "Jul",
      "volume": 159,
      "number": 1,
      "pages": 165
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015MPCPS.159..165G"
    }
  }
}