{
  "id": "1511.04856",
  "version": "v1",
  "published": "2015-11-16T08:01:46.000Z",
  "updated": "2015-11-16T08:01:46.000Z",
  "title": "Minimality of p-adic rational maps with good reduction",
  "authors": [
    "Ai-Hua Fan",
    "Shilei Fan",
    "Lingmin Liao",
    "Yuefei Wang"
  ],
  "comment": "20 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "A rational map with good reduction in the field $\\mathbb{Q}\\_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\\mathbb{P}^1(\\mathbb{Q}\\_p)$ over $\\mathbb{Q}\\_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $\\mathbb{P}^1(\\mathbb{Q}\\_p)$ is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of the periodic orbits and minimal subsystems. For any prime $p$, a criterion of minimality for rational maps with good reduction is obtained. When $p=2$, a complete characterization of minimal rational maps with good reduction is given in terms of their coefficients. It is also proved that a rational map of degree $2$ or $3$ can never be minimal on the whole space $\\mathbb{P}^1(\\mathbb{Q}\\_2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-16T08:01:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "p-adic rational maps",
      "minimality",
      "minimal subsystems",
      "periodic orbits",
      "adic numbers defines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151104856F"
    }
  }
}