{
  "id": "0911.1875",
  "version": "v1",
  "published": "2009-11-10T20:47:16.000Z",
  "updated": "2009-11-10T20:47:16.000Z",
  "title": "A dynamical pairing between two rational maps",
  "authors": [
    "Clayton Petsche",
    "Lucien Szpiro",
    "Thomas J. Tucker"
  ],
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Given two rational maps $\\varphi$ and $\\psi$ on $\\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<\\varphi,\\psi>$ which is closely related to the canonical height functions $h_\\varphi$ and $h_\\psi$ associated to these maps. Our main results show a strong connection between the value of $<\\varphi,\\psi>$ and the canonical heights of points which are small with respect to at least one of the two maps $\\varphi$ and $\\psi$. Several necessary and sufficient conditions are given for the vanishing of $<\\varphi,\\psi>$. We give an explicit upper bound on the difference between the canonical height $h_\\psi$ and the standard height $h_\\st$ in terms of $<\\sigma,\\psi>$, where $\\sigma(x)=x^2$ denotes the squaring map. The pairing $<\\sigma,\\psi>$ is computed or approximated for several families of rational maps $\\psi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-11-10T20:47:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G50",
      "14G40",
      "37P15"
    ],
    "keywords": [
      "rational maps",
      "dynamical pairing",
      "explicit upper bound",
      "canonical height functions",
      "strong connection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.1875P"
    }
  }
}