{
  "id": "1304.2834",
  "version": "v2",
  "published": "2013-04-10T03:12:32.000Z",
  "updated": "2014-10-09T16:55:46.000Z",
  "title": "The McMullen Map in Positive Characteristic",
  "authors": [
    "Alon Levy"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.AG",
    "math.DS"
  ],
  "abstract": "McMullen proved the moduli space of complex rational maps can be parametrized by the spectrum of all periodic-point multipliers up to a finite amount of data, with the well-understood exception of Latt\\`{e}s maps. We generalize his method to large positive characteristic. McMullen's method is analytic; a modified version of the method using rigid analysis works over a function field over a finite field of characteristic larger than the degree of the map. Over a finite field with such characteristic it implies that, generically, rational maps can indeed be parametrized by their multiplier spectra up to a finite-to-one map. Moreover, the set of exceptions, that is positive-dimension varieties in moduli space with identical multipliers, maps to just a finite set of multiplier spectra. We also prove an application, generalizing a result of McMullen over the complex numbers: there is no generally convergent purely iterative root-finding algorithm over a non-archimedean field whose residue characteristic is larger than either the degree of the algorithm or the degree of the polynomial whose roots the algorithm finds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-10T03:12:32.000Z",
      "abstract": "McMullen proved that the moduli space of complex rational maps can be parametrized by the spectrum of multipliers at all periodic points up to a finite amount of data, with the well-understood exception of Latt\\`{e}s maps. We generalize his method to large positive characteristic. The method is analytic and does not work over a finite field. However, a modified version of the method using rigid analysis does work over a function field over a finite field of characteristic larger than the degree of the map; over a finite field with such characteristic it implies that, generically, rational maps can indeed be parametrized by their multiplier spectra up to a finite-to-one map. Moreover, the set of exceptions, that is positive-dimension varieties in moduli space with identical multipliers, maps to just a finite set of multiplier spectra.",
      "comment": "12 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-09T16:55:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P45",
      "14D10"
    ],
    "keywords": [
      "positive characteristic",
      "mcmullen map",
      "finite field",
      "moduli space",
      "multiplier spectra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.2834L"
    }
  }
}