{
  "id": "math/0412438",
  "version": "v1",
  "published": "2004-12-21T19:23:52.000Z",
  "updated": "2004-12-21T19:23:52.000Z",
  "title": "The boundary of the moduli space of quadratic rational maps",
  "authors": [
    "Laura DeMarco"
  ],
  "comment": "38 pages, 3 figures",
  "categories": [
    "math.DS",
    "math.AG"
  ],
  "abstract": "Let $M_2$ be the space of quadratic rational maps $f:{\\bf P}^1\\to{\\bf P}^1$, modulo the action by conjugation of the group of M\\\"obius transformations. In this paper a compactification $X$ of $M_2$ is defined, as a modification of Milnor's $\\bar{M}_2\\iso{\\bf CP}^2$, by choosing representatives of a conjugacy class $[f]\\in M_2$ such that the measure of maximal entropy of $f$ has conformal barycenter at the origin in ${\\bf R}^3$, and taking the closure in the space of probability measures. It is shown that $X$ is the smallest compactification of $M_2$ such that all iterate maps $[f]\\mapsto [f^n]\\in M_{2^n}$ extend continuously to $X \\to \\bar{M}_{2^n}$, where $\\bar{M}_d$ is the natural compactification of $M_d$ coming from geometric invariant theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-12-21T19:23:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45"
    ],
    "keywords": [
      "quadratic rational maps",
      "moduli space",
      "geometric invariant theory",
      "natural compactification",
      "iterate maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....12438D"
    }
  }
}