{
  "id": "math/0403078",
  "version": "v2",
  "published": "2004-03-03T16:47:54.000Z",
  "updated": "2004-12-21T17:47:59.000Z",
  "title": "Iteration at the boundary of the space of rational maps",
  "authors": [
    "Laura DeMarco"
  ],
  "comment": "25 pages",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "Let $Rat_d$ denote the space of holomorphic self-maps of ${\\bf P}^1$ of degree $d\\geq 2$, and $\\mu_f$ the measure of maximal entropy for $f\\in Rat_d$. The map of measures $f\\mapsto\\mu_f$ is known to be continuous on $Rat_d$, and it is shown here to extend continuously to the boundary of $Rat_d$ in $\\bar{Rat}_d \\simeq {\\bf P}^{2d+1}$, except along a locus $I(d)$ of codimension $d+1$. The set $I(d)$ is also the indeterminacy locus of the iterate map $f\\mapsto f^n$ for every $n\\geq 2$. The limiting measures are given explicitly, away from $I(d)$. The degenerations of rational maps are also described in terms of metrics of non-negative curvature on the Riemann sphere: the limits are polyhedral.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-12-21T17:47:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational maps",
      "riemann sphere",
      "holomorphic self-maps",
      "indeterminacy locus",
      "iterate map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......3078D"
    }
  }
}