{
  "id": "1404.7417",
  "version": "v3",
  "published": "2014-04-29T16:13:54.000Z",
  "updated": "2015-06-16T17:11:13.000Z",
  "title": "Bifurcation measures and quadratic rational maps",
  "authors": [
    "Laura DeMarco",
    "Xiaoguang Wang",
    "Hexi Ye"
  ],
  "comment": "Final version, to appear in Proceedings of the LMS",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\\mathbb{P}^1$. We focus on the family of curves, $Per_1(\\lambda)$ for $\\lambda$ in $\\mathbb{C}$, defined by the condition that each $f\\in Per_1(\\lambda)$ has a fixed point of multiplier $\\lambda$. We prove that the curve $Per_1(\\lambda)$ contains infinitely many postcritically-finite maps if and only if $\\lambda = 0$; addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of a map $f$ define distinct bifurcation measures along $Per_1(\\lambda)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-01T14:19:16.000Z",
      "abstract": "We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\\mathbb{P}^1$. We prove that the curve $Per_1(\\lambda)$, consisting of all maps with a fixed point of multiplier $\\lambda$, contains infinitely many postcritically-finite maps if and only if $\\lambda = 0$; this is a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of a map $f$ define distinct bifurcation measures along $Per_1(\\lambda)$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-06-16T17:11:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45",
      "37P30"
    ],
    "keywords": [
      "quadratic rational maps",
      "define distinct bifurcation measures",
      "postcritically-finite maps",
      "special case",
      "moduli space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.7417D"
    }
  }
}