{
  "id": "1308.4324",
  "version": "v2",
  "published": "2013-08-20T14:56:01.000Z",
  "updated": "2016-10-27T16:31:09.000Z",
  "title": "Quasisymmetric geometry of the Julia sets of McMullen maps",
  "authors": [
    "Weiyuan Qiu",
    "Fei Yang",
    "Yongcheng Yin"
  ],
  "comment": "19 pages, 3 figures",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_\\lambda(z)=z^m+\\lambda/z^\\ell$, where $\\lambda\\in\\mathbb{C}\\setminus\\{0\\}$ and $\\ell$, $m$ are positive integers satisfying $1/\\ell+1/m<1$. If the free critical points of $f_\\lambda$ are escaped to the infinity, we prove that the Julia set $J_\\lambda$ of $f_\\lambda$ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpi\\'{n}ski carpet (which is also standard in some sense). If the free critical points are not escaped, we give a sufficient condition on $\\lambda$ such that $J_\\lambda$ is a Sierpi\\'{n}ski carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-20T14:56:01.000Z",
      "title": "A geometric characterization of the Julia sets of McMullen maps",
      "abstract": "We study the geometric properties of the Julia sets of McMullen maps $f_\\lambda(z)=z^m+\\lambda/z^l$, where $\\lambda\\in\\mathbb{C}\\setminus\\{0\\}$ and $l,m$ are both positive integers satisfying $1/l+1/m<1$. If the free critical points of $f_\\lambda$ are escaping, we prove that the Julia set $J_\\lambda$ of $f_\\lambda$ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor circles or a round Sierpi\\'nski carpet (which is also standard in some sense). If the free critical points are not escaping, we give a sufficient condition on $\\lambda$ to grantee that $J_\\lambda$ is a Sierpi\\'nski carpet and prove most of them are quasisymmetrically equivalent to a round one. In particular, there exists non-hyperbolic rational map whose Julia set is quasisymmetrically equivalent to a round carpet.",
      "comment": "14 pages, 3 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-10-27T16:31:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45",
      "37F10",
      "37F25"
    ],
    "keywords": [
      "julia set",
      "mcmullen maps",
      "geometric characterization",
      "free critical points",
      "quasisymmetrically equivalent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.4324Q"
    }
  }
}