{
  "id": "0710.4406",
  "version": "v4",
  "published": "2007-10-24T08:41:47.000Z",
  "updated": "2013-05-29T07:35:51.000Z",
  "title": "Rigidity and non local connectivity of Julia sets of some quadratic polynomials",
  "authors": [
    "Genadi Levin"
  ],
  "comment": "Final version of the Addendum, minor changes. To appear",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "For an infinitely renormalizable quadratic map $f_c: z\\mapsto z^2+c$ with the sequence of renormalization periods ${k_m}$ and rotation numbers ${t_m=p_m/q_m}, we prove that if $\\limsup k_m^{-1}\\log |p_m|>0$, then the Mandelbrot set is locally connected at $c$. We prove also that if $\\limsup |t_{m+1}|^{1/q_m}<1$ and $q_m\\to \\infty$, then the Julia set of $f_c$ is not locally connected and the Mandelbrot set is locally connected at $c$ provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor. Abstract of the Addendum: We improve one of the main results of the above paper.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-05-29T07:35:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45",
      "37F50",
      "37G15",
      "30D05"
    ],
    "keywords": [
      "non local connectivity",
      "julia sets",
      "quadratic polynomials",
      "main results",
      "infinitely renormalizable quadratic map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.4406L"
    }
  }
}