{
  "id": "math/9201285",
  "version": "v1",
  "published": "1991-05-28T00:00:00.000Z",
  "updated": "1991-05-28T00:00:00.000Z",
  "title": "On the Lebesgue measure of the Julia set of a quadratic polynomial",
  "authors": [
    "Mikhail Lyubich"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The goal of this note is to prove the following theorem: Let $p_a(z) = z^2+a$ be a quadratic polynomial which has no irrational indifferent periodic points, and is not infinitely renormalizable. Then the Lebesgue measure of the Julia set $J(p_a)$ is equal to zero. As part of the proof we discuss a property of the critical point to be {\\it persistently recurrent}, and relate our results to corresponding ones for real one dimensional maps. In particular, we show that in the persistently recurrent case the restriction $p_a|\\omega(0)$ is topologically minimal and has zero topological entropy. The Douady-Hubbard-Yoccoz rigidity theorem follows this result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1991-05-28T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "julia set",
      "quadratic polynomial",
      "lebesgue measure",
      "irrational indifferent periodic points",
      "douady-hubbard-yoccoz rigidity theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}