{
  "id": "2202.12372",
  "version": "v1",
  "published": "2022-02-24T21:55:21.000Z",
  "updated": "2022-02-24T21:55:21.000Z",
  "title": "The parabolic and near-parabolic renormalization for a class of polynomial maps and its applications",
  "authors": [
    "X. Zhang"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:math/0605514, arXiv:1510.00043 by other authors",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "For a class of polynomial maps of one variable with a parabolic fixed points and degrees bigger than $21$, the parabolic renormalization is introduced based on Fatou coordinates and horn maps, and a type of maps which are invariant under the parabolic renormalization is also given. For the small perturbation of these kinds of maps, the near-parabolic renormalization is also introduced based on the first return maps defined on the fundamental regions. As an application, we show the existence of non-renormalizable polynomial maps with degrees bigger than $21$ such that the Julia sets have positive Lebesgue measure and Cremer fixed points, this provides a positive answer for the classical Fatou conjecture (the existence of Julia set with positive area) with degrees bigger than $21$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-24T21:55:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "near-parabolic renormalization",
      "degrees bigger",
      "application",
      "julia set",
      "fixed points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}