{
  "id": "1803.04082",
  "version": "v1",
  "published": "2018-03-12T01:19:40.000Z",
  "updated": "2018-03-12T01:19:40.000Z",
  "title": "Real entropy rigidity under quasi-conformal deformations",
  "authors": [
    "Khashayar Filom"
  ],
  "comment": "44 pages, 1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "We set up a real entropy function $h_\\Bbb{R}$ on the space $\\mathcal{M}'_d$ of M\\\"obius conjugacy classes of real rational maps of degree $d$ by assigning to each class the real entropy of a representative $f\\in\\Bbb{R}(z)$; namely, the topological entropy of its restriction $f\\restriction_{\\hat{\\Bbb{R}}}$ to the real circle. We prove a structure theorem for the real Julia set $\\mathcal{J}_\\Bbb{R}(f):=\\mathcal{J}(f)\\cap\\hat{\\Bbb{R}}$ that will be utilized to establish a rigidity result stating that $h_\\Bbb{R}$ is locally constant on the subspace determined by real maps which are quasi-conformally conjugate to $f$. We also compare the real locus $\\mathcal{M}_d(\\Bbb{R})$ of the moduli space $\\mathcal{M}_d(\\Bbb{C})$ of degree $d$ rational maps to the locus $\\mathcal{M}'_d$ of classes with a real representative, and we characterize maps of maximal real entropy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-12T01:19:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B40",
      "37E05",
      "37E10",
      "37F10",
      "37F30",
      "37P45"
    ],
    "keywords": [
      "real entropy rigidity",
      "quasi-conformal deformations",
      "maximal real entropy",
      "real julia set",
      "real entropy function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}