{
  "id": "1806.07152",
  "version": "v1",
  "published": "2018-06-19T10:58:09.000Z",
  "updated": "2018-06-19T10:58:09.000Z",
  "title": "The no lakes of Wada theorem in complex dynamics and a new entry in Sullivan's dictionary",
  "authors": [
    "Yûsuke Okuyama"
  ],
  "comment": "5 pages",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "We show that the Julia set $J(f)$ of a rational function $f$ on $\\mathbb{P}^1$ of degree $>1$ is never the boundary of lakes of Wada. Simultaneously, we also show that with respect to the equilibrium measure $\\mu_f$ of $f$, the residual Julia set $J_0(f)$ of $f$ is an either full or null set, and $\\mu_f(J_0(f))=0$ if and only if there is a Fatou component of $f$ which is totally invariant under $f^2$, and then $J_0(f)=\\emptyset$. This in particular yields the dynamical counterpart to Abikoff's theorem in the theory of Kleinian groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-19T10:58:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex dynamics",
      "sullivans dictionary",
      "wada theorem",
      "residual julia set",
      "kleinian groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}