{
  "id": "1411.2535",
  "version": "v1",
  "published": "2014-11-10T18:58:15.000Z",
  "updated": "2014-11-10T18:58:15.000Z",
  "title": "Complementary components to the cubic Principal Hyperbolic Domain",
  "authors": [
    "Alexander Blokh",
    "Lex Oversteegen",
    "Ross Ptacek",
    "Vladlen Timorin"
  ],
  "comment": "14 pages; one figure. arXiv admin note: substantial text overlap with arXiv:1305.5799",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\\mathcal{P}_\\lambda$ with the slice $\\mathcal{F}_\\lambda$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $\\lambda$ at $0$. We show that any bounded domain $\\mathcal{W}$ of $\\mathcal{F}_\\lambda\\setminus\\mathcal{P}_\\lambda$ consists of $J$-stable polynomials $f$ with connected Julia sets $J(f)$ and is either of \\emph{Siegel capture} type (then $f\\in \\mathcal{W}$ has an invariant Siegel domain $U$ around $0$ and another Fatou domain $V$ such that $f|_V$ is two-to-one and $f^k(V)=U$ for some $k>0$) or of \\emph{queer} type (then at least one critical point of $f\\in \\mathcal{W}$ belongs to $J(f)$, the set $J(f)$ has positive Lebesgue measure, and carries an invariant line field).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-10T18:58:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45",
      "37F10",
      "37F20",
      "37F50"
    ],
    "keywords": [
      "cubic principal hyperbolic domain",
      "complementary components",
      "invariant line field",
      "invariant siegel domain",
      "positive lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.2535B"
    }
  }
}