{
  "id": "1809.02586",
  "version": "v1",
  "published": "2018-09-07T17:25:14.000Z",
  "updated": "2018-09-07T17:25:14.000Z",
  "title": "Location of Siegel capture polynomials in parameter spaces",
  "authors": [
    "Alexander Blokh",
    "Arnaud Cheritat",
    "Lex OVersteegen",
    "Vladlen Timorin"
  ],
  "comment": "26 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "A cubic polynomial $f$ with a periodic Siegel disk containing an eventual image of a critical point is said to be a \\emph{Siegel capture polynomial}. If the Siegel disk is invariant, we call $f$ a \\emph{IS-capture polynomial} (or just an IS-capture; IS stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials. In particular, we show that any IS-capture is on the boundary of a unique hyperbolic component determined by the rational lamination of the map. We also relate IS-captures to the cubic Principal Hyperbolic Domain and its closure (by definition, the \\emph{cubic Principal Hyperbolic Domain} consists of cubic hyperbolic polynomials with Jordan curve Julia sets). We prove that, in the slice of cubic polynomials given by a fixed multiplier at one of the fixed points, the closure of the cubic principal hyperbolic domain cannot have bounded complementary domains containing IS-captures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-07T17:25:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45",
      "37F10",
      "37F20",
      "37F50"
    ],
    "keywords": [
      "siegel capture polynomials",
      "parameter space",
      "cubic principal hyperbolic domain",
      "complementary domains containing is-captures",
      "cubic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}