{
  "id": "1405.4287",
  "version": "v2",
  "published": "2014-05-16T19:59:53.000Z",
  "updated": "2015-03-01T20:12:57.000Z",
  "title": "Combinatorial models for spaces of cubic polynomials",
  "authors": [
    "Alexander Blokh",
    "Lex Oversteegen",
    "Ross Ptacek",
    "Vladlen Timorin"
  ],
  "comment": "52 pages, 12 figures (in the new version a few typos have been corrected and some proofs have been expanded). arXiv admin note: substantial text overlap with arXiv:1401.5123",
  "categories": [
    "math.DS"
  ],
  "abstract": "A model for the Mandelbrot set is due to Thurston and is stated in the language of geodesic laminations. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials, even conjectural models are missing, one possible reason being that the higher degree analog of the MLC conjecture is known to be false. We provide a combinatorial model for an essential part of the parameter space of complex cubic polynomials, namely, for the space of all cubic polynomials with connected Julia sets all of whose cycles are repelling (we call such polynomials \\emph{dendritic}). The description of the model turns out to be very similar to that of Thurston.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-16T19:59:53.000Z",
      "abstract": "To construct a model for a connectedness locus of polynomials of degree $d\\ge 3$ (cf with Thurston's model of the Mandelbrot set), we define \\emph{linked} geolaminations $\\mathcal{L}_1$ and $\\mathcal{L}_2$. An \\emph{accordion} is defined as the union of a leaf $\\ell$ of $\\mathcal{L}_1$ and leaves of $\\mathcal{L}_2$ crossing $\\ell$. We show that any accordion behaves like a gap of one lamination and prove that the maximal \\emph{perfect} (without isolated leaves) sublaminations of $\\mathcal{L}_1$ and $\\mathcal{L}_2$ coincide. In the cubic case let $\\mathcal{D}_3\\subset \\mathcal{M}_3$ be the set of all \\emph{dendritic} (with only repelling cycles) polynomials. Let $\\mathcal{MD}_3$ be the space of all \\emph{marked} polynomials $(P, c, w)$, where $P\\in \\mathcal{D}_3$ and $c$, $w$ are critical points of $P$ (perhaps, $c=w$). Let $c^*$ be the \\emph{co-critical point} of $c$ (i.e., $P(c^*)=P(c)$ and, if possible, $c^*\\ne c$). By Kiwi, to $P\\in \\mathcal{D}_3$ one associates its lamination $\\sim_P$ so that each $x\\in J(P)$ corresponds to a convex polygon $G_x$ with vertices in $\\mathbb{S}$. We relate to $(P, c, w)\\in \\mathcal{MD}_3$ its \\emph{mixed tag} $\\mathrm{Tag}(P, c, w)=G_{c^*}\\times G_{P(w)}$ and show that mixed tags of distinct marked polynomials from $\\mathcal{MD}_3$ are disjoint or coincide. Let $\\mathrm{Tag}(\\mathcal{MD}_3)^+ = \\bigcup_{\\mathcal{D}_3}\\mathrm{Tag}(P,c,w)$. The sets $\\mathrm{Tag}(P, c, w)$ partition $\\mathrm{Tag}(\\mathcal{MD}_3)^+$ and generate the corresponding quotient space $\\mathrm{MT}_3$ of $\\mathrm{Tag}(\\mathcal{MD}_3)^+$. We prove that $\\mathrm{Tag}:\\mathcal{MD}_3\\to \\mathrm{MT}_3$ is continuous so that $\\mathrm{MT}_3$ serves as a model space for $\\mathcal{MD}_3$.",
      "comment": "49 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1401.5123",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-01T20:12:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F20",
      "37F10",
      "37F50"
    ],
    "keywords": [
      "cubic polynomials",
      "combinatorial models",
      "connectedness locus",
      "distinct marked polynomials",
      "convex polygon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.4287B"
    }
  }
}