{
  "id": "2202.06734",
  "version": "v1",
  "published": "2022-02-14T14:15:41.000Z",
  "updated": "2022-02-14T14:15:41.000Z",
  "title": "Lavaurs algorithm for cubic symmetric polynomials",
  "authors": [
    "Alexander Blokh",
    "Lex G. Oversteegen",
    "Nikita Selinger",
    "Vladlen Timorin",
    "Sandeep Chowdary Vejandla"
  ],
  "comment": "27 pages, 3 figures. arXiv admin note: text overlap with arXiv:2201.11434",
  "categories": [
    "math.DS"
  ],
  "abstract": "To investigate the degree $d$ connectedness locus, Thurston studied \\emph{$\\sigma_d$-invariant laminations}, where $\\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f_c(z) = z^2 +c$. In the same spirit, we consider the space of all \\emph{cubic symmetric polynomials} $f_\\lambda(z)=z^3+\\lambda^2 z$ in three articles. In the first one we construct the lamination $C_sCL$ together with the induced factor space $\\mathbb{S}/C_sCL$ of the unit circle $\\mathbb{S}$. As will be verified in the third paper, $\\mathbb{S}/C_sCL$ is a monotone model of the \\emph{cubic symmetric connectedness locus}, i.e., the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for constructing $C_sCL$ analogous to the Lavaurs algorithm for constructing a combinatorial model $\\mathcal{M}^{comb}_2$ of the Mandelbrot set $\\mathcal{M}_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-14T14:15:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F20",
      "37F10"
    ],
    "keywords": [
      "cubic symmetric polynomials",
      "lavaurs algorithm",
      "unit circle",
      "symmetric connectedness locus",
      "invariant laminations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}