{
  "id": "2411.04938",
  "version": "v1",
  "published": "2024-11-07T18:14:57.000Z",
  "updated": "2024-11-07T18:14:57.000Z",
  "title": "Baby Mandelbrot sets and Spines in some one-dimensional subspaces of the parameter space for generalized McMullen Maps",
  "authors": [
    "Suzanne Boyd",
    "Matthew Hoeppner"
  ],
  "comment": "38 pages, 13 figures with 32 subfigures",
  "categories": [
    "math.DS"
  ],
  "abstract": "For the family of complex rational functions of the form $R_{n,c,a}(z) = z^n + \\dfrac{a}{z^n}+c$, known as ``Generalized McMullen maps'', for $a\\neq 0$ and $n \\geq 3$ fixed, we study the boundedness locus in some one-dimensional slices of the $(a,c)$-parameter space, by fixing a parameter or imposing a relation. First, if we fix $c$ with $|c|\\geq 6$ while allowing $a$ to vary, assuming a modest lower bound on $n$ in terms of $|c|$, we establish the location in the $a$-plane of $n$ ``baby\" Mandelbrot sets, that is, homeomorphic copies of the original Mandelbrot set. We use polynomial-like maps, introduced by Douady and Hubbard and applied for the subfamily $R_{n,a,0}$ by Devaney. Second, for slices in which $c=ta$, we again observe what look like baby Mandelbrot sets within these slices, and begin the study of this subfamily by establishing a neighborhood containing the boundedness locus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-07T18:14:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "32A20",
      "32A19"
    ],
    "keywords": [
      "baby mandelbrot sets",
      "generalized mcmullen maps",
      "parameter space",
      "one-dimensional subspaces",
      "boundedness locus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}