{
  "id": "2210.06619",
  "version": "v1",
  "published": "2022-10-12T22:56:31.000Z",
  "updated": "2022-10-12T22:56:31.000Z",
  "title": "Genus $g$ Cantor sets and garmane Julia sets",
  "authors": [
    "Alastair N. Fletcher",
    "Daniel Stoertz",
    "Vyron Vellis"
  ],
  "comment": "26 pages, 10 figures",
  "categories": [
    "math.DS",
    "math.GN"
  ],
  "abstract": "The main aim of this paper is to give topological obstructions to Cantor sets in $\\mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our main tool is the genus of a Cantor set. We give a new construction of a genus $g$ Cantor set, the first for which the local genus is $g$ at every point, and then show that this Cantor set can be realized as the Julia set of a uniformly quasiregular mapping. These are the first such Cantor Julia sets constructed for $g\\geq 3$. We then turn to our dynamical applications and show that every Cantor Julia set of a hyperbolic uniformly quasiregular map has a finite genus $g$; that a given local genus in a Cantor Julia set must occur on a dense subset of the Julia set; and that there do exist Cantor Julia sets where the local genus is non-constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-12T22:56:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C50",
      "30C65",
      "37F10"
    ],
    "keywords": [
      "cantor set",
      "garmane julia sets",
      "cantor julia set",
      "local genus",
      "hyperbolic uniformly quasiregular map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}