{
  "id": "1704.06624",
  "version": "v1",
  "published": "2017-04-21T16:39:52.000Z",
  "updated": "2017-04-21T16:39:52.000Z",
  "title": "Natural extensions of unimodal maps: prime ends of planar embeddings and semi-conjugacy to sphere homeomorphisms",
  "authors": [
    "Philip Boyland",
    "André de Carvalho",
    "Toby Hall"
  ],
  "comment": "72 pages, 14 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $f\\colon I\\to I$ be a unimodal map with topological entropy $h(f)>\\frac12\\log2$, and let $\\widehat{f}\\colon\\widehat{I}\\to\\widehat{I}$ be its natural extension, where $\\widehat{I}=\\varprojlim(I,f)$. Subject to some regularity conditions, which are satisfied for tent maps and quadratic maps, we give a complete description of the prime ends of the Barge-Martin embedding of $\\widehat{I}$ into the disk, and identify the prime ends rotation number with the height of $f$. We also show that $\\widehat{f}$ is semi-conjugate to a sphere homeomorphism by a semi-conjugacy for which all fibers except one contain at most three points. In the case where $f$ is a post-critically finite tent map, we show that the corresponding sphere homeomorphism is a generalized pseudo-Anosov map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-21T16:39:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "37E30",
      "37B45"
    ],
    "keywords": [
      "sphere homeomorphism",
      "unimodal map",
      "natural extension",
      "planar embeddings",
      "semi-conjugacy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 72,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}