{
  "id": "1707.01843",
  "version": "v1",
  "published": "2017-07-06T15:52:11.000Z",
  "updated": "2017-07-06T15:52:11.000Z",
  "title": "Non-escaping endpoints do not explode",
  "authors": [
    "Vasiliki Evdoridou",
    "Lasse Rempe-Gillen"
  ],
  "comment": "18 pages, 2 figures",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "The family of exponential maps $f_a(z)= e^z+a$ is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set $J(f_a)$. When $a\\in (-\\infty,-1)$, and more generally when $a$ belongs to the Fatou set of $f_a$, it is known that $J(f_a)$ can be written as a union of \"hairs\" and \"endpoints\" of these hairs. Alhabib and the second author, extending a result of Mayer, recently proved that, while the set of endpoints is totally separated, its union with infinity is a connected set. In fact this is true even for the smaller set of all escaping endpoints. We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. It turns out that this property is closely related to a topological structure known as a \"spider's web\"; in particular we give a new topological characterisation of spiders' webs that may be of independent interest. We also show how our results can be applied to Fatou's function, $z\\mapsto z + 1 + e^{-z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-06T15:52:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "30D05",
      "54G15"
    ],
    "keywords": [
      "non-escaping endpoints",
      "topological structure",
      "spiders web",
      "fundamental importance",
      "smaller set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}