{
  "id": "1112.0531",
  "version": "v3",
  "published": "2011-12-02T18:20:58.000Z",
  "updated": "2014-07-04T14:25:54.000Z",
  "title": "A separation theorem for entire transcendental maps",
  "authors": [
    "Anna Miriam Benini",
    "Nuria Fagella"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\\in\\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their landing points, separate the plane into finitely many regions, each containing exactly one interior $p-$periodic point or one parabolic immediate basin invariant under $f^p$. This result generalizes the Goldberg-Milnor Separation Theorem for polynomials, and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel discs; that \"hidden components\" of a bounded Siegel disc have to be either wandering domains or preperiodic to the Siegel disc itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-04T14:25:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "37F20"
    ],
    "keywords": [
      "entire transcendental maps",
      "periodic point",
      "siegel disc",
      "singular values",
      "parabolic immediate basin invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.0531B"
    }
  }
}