{
  "id": "0708.3187",
  "version": "v4",
  "published": "2007-08-23T15:15:19.000Z",
  "updated": "2011-05-10T18:15:33.000Z",
  "title": "Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups",
  "authors": [
    "Rich Stankewitz",
    "Hiroki Sumi"
  ],
  "comment": "To appear in Trans. Amer. Math. Soc",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \\in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of $G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $g \\in G$ are distributed within the Julia set of the entire semigroup $G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-05-10T18:15:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "37F50",
      "30D05"
    ],
    "keywords": [
      "julia set",
      "postcritically bounded polynomial semigroups",
      "dynamical properties",
      "distinct fatou components",
      "complex polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.3187S"
    }
  }
}