{
  "id": "0911.3727",
  "version": "v6",
  "published": "2009-11-19T09:18:15.000Z",
  "updated": "2012-08-30T02:41:11.000Z",
  "title": "Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups",
  "authors": [
    "Hiroki Sumi",
    "Mariusz Urbanski"
  ],
  "comment": "17 pages, 2 figures. Published in Discrete and Continuous Dynamical Systems Ser. A, 32 (2012), no. 7, 2591--2606",
  "journal": "Discrete and Continuous Dynamical Systems Ser. A, 32 (2012), no. 7, 2591--2606",
  "categories": [
    "math.DS",
    "math.CV",
    "math.PR"
  ],
  "abstract": "We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We estimate the Bowen parameters (zeros of the pressure functions) and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $F$ and the Lyapunov exponent of $F$ with respect to the maximal entropy measure for $F$. Moreover, we show that the equality holds if and only if the generators $f_{j}$ are simultaneously conjugate to the form $a_{j}z^{\\pm d}$ by a linear fractional transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than two.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2012-08-30T02:41:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F35",
      "37F15"
    ],
    "keywords": [
      "bowen parameter",
      "expanding rational semigroups",
      "hausdorff dimension",
      "expanding finitely generated rational semigroups",
      "skew product map"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.3727S"
    }
  }
}