{
  "id": "0811.1809",
  "version": "v7",
  "published": "2008-11-12T03:06:25.000Z",
  "updated": "2011-02-15T10:06:57.000Z",
  "title": "Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups",
  "authors": [
    "Hiroki Sumi",
    "Mariusz Urbanski"
  ],
  "comment": "Published in Discrete and Continuous Dynamical Systems Ser. A., Vol 30, No. 1, 2011, 313--363. 50 pages, 2 figures",
  "journal": "Discrete and Continuous Dynamical Systems Ser. A., Vol 30, No. 1, 2011, 313--363",
  "categories": [
    "math.DS",
    "math.CV",
    "math.PR"
  ],
  "abstract": "We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.",
  "revisions": [
    {
      "version": "v7",
      "updated": "2011-02-15T10:06:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F35",
      "37F15"
    ],
    "keywords": [
      "julia set",
      "semi-hyperbolic rational semigroups",
      "probability absolutely continuous invariant",
      "nice open set condition holds",
      "borel probability absolutely continuous"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.1809S"
    }
  }
}