{
  "id": "math/9509221",
  "version": "v1",
  "published": "1995-09-07T00:00:00.000Z",
  "updated": "1995-09-07T00:00:00.000Z",
  "title": "On measure and Hausdorff dimension of Julia sets for holomorphic Collet--Eckmann maps",
  "authors": [
    "Feliks Przytycki"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $f:\\bar\\bold C\\to\\bar\\bold C$ be a rational map on the Riemann sphere , such that for every $f$-critical point $c\\in J$ which forward trajectory does not contain any other critical point, $|(f^n)'(f(c))|$ grows exponentially fast (Collet--Eckmann condition), there are no parabolic periodic points, and else such that Julia set is not the whole sphere. Then smooth (Riemann) measure of the Julia set is 0. For $f$ satisfying additionally Masato Tsujii's condition that the average distance of $f^n(c)$ from the set of critical points is not too small, we prove that Hausdorff dimension of Julia set is less than 2. This is the case for $f(z)=z^2+c$ with $c$ real, $0\\in J$, for a positive line measure set of parameters $c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1995-09-07T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "julia set",
      "holomorphic collet-eckmann maps",
      "hausdorff dimension",
      "critical point",
      "satisfying additionally masato tsujiis condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1995math......9221P"
    }
  }
}