{
  "id": "math/0408290",
  "version": "v1",
  "published": "2004-08-21T21:02:39.000Z",
  "updated": "2004-08-21T21:02:39.000Z",
  "title": "Hausdorff dimension and conformal measures of Feigenbaum Julia sets",
  "authors": [
    "Artur Avila",
    "Mikhail Lyubich"
  ],
  "comment": "Latex, 51 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon'', there exist many Feigenbaum Julia sets $J(f)$ whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincar\\'e critical exponent $\\de_\\crit$ is equal to the hyperbolic dimension $\\HD_\\hyp(J(f))$. Moreover, if $\\area J(f)=0$ then $\\HD_\\hyp (J(f))=\\HD(J(f))$. In the stationary case, the last statement can be reversed: if $\\area J(f)> 0$ then $\\HD_\\hyp (J(f))< 2$. We also give a new construction of conformal measures on $J(f)$ that implies that they exist for any $\\de\\in [\\de_\\crit, \\infty)$, and analyze their scaling and dissipativity/conservativity properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-08-21T21:02:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F35"
    ],
    "keywords": [
      "feigenbaum julia set",
      "conformal measures",
      "hausdorff dimension",
      "dissipativity/conservativity properties",
      "hairiness phenomenon"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......8290A"
    }
  }
}