{
  "id": "1203.3145",
  "version": "v1",
  "published": "2012-03-14T17:05:53.000Z",
  "updated": "2012-03-14T17:05:53.000Z",
  "title": "Equilibrium measures on saddle sets of holomorphic maps on P^2",
  "authors": [
    "John Erik Fornaess",
    "Eugen Mihailescu"
  ],
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "We consider the case of hyperbolic basic sets $\\Lambda$ of saddle type for holomorphic maps $f: \\mathbb P^2\\mathbb C \\to \\mathbb P^2\\mathbb C$. We study equilibrium measures $\\mu_\\phi$ associated to a class of H\\\"older potentials $\\phi$ on $\\Lambda$, and find the measures $\\mu_\\phi$ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $\\delta_{\\mu_\\phi}$ of $\\mu_\\phi$ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $\\mu_\\phi$ in the case when the preimage counting function is constant on $\\Lambda$. For terminal/minimal saddle sets we prove that an invariant measure $\\nu$ obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the \\textit{restriction} $f|_\\Lambda$. This allows then to obtain formulas for the measure $\\nu$ of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of $\\nu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-03-14T17:05:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F15",
      "37D35",
      "37C45",
      "37F10",
      "37F35",
      "37A35"
    ],
    "keywords": [
      "holomorphic maps",
      "hausdorff dimension",
      "study equilibrium measures",
      "terminal/minimal saddle sets",
      "arbitrary bowen balls"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.3145F"
    }
  }
}