{
  "id": "1705.05276",
  "version": "v1",
  "published": "2017-05-15T14:51:52.000Z",
  "updated": "2017-05-15T14:51:52.000Z",
  "title": "Hyperbolic components of rational maps: {Q}uantitative equidistribution and counting",
  "authors": [
    "Thomas Gauthier",
    "Yûsuke Okuyama",
    "Gabriel Vigny"
  ],
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "Let $\\Lambda$ be a quasi-projective variety and assume that, either $\\Lambda$ is a subvariety of the moduli space $\\mathcal{M}_d$ of degree $d$ rational maps, or $\\Lambda$ parametrizes an algebraic family $(f_\\lambda)_{\\lambda\\in\\Lambda}$ of degree $d$ rational maps on $\\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $\\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the $\\log^+$ of the modulus of the multipliers of periodic points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-15T14:51:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational map",
      "hyperbolic components",
      "uantitative equidistribution",
      "distinct neutral cycles",
      "moduli space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}