Hyperbolic components of rational maps: Quantitative equidistribution and counting

  • Thomas Gauthier

    Université de Picardie - Jules Verne, Amiens Cedex 1, France and École Polytechnique, Palaiseau, France
  • Yûsuke Okuyama

    Kyoto Institute of Technology, Japan
  • Gabriel Vigny

    Université de Picardie - Jules Verne, Amiens, France
Hyperbolic components of rational maps: Quantitative equidistribution and counting cover
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Abstract

Let be a quasi-projective variety and assume that, either is a subvariety of the moduli space of degree rational maps, or parametrizes an algebraic family of degree rational maps on . We prove the equidistribution of parameters having distinct neutral cycles towards the bifurcation current letting the periods of the cycles go to , with an exponential speed of convergence. Several consequences of this result are:

  • a precise asymptotic of the number of hyperbolic components of parameters admitting distinct attracting cycles of exact periods as in term of the mass of the bifurcation measure and compute that mass in the case where . In particular, in , the number of such components is asymptotic to , provided that min is large enough.

  • in the moduli space of polynomials of degree , among hyperbolic components such that all (finite) critical points are in the immediate basins of (not necessarily distinct) attracting cycles of respective exact periods , the proportion of those components, counted with multiplicity, having at least two critical points in the same basin of attraction is exponentially small.

  • in , we prove the equidistribution of the centers of the hyperbolic components admitting distinct attracting cycles of exact periods towards the bifurcation measure with an exponential speed of convergence.

  • we have equidistribution, up to extraction, of the parameters having distinct cycles of given multipliers towards the bifurcation current outside a pluripolar set of multipliers as the minimum of the periods of the cycles goes to .

As a by-product, we also get the weak genericity of hyperbolic postcritically finiteness in the moduli space of rational maps. 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 of the moduli of the multipliers of periodic points.

Cite this article

Thomas Gauthier, Yûsuke Okuyama, Gabriel Vigny, Hyperbolic components of rational maps: Quantitative equidistribution and counting. Comment. Math. Helv. 94 (2019), no. 2, pp. 347–398

DOI 10.4171/CMH/462