# Commentarii Mathematici Helvetici

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**Volume 94, Issue 2, 2019, pp. 347–398**

**DOI: 10.4171/CMH/462**

Published online: 2019-04-17

Hyperbolic components of rational maps: Quantitative equidistribution and counting

Thomas Gauthier^{[1]}, Yûsuke Okuyama

^{[2]}and Gabriel Vigny

^{[3]}(1) Université de Picardie - Jules Verne, Amiens Cedex 1, France and École Polytechnique, Palaiseau, France

(2) Kyoto Institute of Technology, Japan

(3) Université de Picardie - Jules Verne, Amiens, France

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 bifurcation current $T_{\mathrm {bif}}^p$ letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. Several consequences of this result are:

- a precise asymptotic of the number of hyperbolic components of parameters admitting $2d-2$ distinct attracting cycles of exact periods $n_1,\dots, n_{2d-2}$ as $\min_j n_j \to \infty$ in term of the mass of the bifurcation measure and compute that mass in the case where $d=2$. In particular, in $\mathcal{M}_d$, the number of such components is asymptotic to $d^{n_1+\cdots+n_{2d-2}}$, provided that min$_j n_j$ is large enough.

- in the moduli space $\mathcal{P}_d$ of polynomials of degree $d$, among hyperbolic components such that all (finite) critical points are in the immediate basins of (not necessarily distinct) attracting cycles of respective exact periods $n_1,\ldots,n_{d-1}$, the proportion of those components, counted with multiplicity, having at least two critical points in the same basin of attraction is exponentially small.

\item in $\mathcal M_d$, we prove the equidistribution of the centers of the hyperbolic components admitting $2d-2$ distinct attracting cycles of exact periods $n_1,\dots, n_{2d-2}$ towards the bifurcation measure $\mu_{\mathrm {bif}}$ with an exponential speed of convergence.

- we have equidistribution, up to extraction, of the parameters having $p$ distinct cycles of given multipliers towards the bifurcation current $T_{\mathrm {bif}}^p$ outside a pluripolar set of multipliers as the minimum of the periods of the cycles goes to $\infty$.

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 $\log^+$ of the moduli of the multipliers of periodic points.

*Keywords: *Hyperbolic component, Lyapunov exponent, bifurcation measure, equidistribution

Gauthier Thomas, Okuyama Yûsuke, Vigny Gabriel: Hyperbolic components of rational maps: Quantitative equidistribution and counting. *Comment. Math. Helv.* 94 (2019), 347-398. doi: 10.4171/CMH/462