Hyperbolic components of rational maps: Quantitative equidistribution and counting

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 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 $n_1,\dots ,n_{d-1}$, the proportion of those components, counted with multiplicity, having at least two critical points in the same basin of attraction is exponentially small.

• 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.