Hyperbolic components of rational maps: quantitative equidistribution and counting (Q2421727)

The paper focuses on the parameter space of rational functions. Let \(\Lambda\) be a quasi-projective variety and assume that, either \(\Lambda\) is a subvariety of the moduli space \(M_d\) of degree \(d\) rational maps, or \(\Lambda\) parametrizes an algebraic family \((f_\lambda)_{\lambda\in \Lambda}\) of degree \(d\) rational maps. It is proved that the equidistribution of parameters having \(p\) distinct neutral cycles towards the bifurcation current \(T_{bif}^p\) letting the periods of the cycles go to \(\infty\), with an exponential speed of convergence. This enables the authors to give: (1) a precise asymptotic estimate of the number of hyperbolic components of parameters admitting \(2d-2\) distinct attracting cycles of exact periods \(n_1, \ldots, d_{2d-2}\) as \(\min_j n_j\to \infty\); (2) a characterization of hyperbolic components such that all (finite) critical points are in the immediate basins of (not necessarily distinct) attracting cycles of respective exact periods \(n_1, \ldots, d_{2d-2}\). The number of these components, counted with multiplicity, having at least two critical points are in the same basin of attraction; (3) a proof of the equidistribution of the centers of the hyperbolic components admitting \(2d-2\) distinct attracting cycles of exact periods \(n_1, \ldots, d_{2d-2}\) towards the bifurcation measure with an exponential speed of convergence; (4) a proof of the equidistribution of the parameters having \(p\) distinct cycles of given multipliers towards the bifurcation current \(T_{bif}^p\).