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Hyperbolic components of rational maps: quantitative equidistribution and counting. (English) Zbl 1431.37041
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 \(T_{bif}^p\).

MSC:
37F12 Critical orbits for holomorphic dynamical systems
37F46 Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets
32U40 Currents
37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems
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