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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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