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Distribution of postcritically finite polynomials. II: Speed of convergence. (English) Zbl 1419.37040
In the moduli space of degree \(d\) polynomials, the authors prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, they prove the exponential speed of convergence for \(\mathcal{C}^2\)-observables.
This improves results obtained with arithmetic methods in the uncritical family and in the space of degree \(d\) polynomials, see [C. Favre et al., Math. Ann., 335, 311–361 (2006; Zbl 1175.11029)]; Israel Journal of Mathematics, 209, 235–292 (2015; Zbl 1352.37202)].
The techniques allow the deduction from that the equidistribution of hyperbolic parameters with (\(d-1\)) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for \(\mathcal{C}^1\)-observables. As an application, the equidistribution (up to an explicit extraction) of parameters with (\(d-1\)) distinct cycles with prescribed multiplier toward the bifurcation measure for any (\(d-1\)) multipliers outside a pluripolar set, is proved.

MSC:
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
32U40 Currents
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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