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Distribution of points with prescribed derivative in polynomial dynamics. (English) Zbl 1403.37051
Let $$f$$ be a polynomial map of degree $$d\geq2$$ of the complex plane into itself. The classical Brolin’s theorem states that the iterated preimages of all points $$a$$, except for at most one, converge to the equilibrium measure $$\mu_f$$ in the sense $\lim_{n\to\infty} \frac1{d^n}\sum_{f^n(z)=a} \delta_z=\mu_f.$ Here $$f^n$$ is the $$n$$-th iteration of $$f$$. The authors prove a similar statement for the preimages of the derivative of $$f$$: $\lim_{n\to\infty} \frac1{d^n-1}\sum_{(f^n)'(z)=\lambda} \delta_z=\mu_f$ for all $$\lambda$$ outside a polar set $$E$$. Furthermore, the set $$E$$ is empty if $$f$$ has no Siegel disk or is hyperbolic.
In addition, the equidistribution is studied for parameters in the space of degree-$$d$$ polynomials with $$d-1$$ marked critical points.
MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) 32U40 Currents
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