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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.
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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