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Quadratic polynomials, multipliers and equidistribution. (English) Zbl 1334.37038
For \(\varrho\in \mathbb{C}\) and \(n\in \mathbb{N}\) let \(\nu_{n,\varrho}\) be the probability measure having point masses of equal size at all points \(c\in \mathbb{C}\), for which the polynomial \(z^2+c\) has a periodic point of period \(n\) and multiplier \(\varrho\). The authors consider the limiting behavior of the sequence \((\nu_{n,\varrho_n})\) for a sequence \((\varrho_n)\) in \(\mathbb{C}\). They show that if \((\log|\varrho_n|)/n \to L\in [-\infty,\infty)\), then \((\nu_{n,\varrho_n})\) converges weakly to some measure \(\mu\). Moreover, they show that with \(\eta=\max\{0,2L-2\log 2\}\) we have \(\mu=\Delta \max\{g_M,\eta\}\), where \(\Delta\) stands for the generalized Laplacian and where \(g_M\) is the Green function of the complement of the Mandelbrot set.

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
Full Text: DOI
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