Quadratic polynomials, multipliers and equidistribution. (English) Zbl 1334.37038

Buff, Xavier; Gauthier, Thomas

Proc. Am. Math. Soc. 143, No. 7, 3011-3017 (2015).

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.

Reviewer: Walter Bergweiler (Kiel)

MSC:

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

Keywords:

Mandelbrot set; periodic point; multiplier; Green function

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