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arXiv · 1109.2166

Geometric limits of Mandelbrot and Julia sets under degree growth

Suzanne Hruska Boyd·Michael J. Schulz
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Abstract

First, for the family P_{n,c}(z) = z^n + c, we show that the geometric limit of the Mandelbrot sets M_n(P) as n tends to infinity exists and is the closed unit disk, and that the geometric limit of the Julia sets J(P_{n,c}) as n tends to infinity is the unit circle, at least when the modulus of c is not one. Then we establish similar results for some generalizations of this family; namely, the maps F_{t,c} (z) = z^t+c for real t>= 2, and the rational maps R_{n,c,a} (z) = z^n + c + a/z^n.

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Suzanne Hruska Boyd, Michael J. Schulz. 2011-09-09. Geometric limits of Mandelbrot and Julia sets under degree growth. https://arxiv.org/abs/1109.2166

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