Geometric limits of Mandelbrot and Julia sets under degree growth

Details Geometric limits of Mandelbrot and Julia sets under degree growth Geometric limits of Mandelbrot and Julia sets under degree growth

2011-09-09

arxiv.org/abs/1109.2166v1

Mathematics

Dynamical Systems

29 pages, 16 figures (34 pic files), submitted

Scientific paper

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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