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My third question about Shishikura's result :

Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper1. The Mandelbrot set is defined by iterating to infinity the z^2+c map.

Does his result also apply for higher powers, such as z^8 + c ?

Thanks again.

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Yes, it does. See the full statement of Theorem 2 on page 6. The assumptions of the theorem are:

Suppose that a rational map $f_0$ of degree $d\ (> 1)$ has a parabolic fixed point ζ with multiplier exp(2πip/q) ($p, q \in\mathbb{Z}, \mathit{gcd}(p, q) = 1$) and that the immediate parabolic basin of ζ contains only one critical point of $f_0$.

This is the case for $z^d+c$.

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  • $\begingroup$ Indeed - and by universality of the Mandelbrot and Mulitbrot sets (proved by McMullen), the same will be true in any non-trivial one-dimensional family you can think up. $\endgroup$
    – Lasse Rempe
    Commented Aug 21, 2015 at 15:59

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