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 paper](http://arxiv.org/abs/math/9201282)1. 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.



Hausdorff dimension of higher powers of the Mandebrot set ?
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 $z^d+c$.


