Question

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.

Answer 1

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 $z^d+c$.