Hausdorff dimension of higher powers of the Mandebrot set ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T13:34:54Z http://mathoverflow.net/feeds/question/37230 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37230/hausdorff-dimension-of-higher-powers-of-the-mandebrot-set Hausdorff dimension of higher powers of the Mandebrot set ? Alexis Monnerot-Dumaine 2010-08-31T08:19:34Z 2010-08-31T13:03:14Z <p>My third question about Shishikura's result :</p> <p>Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in <a href="http://arxiv.org/abs/math/9201282" rel="nofollow">this paper</a>1. The Mandelbrot set is defined by iterating to infinity the z^2+c map.</p> <p>Does his result also apply for higher powers, such as z^8 + c ?</p> <p>Thanks again.</p> http://mathoverflow.net/questions/37230/hausdorff-dimension-of-higher-powers-of-the-mandebrot-set/37261#37261 Answer by Jacques Carette for Hausdorff dimension of higher powers of the Mandebrot set ? Jacques Carette 2010-08-31T13:03:14Z 2010-08-31T13:03:14Z <p>Yes, it does. See the full statement of Theorem 2 on page 6. The assumptions of the theorem are:</p> <blockquote> <p>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 $f_0$.</p> </blockquote> <p>This is the case for $z^d+c$.</p>