%0 Journal Article
%T On measure and Hausdorff dimension of Julia sets for holomorphic Collet--Eckmann maps
%A Feliks Przytycki
%J Mathematics
%D 1995
%I arXiv
%X Let $f:\bar\bold C\to\bar\bold C$ be a rational map on the Riemann sphere , such that for every $f$-critical point $c\in J$ which forward trajectory does not contain any other critical point, $|(f^n)'(f(c))|$ grows exponentially fast (Collet--Eckmann condition), there are no parabolic periodic points, and else such that Julia set is not the whole sphere. Then smooth (Riemann) measure of the Julia set is 0. For $f$ satisfying additionally Masato Tsujii's condition that the average distance of $f^n(c)$ from the set of critical points is not too small, we prove that Hausdorff dimension of Julia set is less than 2. This is the case for $f(z)=z^2+c$ with $c$ real, $0\in J$, for a positive line measure set of parameters $c$.
%U http://arxiv.org/abs/math/9509221v1