## Duke Mathematical Journal

### Karpińska's paradox in dimension 3

Walter Bergweiler

#### Abstract

It was proved by Devaney and Krych, by McMullen, and by Karpińska that, for $0\lt\lambda\lt 1/e$, the Julia set of $\lambda e^z$ is an uncountable union of pairwise disjoint simple curves tending to infinity, and the Hausdorff dimension of this set is $2$, but the set of curves without endpoints has Hausdorff dimension $1$. We show that these results have $3$-dimensional analogues when the exponential function is replaced by a quasi-regular self-map of ${\mathbb R}^3$ introduced by Zorich.

#### Article information

Source
Duke Math. J. Volume 154, Number 3 (2010), 599-630.

Dates
First available in Project Euclid: 7 September 2010

https://projecteuclid.org/euclid.dmj/1283865314

Digital Object Identifier
doi:10.1215/00127094-2010-047

Mathematical Reviews number (MathSciNet)
MR2730579

Zentralblatt MATH identifier
1218.37057

#### Citation

Bergweiler, Walter. Karpińska's paradox in dimension 3. Duke Math. J. 154 (2010), no. 3, 599--630. doi:10.1215/00127094-2010-047. https://projecteuclid.org/euclid.dmj/1283865314.