Duke Mathematical Journal

Karpińska's paradox in dimension 3

Walter Bergweiler

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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

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

First available in Project Euclid: 7 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F35: Conformal densities and Hausdorff dimension
Secondary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]


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.

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