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<λ<1/e, the Julia set of λez 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 R3 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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