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arXiv · 0902.2686

Karpi\'nska's paradox in dimension three

Walter Bergweiler
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Abstract

For 0 < c < 1/e the Julia set of f(z) = c exp(z) is an uncountable union of pairwise disjoint simple curves tending to infinity [Devaney and Krych 1984], the Hausdorff dimension of this set is two [McMullen 1987], but the set of curves without endpoints has Hausdorff dimension one [Karpinska 1999]. We show that these results have three-dimensional analogues when the exponential function is replaced by a quasiregular self-map of three-space introduced by Zorich.

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Walter Bergweiler. 2009-02-16. Karpi\'nska's paradox in dimension three. https://doi.org/10.1215/00127094-2010-047

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