Karpińska's paradox in dimension 3
Bergweiler, Walter
Duke Math. J., Tome 151 (2010) no. 1, p. 599-630 / Harvested from Project Euclid
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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.
Publié le : 2010-09-15
Classification:  37F35,  30C65,  30D05,  37F10 
@article{1283865314,
     author = {Bergweiler, Walter},
     title = {Karpi\'nska's paradox in dimension 3},
     journal = {Duke Math. J.},
     volume = {151},
     number = {1},
     year = {2010},
     pages = { 599-630},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1283865314}
}

Bergweiler, Walter. Karpińska's paradox in dimension 3. Duke Math. J., Tome 151 (2010) no. 1, pp.  599-630. http://gdmltest.u-ga.fr/item/1283865314/