Karpi\'nska's paradox in dimension three
Dynamical Systems
2019-12-19 v1 Complex Variables
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.
Cite
@article{arxiv.0902.2686,
title = {Karpi\'nska's paradox in dimension three},
author = {Walter Bergweiler},
journal= {arXiv preprint arXiv:0902.2686},
year = {2019}
}
Comments
21 pages