Duke Mathematical Journal
- Duke Math. J.
- Volume 154, Number 3 (2010), 599-630.
Karpińska's paradox in dimension 3
It was proved by Devaney and Krych, by McMullen, and by Karpińska that, for , the Julia set of is an uncountable union of pairwise disjoint simple curves tending to infinity, and the Hausdorff dimension of this set is , but the set of curves without endpoints has Hausdorff dimension . We show that these results have -dimensional analogues when the exponential function is replaced by a quasi-regular self-map of introduced by Zorich.
Duke Math. J. Volume 154, Number 3 (2010), 599-630.
First available in Project Euclid: 7 September 2010
Permanent link to this document
Digital Object Identifier
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