Karpi\'nska's paradox in dimension three

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.Comment: 21 page

Iteration of meromorphic functions

This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).Comment: 38 pages. Abstract added in migration. See http://analysis.math.uni-kiel.de/bergweiler/ for recent comments and correction

Green's function and anti-holomorphic dynamics on a torus

We give a new, simple proof of the fact recently discovered by C.-S. Lin and C.-L. Wang that the Green function of a torus has either three or five critical points, depending on the modulus of the torus. The proof uses anti-holomorphic dynamics. As a byproduct we find a one-parametric family of anti-holomorphic dynamical systems for which the parameter space consists only of hyperbolic components and analytic curves separating them.Comment: 17 pages, 3 figures (some details added, some overall revision