123 research outputs found
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
Fixed points of composite entire and quasiregular maps
We give a new proof of the result that if f and g are transcendental entire
functions, then the composite function f(g) has infinitely many fixed points.
The method yields a number of generalization of this result. In particular, it
extends to quasiregular maps in space.Comment: 15 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
- …