05-8   Berichtsreihe des Mathematischen Seminars der Universität Kiel

Heike Siebert:

Fixed Points and Normal Families of Quasiregular Mappings

In this paper we consider families of K-quasiregular mappings. We show that such a family is normal, if for every of its mappings f there is k > 1 such that the k-th iterate of f has no fixed point. Moreover we examine to what extent the result holds if we consider only repelling fixed points rather than fixed points in general. Furthermore, we prove that a family of K-quasiregular mappings is quasinormal, if it contains only mappings that do not have periodic points of some period greater than one. This implies that a quasiregular mapping, which is defined in the whole euclidean n-space and which has an essential singularity in infinity, has infinitely many periodic points of any period greater than one.

The results generalize results of M. Essen, S. Wu, D. Bargmann and W. Bergweiler for holomorphic functions.

Mathematics Subject Classification (1991): 30C65 (primary classification), 30D45, 37C25 (secondary classification)

Keywords: Quasiregular mappings, normal families, fixed points, periodic points

Mail an Jens Burmeister
[Thu Feb 19 18:56:36 2009]