Fixed curves near fixed points

Abstract

Let H be a composition of an R-linear planar mapping and z → z n . We classify the dynamics of H in terms of the parameters of the R-linear mapping and the degree by associating a certain finite Blaschke product. We apply this classification to this situation where z 0 is a fixed point of a planar quasiregular mapping with constant complex dilatation in a neighbourhood of z 0 . In particular we find how many curves there are that are fixed by f and that land at z 0 . arXiv:1504.05463v1 [math.DS]

Unicritical Blaschke Products and Domains of Ellipticity

“…An application of these results will be given in [7]. There, it will be shown that in the neighbourhood of a fixed point of a quasiregular mapping in the plane with constant complex dilatation and of any local index, the behaviour of the iterates can be determined by a conjugate of a Blaschke product.…”

Epicycloids and Blaschke products

“…The motivation for studying unicritical Blaschke products originally arose in classifying the dynamics of quasiregular mappings of constant complex dilation near fixed points [5]. In [4], it was shown that every unicritical Blaschke product of degree d is conjugate to a unique element of B d and that the connectedness locus C d consists of E d and one point on the relative boundary in…”