Fixed curves near fixed points

Abstract

Let $H$ be a composition of an $\mathbb{R}$-linear planar mapping and $z\mapsto z^{n}$. We classify the dynamics of $H$ in terms of the parameters of the $\mathbb{R}$-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 $z_{0}$.