Fixed curves near fixed points

TL;DR

This paper classifies the local dynamics near fixed points of certain planar quasiregular mappings by relating them to finite Blaschke products, revealing the structure and number of fixed curves landing at the fixed point.

Contribution

It introduces a classification method for fixed points of quasiregular mappings using finite Blaschke products, connecting complex dynamics with geometric properties.

Findings

Determines the number of fixed curves landing at a fixed point.

Provides a classification of dynamics based on parameters of the linear part.

Establishes a link between quasiregular mappings and finite Blaschke products.

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 $z_0$.