# Constructing a quasiregular analogue of $z \exp(z)$ in dimension 3

## TL;DR

This paper constructs explicit examples of quasiregular mappings in three dimensions that mimic the complex function z exp(z), including those with unique zeros and specific dynamical properties, advancing the understanding of higher-dimensional quasiregular dynamics.

## Contribution

It introduces the first explicit quasiregular analogue of z exp(z) in dimension 3 with a unique zero and explores their dynamics, also constructing quasimeromorphic mappings with finite backward orbits.

## Key findings

- First explicit quasiregular analogue of z exp(z) in dimension 3.
- Construction of quasiregular mappings with exactly one zero.
- Development of quasimeromorphic mappings with finite backward orbit of infinity.

## Abstract

We construct a quasiregular analogue of the function $z\exp(z)$ in dimension 3, which gives the first explicit example of a quasiregular mapping of transcendental type that has exactly one zero. We then modify the construction to create a family of such quasiregular mappings and study their dynamics. From this, we also construct the first quasimeromorphic mappings with an essential singularity at infinity where the backward orbit of infinity is non-empty and finite.