# On slow escaping and non-escaping points of quasimeromorphic mappings

## TL;DR

This paper investigates the behavior of quasimeromorphic mappings near essential singularities, showing the existence of points with arbitrarily slow escape to infinity and extending known results from complex dynamics to higher dimensions.

## Contribution

It extends results on slow escaping points and growth rates from complex functions to quasimeromorphic mappings in higher dimensions.

## Key findings

- Existence of points with arbitrarily slow escape to infinity.
- New growth rate estimates near essential singularities.
- Extension of bounded orbit and bungee set results to quasimeromorphic mappings.

## Abstract

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.