The bungee set in quasiregular dynamics

TL;DR

This paper investigates the properties of the bungee set in quasiregular transcendental maps, revealing its infinitude and similarities to entire functions, along with novel behaviors unique to quasiregular dynamics.

Contribution

It is the first study of the bungee set in quasiregular maps, demonstrating its infiniteness and contrasting properties with analytic maps.

Findings

The bungee set of quasiregular maps is infinite.

The bungee set shares properties with that of transcendental entire functions.

Existence of quasiconformal maps with non-empty bungee sets, unlike analytic homeomorphisms.

Abstract

In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that this set is infinite, and shares many properties with the bungee set of a transcendental entire function. By way of contrast, we give examples of novel properties of this set in the quasiregular setting. In particular, we give an example of a quasiconformal map of the plane with a non-empty bungee set; this behaviour is impossible for an analytic homeomorphism.