# Dynamics of generalised exponential maps

## TL;DR

This paper extends the study of exponential map dynamics from complex functions to a broader class of continuous maps in c2, demonstrating that key properties are due to their elementary structure rather than analyticity.

## Contribution

The paper generalizes the dynamical properties of exponential maps b5_a to a large class of continuous, not necessarily quasiregular, maps in c2, highlighting the role of elementary function structure.

## Key findings

- Similar dynamical properties observed in broader class of maps
- Dynamical complexity arises from elementary structure, not analyticity
- Extension of Julia set concepts to non-quasiregular maps

## Abstract

Since 1984, many authors have studied the dynamics of maps of the form $\mathcal{E}_a(z) = e^z - a$, with $a > 1$. It is now well-known that the Julia set of such a map has an intricate topological structure known as a Cantor bouquet, and much is known about the dynamical properties of these functions. In recent papers some of these ideas have been generalised to a class of quasiregular maps in $\mathbb{R}^3$, which, in a precise sense, is analogous to the class of maps of the form $\mathcal{E}_a$. Our goal in this paper is to make similar generalisations in $\mathbb{R}^2$. In particular, we show that there is a large class of continuous maps, which, in general, are not even quasiregular, but are closely analogous to the map $\mathcal{E}_a$, and have very similar dynamical properties. In some sense this shows that many of the interesting dynamical properties of the map $\mathcal{E}_a$ arise from its elementary function theoretic structure, rather than as a result of analyticity.