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Dynamics of generalised exponential maps
Published online by Cambridge University Press: 21 April 2022
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.
It is rather surprising that many of the interesting dynamical properties of the maps 
$\mathcal{E}_a$ actually arise from their elementary function theoretic structure, rather than as a result of analyticity. We show this by studying a large class of continuous 
$\mathbb{R}^2$ maps, which, in general, are not even quasiregular, but are somehow analogous to 
$\mathcal{E}_a$. We define analogues of the Fatou and the Julia set and we prove that this class has very similar dynamical properties to those of 
$\mathcal{E}_a$, including the Cantor bouquet structure, which is closely related to several topological properties of the endpoints of the Julia set.
Information
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 1 , January 2023 , pp. 123 - 136
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society
Footnotes
†
Supported by Engineering and Physical Sciences Research Council grant EP/R010560/1.
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