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Dynamics of generalised exponential maps

Part of: Complex dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  21 April 2022

PATRICK COMDÜHR ,
VASILIKI EVDORIDOU  and
DAVID J. SIXSMITH
PATRICK COMDÜHR
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany e-mail: comduehr@math.uni-kiel.de
VASILIKI EVDORIDOU
Affiliation:
School of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA e-mails: vasiliki.evdoridou@open.ac.uk, david.sixsmith@open.ac.uk
DAVID J. SIXSMITH
Affiliation:
School of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA e-mails: vasiliki.evdoridou@open.ac.uk, david.sixsmith@open.ac.uk
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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.

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
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

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Footnotes

†

Supported by Engineering and Physical Sciences Research Council grant EP/R010560/1.

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