Dynamics of generalised exponential maps

Comdühr, Patrick; Evdoridou, Vasiliki and Sixsmith, Dave (2023). Dynamics of generalised exponential maps. Mathematical Proceedings of the Cambridge Philosophical Society, 174(1) pp. 123–136.

DOI: https://doi.org/10.1017/S0305004122000160

Abstract

Since 1984, many authors have studied the dynamics of maps of the form ε_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 ε_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 ℝ² maps, which, in general, are not even quasiregular, but are somehow analogous to ε_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 ε_a , including the Cantor bouquet structure, which is closely related to several topological properties of the endpoints of the Julia set.

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