Dynamics of generalised exponential maps

Comdühr, Patrick; Evdoridou, Vasiliki and Sixsmith, Dave (2022). Dynamics of generalised exponential maps. Mathematical Proceedings of the Cambridge Philosophical Society (Early Access).

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

Abstract

Since 1984, many authors have studied the dynamics of maps of the form εa(z) = ez - 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 εa 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 εa . We define analogues of the Fatou and the Julia set and we prove that this class has very similar dynamical properties to those of εa , including the Cantor bouquet structure, which is closely related to several topological properties of the endpoints of the Julia set.

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