Devaney and Krych showed that for the exponential family λe z , where 0 < λ < 1/e, the Julia set consists of uncountably many pairwise disjoint simple curves tending to ∞. Viana proved that these curves are smooth. In this article we consider a quasiregular counterpart of the exponential map, the so-called Zorich maps, and generalize Viana's result to these maps.2010 Mathematics Subject Classification. 37F10 (primary), 30C65, 30D05 (secondary).
Since 1984, many authors have studied the dynamics of maps of the form 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 R 3 , which, in a precise sense, is analogous to the class of maps of the form E a . Our goal in this paper is to make similar generalisations in 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 E a , and have very similar dynamical properties. In some sense this shows that many of the interesting dynamical properties of the map E a arise from its elementary function theoretic structure, rather than as a result of analyticity.
In 1984 Devaney and Krych showed that for the exponential family λe z , where 0 < λ < 1/e, the Julia set consists of uncountably many pairwise disjoint simple curves tending to ∞, which they called hairs. Viana proved that these hairs are smooth. Barański as well as Rottenfußer, Rückert, Rempe and Schleicher gave analogues of the result of Devaney and Krych for more general classes of functions. In contrast to Viana's result we construct in this article an entire function, where the Julia set consists of hairs, which are nowhere differentiable.2010 Mathematics Subject Classification. 30D05 (primary), 37F10, 30C65 (secondary).
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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.
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