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.

Although detailed descriptions of the possible types of behaviour inside periodic Fatou components have been known for over 100 years, a classification of wandering domains has only recently been given. Recently, simply connected wandering domains were classified into nine possible types and examples of escaping wandering domains of each of these types were constructed. Here we consider the case of oscillating wandering domains, for which only six of these types are possible. We use a new technique based on approximation theory to construct examples of all six types of oscillating simply connected wandering domains. This requires delicate arguments since oscillating wandering domains return infinitely often to a bounded part of the plane. Our technique is inspired by that used by Eremenko and Lyubich to construct the first example of an oscillating wandering domain, but with considerable refinements which enable us to show that the wandering domains are bounded, to specify the degree of the mappings between wandering domains and to give precise descriptions of the dynamical behaviour of these mappings.