While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

Let f be Fatou's function, that is, f (z) = z + 1 + e −z. We prove that the escaping set of f has the structure of a`spider's web' and we show that this result implies that the non-escaping endpoints of the Julia set of f together with innity form a totally disconnected set. We also present a well-known transcendental entire function, due to Bergweiler, for which the escaping set is a spider's web and we point out that the same property holds for some families of functions.

The family of exponential maps fa(z) = e z + a is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set J(fa). When a ∈ (−∞, −1), and more generally when a belongs to the Fatou set F (fa), it is known that J(fa) can be written as a union of hairs and endpoints of these hairs. In 1990, Mayer proved for a ∈ (−∞, −1) that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to the case where a ∈ F (fa), and showed that it holds even for the smaller set of all escaping endpoints.We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. It turns out that this property is closely related to a topological structure known as a 'spider's web'; in particular we give a new topological characterisation of spiders' webs that may be of independent interest. We also show how our results can be applied to Fatou's function,Still assuming that a ∈ (−∞, −1), the complementary set of non-escaping endpoints of f a satisfies the following identities:see Corollary 2.2 and Proposition 2.4. Here J r (f a ) is the radial Julia set, a set of particular importance. The results of [2] naturally suggest the question whether ∞ is an explosion point for J r (f a ) also. It is known [36, Section 2] that J r (f a ) has Hausdorff dimension strictly greater than one, which is compatible with this possibility. Nonetheless, we prove here that the sets of escaping and non-escaping endpoints are topologically very different from each other.Theorem 1.2 (Non-escaping endpoints do not explode). Let a ∈ (−∞, −1). Then the set J r (f a ) ∪ {∞} is totally separated.

Let f be a function in the Eremenko-Lyubich class B, and let U be an unbounded, forward invariant Fatou component of f . We relate the number of singularities of an inner function associated to f | U with the number of tracts of f . In particular, we show that if f lies in either of two large classes of functions in B, and also has finitely many tracts, then the number of singularities of an associated inner function is at most equal to the number of tracts of f .Our results imply that for hyperbolic functions of finite order there is an upper bound -related to the order -on the number of singularities of an associated inner function.arXiv:1807.07270v1 [math.DS]

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.