David J. Sixsmith

On the set where the iterates of an entire function are neither escaping nor bounded

For a transcendental entire function f, we study the set of points BU(f) whose iterates under f neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if U is a Fatou component that meets BU(f), then most boundary points of U (in the sense of harmonic measure) lie in BU(f). We prove this using a new result concerning the set of limit points of the iterates of f on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.

Hollow quasi-Fatou components of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from

Dynamical sets whose union with infinity is connected

\mathbb{R}^d

On permutable meromorphic functions

to

Periodic domains of quasiregular maps

\mathbb{R}^d

Julia and Escaping Set Spiders’ Webs of Positive Area

. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set. Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasi-Fatou components. First, we study the number of complementary components of quasi-Fatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasi-Fatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using novel techniques, and may be of interest even in the case of transcendental entire functions.

The size and topology of quasi-Fatou components of quasiregular maps

Suppose that

Escaping sets of continuous functions

f

The topology of the set of non-escaping endpoints

is a transcendental entire function. In 2014, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected, and an example a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class. It is known that for many transcendental entire functions the escaping set has a topological structure known as a spiders web. We use our results to give a large class of functions in the Eremenko-Lyubich class for which the escaping set is not a spiders web. Finally we give a novel topological criterion for certain sets to be a spiders web.

The bungee set in quasiregular dynamics.

We study the class