Cantor Bouquets and Wandering Domains for a Class of Entire Functions

Dourekas, Yannis (2020). Cantor Bouquets and Wandering Domains for a Class of Entire Functions. PhD thesis The Open University.

Abstract

In this thesis we focus on different topological structures that arise as a result of the iteration of functions in a class of sums of exponentials, along with the different Fatou components that exist for functions in this class, making particular reference to wandering domains.

For many transcendental entire functions, the escaping set has the structure of a Cantor bouquet, consisting of uncountably many disjoint curves. Rippon and Stallard showed that there are many functions for which the escaping set has a new connected structure known as an infinite spider’s web. We investigate a connection between these two topological structures for functions in our class.

The issue of whether an analytic function has wandering domains has long been of interest in complex dynamics. Sullivan proved in 1985 that rational maps do not have wandering domains. On the other hand, several transcendental entire functions have wandering domains. Using recent results on the relationship between Fatou components and the postsingular set, we prove that functions in a subset of our class do not have wandering domains. We also prove that for many of the functions the Julia set is the whole plane.

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