Phil J. Rippon : Topics in the iteration of entire functions

The study of the iteration of entire and meromorphic functions began in the late 19th century and there were major developments in the early 20th century with the work of Fatou and Julia. Following the rekindling of widespread interest in the subject in the last 20 years, a vast literature has developed. Many new techniques have been introduced and the subject has drawn heavily on results from classical complex analysis, often closely related to the hyperbolic metric.

This course will provide an introduction to many of the key techniques used in complex iteration, illustrating these techniques by applying them to prove a number of milestone results in the subject.

For simplicity, the course will concentrate on the iteration of entire functions, but with an emphasis on transcendental entire functions. Amongst the topics that it is hoped to cover are:

• behaviour near fixed points and periodic cycles of various types;
• the domain of normality, also called the Fatou set, and its complement, the Julia set;
• classification of the components of the domain of normality (attracting components, parabolic components, Siegel discs, Baker domains and wandering domains);
• the role of inverse function singularities;
• use of quasiconformal mappings to prove results on the non-existence of wandering domains;
• results on the existence and non-existence of Baker domains;
• use of approximation theory to construct examples;
• Hausdorff dimension of Julia sets;
• bifurcation diagrams for families of functions.

As background, the course will assume a sound knowledge of complex analysis, up to and including the Riemann mapping theorem. A number of more advanced results from complex analysis will also be needed, and for these careful statements with indications of proof will be supplied.

The Julia set of a function with infinitely many Baker domains :