Introductory description

Complex Dynamics is a very active area of the field of Dynamical Systems. This course will be an introduction to the subject focusing on the dynamics of complex quadratic polynomials. This family of examples will be studied using a variety of tools coming from classical and modern techniques in complex analysis, topology, geometry and dynamical systems.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

We will cover some of the following topics:

1. Local dynamics of holomorphic maps
2. Expanding maps, shadowing, closing lemmas.
3. The theory of external rays
4. Global dynamical behaviour of hyperbolic Julia sets, Markov partitions and symbolic dynamics.
5. Global behaviour of arbitrary Julia sets.
6. Structural stability, shadowing, closing lemmas,
7. Global properties of parameter space, the Mandelbrot set and renormalisation

Learning outcomes

By the end of the module, students should be able to:

• Use a variety of techniques to analyse complex dynamical systems
• Understand the role of structural stability in dynamical systems.
• Understand the role of renormalisation in dynamical systems.
• Understand how Markov partitions can be used to understand the behaviour of orbits in dynamical systems.

Subject specific skills

Students will have an in depth knowledge of the behaviour of a beautiful family of dynamical systems. They will be equipped to use a variety of analytic and dynamical techniques to investigate and understand a range of dynamical systems exhibiting chaotic behaviour. They will have the background to study in the area, at graduate level, and to apply the techniques they have learned to various areas of applications where dynamical systems appear.

Transferable skills

Students will be equipped to understand a range of dynamical systems exhibiting complex behaviour. They will have the background to study various areas of applications where dynamical systems appear, such as control systems, engineering and meteorology. More generally, they will have had the opportunity to develop their analytic skills through the study of complex and abstract systems.