Aims: 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.

The course will have three main themes. Firstly, to understand the local behaviour of holomorphic transformations in one complex variable. Second, to understand the global behaviour of holomorphic maps focussing on complex quadratic polynomials. These exhibit dynamically important features such as chaotic behaviour. Third, we explore the parameter space of quadratic polynomials and the Mandelbrot set. Here we see examples of structural stability, structural instability and renormalisation behaviour.

Content:

We will cover some of the following topics:

• Local dynamics of holomorphic maps
• Expanding maps, shadowing, closing lemmas
• The theory of external rays
• Global dynamical behaviour of hyperbolic Julia sets, Markov partitions and symbolic dynamics
• Global behaviour of arbitrary Julia sets
• Structural stability, shadowing, closing lemmas
• Global properties of parameter space, the Mandelbrot set and renormalisation

Learning outcomes:

• 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