1. To introduce students to the theory of the iteration of functions of one complex variable, and its fundamental objects;

2. To introduce students to some topics of current and recent research in the field;

3. To study various advanced results from complex analysis, and show how to apply these in a dynamical setting;

4. To illustrate that many results in complex analysis are "magic", in that there is no reason to expect them in a real-variable context, and the implications of this in complex dynamics;

5. To explain how complex-variable methods have been instrumental in questions purely about real-valued one-dimensional dynamical systems, such as the logistic family.

6. To deepen students' appreciations for formal reasoning and proof. After completing the module, students should be able to:
1. understand the compactification of the complex plane to the Riemann sphere, and use spherical distances and derivatives.
2. us e Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.
3. state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.
4. determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.
5. apply advanced results from complex analysis in the setting of complex dynamics.
6. determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.

(LO1) To understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives.

(LO2) To be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.

(LO3) To be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.

(LO4) To be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.

(LO5) To know how to apply advanced results from complex analysis in a dynamical setting.

(LO6) To be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.

(S1) Problem solving/ critical thinking/ creativity analysing facts and situations and applying creative thinking to develop appropriate solutions.

(S2) Problem solving skills

Syllabus:
1. Introduction; review of complex functions and related topics; iteration.
2. The Riemann sphere: spherical distance, spherical derivative, Möbius transformations.
3. The Julia and Fatou sets; (in)stability of periodic cycles; filled Julia set of a polynomial; simple examples.
4. Normal families: Weierstraß theorem, Hurwitz theorem, Marty's theorem, Montel's theorem, Picard's theorem
5. Properties of the Julia set, dense orbits and sensitive dependence on initial conditions.
6. Zalcman's lemma and chaos of the Julia set.
7. The Mandelbrot set: Definition, characterisation, connection with the logistic family.