The next lecture in this series is here.
The series will consist of 6 one-hour lectures which will focus on the iteration of entire functions. We explore, among other things, some famous fractal Julia sets and the well-known Mandelbrot set. In particular, we will cover the following topics:
- Equicontinuity, normal families, Montel’s theorem, Riemann mapping theorem, the Riemann sphere.
- Iteration of polynomials. Definition of the Fatou set and the Julia set for a polynomial. Examples.
- The filled Julia set. Fixed and periodic points.
- An introduction to the properties of the Fatou set and the Julia set.
- The Mandelbrot set: its definition and properties.
- Introduction to the iteration of transcendental entire functions.
- Similarities and differences between polynomials and transcendental entire functions.
- The escaping set: definition, properties, and its important role.
- Examples of the Fatou, Julia and escaping sets for transcendental entire functions.
The lecture series is addressed to PhD students from diverse mathematical backgrounds. We shall assume a basic knowledge of complex analysis and a little topology. Some more advanced background in complex analysis will be covered in the first lecture. No knowledge of dynamics will be assumed.
This video is part of the London Mathematical Society‘s Online Graduate Lecture Series. These are supported by the LMS, and organized by the Open University Dynamical Systems group.