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.
- Dave Sixsmith: Background in complex analysis
- Vasiliki Evdoridou: Introduction to holomorphic dynamics
- Vasiliki Evdoridou: Iteration of polynomials
- Vasiliki Evdoridou: The Mandelbrot set
- Dave Sixsmith: Introduction to transcendental dynamics
- Dave Sixsmith: Examples of transcendental dynamics
These videos are part of the
London Mathematical Society‘s
Online Graduate Lecture Series. These are supported by the LMS, and organized by WOMBL.
