Fall 2016 - M 597 - Complex Dynamics

Resources

Class Materials

Basics

Cubic Newton parameter space zoom-inThe following four textbooks are great introductions to the topic, each with a slightly different focus and different flavors of presentation. The book of Milnor will be the main reference for the first part of the course. All of these should be available through the library.

Background

We will use many results from complex analysis without proof. These are good references for this background material.

History

It is often interesting to see how and why certain fields in mathematics evolved. These two books cover the early history up to the 1940s.

Special Topics

Julia set on the Riemann sphere
  • Curtis T. McMullen:  Complex Dynamics and Renormalization - Great introduction to renormalization techniques for quadratic polynomials, also very valuable as a quick reference for important background results from complex analysis, conformal geometry, and basic complex dynamics. Parts of this should be covered in the second half of the course.
  • Bodil Branner, Núria Fagella: Quasiconformal Surgery in Holomorphic Dynamics - Quasiconformal surgery was introduced into complex dynamics in the 1980s by Dennis Sullivan in the proof of the No-Wandering-Domains Theorem, and has since then been one of the most important and versatile tools in the theory. This book gives a well-written introduction to the technique and many of its applications. Parts of this should be covered in the second half of the course.
  • Joseph H. Silverman: The Arithmetic of Dynamical Systems - A very different approach to complex dynamics, from the point of view of algebra and number theory. This field has been very active in recent years, but we will probably not cover it in class.

Software

  • Mandel by Wolf Jung - Easily the best software to explore Julia sets and the Mandelbrot sets, and it is free and open source, too.