# Fall 2016 - M 597 - Complex Dynamics

# Resources

## Class Materials

- Syllabus
- Lecture notes (11/16/2016). These notes are a frequently updated draft, the date is in the document. Please let me know of any mistakes you find or suggestions you have.
- Normal forms via uniformization (10/28/2016). This is the draft of the proof of local normal forms at attracting, repelling, and parabolic fixed points using the uniformization theorem, handed out in class. It will eventually be incorporated into the lecture notes.
- Homework

## Basics

The 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.

- John Milnor: Dynamics in One Complex Variable, Introductory Lectures - Electronic version is freely accessible from campus computers. An early version is available for free at https://arxiv.org/abs/math/9201272
- Lennart Carleson, Theodore W. Gamelin: Complex Dynamics - Electronic version is also freely available on campus.
- Alan F. Beardon: Iteration of Rational Functions
- Norbert Steinmetz: Rational Iteration

## Background

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

- Theodore W. Gamelin: Complex Analysis - Covers all the complex analysis background up to Riemann surfaces and the uniformization theorem. The book even covers a little bit of complex dynamics in Chapter XII. Electronic version is freely accessible from campus.
- Olli Lehto, K. I. Virtanen: Quasiconformal Mappings in the Plane - The standard reference for quasiconformal maps.

## 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.

- Daniel S. Alexander: A History of Complex Dynamics, From Schröder to Fatou and Julia - Early history of complex dynamics
- Daniel S. Alexander, Felice Iavernaro, Alessandro Rosa: Early Days in Complex Dynamics: A history of complex dynamics in one variable during 1906–1942 - A follow-up, covering a somewhat later (but still early) period.

## Special Topics

- 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.