Math 189A: Special Topics in Complex Dynamics
Spring 2001
Prof Lesley Ward

# Welcome to the Math 189A Homepage!

Math 189A, Complex Dynamics, is a new half course at HMC; it will run for the first time in the second half of Spring 2001. The course meets on MW at 2:45pm, and lectures will run from Monday 5 March to Wednesday 2 May. The text is Iteration of Rational Functions, by Alan F. Beardon; it is available at Huntley.

The course is about the iteration of rational maps (quotients of polynomials) of the complex plane to itself. Each rational map divides the complex plane into two complementary sets: the Fatou set and the Julia set, named for two of the pioneers of dynamical systems. On the Fatou set, the dynamics of the rational map is regular, and iterates of nearby starting points have the same longterm behaviour. On the Julia set, the dynamics is chaotic, nearby points are driven apart by iteration, and longterm behaviour is sensitively dependent on the choice of initial point.

We will study fixed points and periodic points, and their attracting, repelling, or neutral nature. We will discuss the structure and properties of the Fatou and Julia sets. For example, the repelling periodic points of a given rational mapping are dense in the Julia set, and the Julia set can also be generated as the set of accumulation points of the backwards orbit of a generic point in the Fatou set. Other topics include critical points, conjugacy, equicontinuity, and normal families of maps. We will also study the Mandelbrot set, and its role as a guide to the behaviour under iteration of the model family of quadratic maps f(z) = z2 + c.

Office hours
• Mondays 10am -- noon, Olin 1281
• Tuesdays 10 -- 11am, Olin 1281
Also by appointment.

Weekly homework assignments

Due in class on Wednesdays. Available here in postscript format:
• Homework 1 Due Tues 20 March. Julia and Fatou sets, periodic points, conjugation.
• Homework 2 Due Wed 28 March. Rational maps, and the iteration of z -> z2.
• Homework 3 Due Tues 3 April. Iteration of z -> z2 + c.
• Homework 4 Due Wed 11 April. Prepare presentation on article.
• Homework 5 Due Wed 18 April. Fatou and Julia sets, equicontinuity, normal families.

Course Outline