Math 189A: Special Topics in Complex Dynamics
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.
Also by appointment.
- Mondays 10am -- noon, Olin 1281
- Tuesdays 10 -- 11am, Olin 1281
Weekly homework assignments
Due in class on Wednesdays. Available here in postscript
- 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.
- 60% for Homework: weekly, due on Wednesdays in class.
- 10% for First Project and report.
- 30% for Second Project and report.
Last modified April 13, 2001, by Prof Lesley Ward.
| Math Department |
Last modified: April 2001