MAT 748: Topics in Complex Dynamics
Spring 2002
Course Description:
This course is an introduction to the mathematical tools that are
essential in the modern theory of holomorphic dynamical systems. The
emphasis will be on techniques from geometric complex analysis and
quasiconformal theory in dimension 1. Along the way we will discuss
some concrete applications of these ideas in dynamics of
Kleinian groups and rational maps of the sphere, in particular the
quadratic family and the ubiquitous Mandelbrot set.
Here is a rough sketch of possible topics:
- Some geometric function theory, uniformization of Riemann surfaces,
the hyperbolic metric, Schwarz lemma, classical Koebe distortion theorems,
modulus and extremal length, length-area inequalities.
- Quasiconformal maps in the plane and the measurable Riemann mapping
theorem of Morrey-Ahlfors-Bers.
- Invariant line fields, ergodic properties of rational maps of the sphere.
- Holomorphic motions, lambda-lemma and J-stability.
- Introduction to the Mandelbrot set and the quadratic family,
uniformization of the hyperbolic components.
General prerequisite: A good knowledge of complex analysis (Math 609 would
be sufficient). For the most part the course does not require any
background in dynamics, but ocassionally some knowledge of complex dynamics
will be helpful (Math 748 Fall 2000 would be more than enough).
Lecturer:
Saeed Zakeri
Office: DRL 4N59, Tel. 3-9074.
Office Hours: by appointment
Class meetings: Tuesdays and Thursdays
3:00-4:30 pm in DRL 4C4.
Email:
zakeri@math.upenn.edu
Saeed Zakeri
References