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MA 871: Complex Dynamical Systems. Fall 2003.

Robert L. Devaney

MW-2-3:30 PM

The aim of this course is to introduce mathematicians
to current research in complex dynamics. We will begin with
an overview of the classical work of Julia and Fatou on the
dynamics of complex analytic maps. Much of the course will be devoted
to the study of the dynamics of complex transcendental functions.
The main goal will be to understand in depth the
dynamics of quadratic functions and the geometry of the Mandelbrot
set, followed by the very different dynamics of the complex exponential.

This course is a graduate level seminar aimed at graduate students in
mathematics. Prerequisites include a background in discrete dynamical
systems (at the level of MA 771 at Boston University) and a graduate
level course in complex analysis (at the level of MA 713). Students should
be familiar with the following topics from dynamics: iteration,
local behavior near periodic points, chaotic behavior, symbolic dynamics,
and some elementary bifurcation theory. From complex analysis,
students should know the Riemann mapping theorem, Schwarz Lemma,
and properties of analytic mappings.

Students who register for the course will be required to make several
presentations during the course on a topic of current
interest in the literature.

There is no textbook for the course. Lectures will be
drawn mainly from the research literature. Some background sources
include:

- Alexander, Daniel S.
** A History of Complex Dynamics **
Vieweg, 1994.
- Carleson, L. and Gamelin, T. W.
** Complex Dynamics**.
- Devaney, R. L.
**An Introduction to Chaotic
Dynamical Systems**,
Perseus Press (Westview Press) (chapter 3 only).
- McMullen, C.
** Complex Dynamics and Renormalization**,
Princeton University Press.
- Milnor, J.
** Dynamics in One Complex Variable**,
Vieweg.

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Topics to be covered include:

Periodic points of analytic maps
Geometry of the Julia set
Stable domains
Role of critical points
The Mandelbrot set
Quasiconformal maps
Dynamics of transcendental functions
Hairs and external rays
The parameter plane for the complex exponential
Knaster continua
Structural instability of complex maps