MAT 656: Topics in Dynamical Systems
Spring 2003
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 of geometric complex analysis and
quasiconformal theory in dimension 1. Along the way we will discuss
concrete applications of these techniques in the dynamical study of
rational maps of the sphere and, when appropriate, Kleinian groups.
Here is an outline of possible topics:
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Uniformization of Riemann surfaces, 2D hyperbolic geometry,
Schwarz lemma, normal families, classical Koebe distortion
theorems, modulus and extremal length, length-area inequalities.
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Basic properties of Julia sets (of rational maps) and limit sets
(of Kleinian groups).
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Quasiconformal maps in the plane, Beltrami differentials, conformal
structures, the measurable Riemann mapping theorem, deformations,
Sullivan's no wandering domain theorem, basic quasiconformal surgery.
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Holomorphic motions and lambda-lemma, applications in stability
theory of holomorphic dynamical systems.
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Introduction to the quadratic family and the Mandelbrot set.
General prerequisite: A good knowledge of complex analysis (Math 543
would be sufficient) and curiosity to learn about a beautiful piece of
modern mathematics. For the most part the course will not require any
background in dynamical systems, but some knowledge of it will be
helpful.
Lecturer:
Saeed Zakeri
Office: Math Tower 4-114, Tel. 2-8276.
Office Hours: by appointment
Class meetings: Tuesdays and Thursdays
2:20-3:40 pm in Physics P-124.
Email:
zakeri@math.sunysb.edu
Saeed Zakeri