MATH 722 — Spring 2000
Topics in Complex Analysis: Complex Dynamics & Teichmüller Theory

 
Instructor: Adam Epstein
Time: TR 1:25-2:40
Room: MT 205

This course will be (somewhat of) a continuation of Prof. Hubbard's Math 613. The central focus will be Thurston's Existence and Uniqueness Theorems for the realization of postcritcally finite branched covers of S^2 as rational maps of P. In order to properly state and prove these results we will need to develop/review some basic Teichmüller Theory (perhaps covered in Math 613):

  • Meromorphic Quadratic Differentials
  • Teichmüller's Existence and Uniqueness Theorems

Appropriate infinitesimal versions of Thurston Rigidity and the relevant formalism can be used to prove other important results in complex dynamics:

  • Finiteness of the set of nonrepelling cycles
  • Smoothness and Transversality of dynamically significant subloci of parameter space
  • Sullivan's No Wandering Domains Theorem

We will prove these results for rational, and more generally for finite type analytic maps: for example, $ez, \wp(z), ....$ For completeness, we will develop/review basic complex dynamics in this more general context.

Prerequisites: Math 613 or a course in Riemann surfaces or a course in complex dynamics.

Last modified: April 7, 2003