MAT 685     Fall 2006

Holomorphic Dynamics

Mikhail Lyubich

Holomorphic dynamics studies iterates of rational endomorphisms of the Riemann sphere. The main object associated to such a dynamical system is called the Julia set: it is an intricate fractal subset of the sphere that depends very sensitively on the parameters. It is nowhere dense unless it coincides with the whole sphere. Until the 1990's, it was believed that once the Julia set is not the whole sphere, it must have zero area. The belief changed, however, and a substantial effort was then made to construct a Julia set with positive area. One strategy for constructing such a set was laid out by Adrien Douady. Recently this strategy was successfully carried out by Buff and Cheritat, based on previous work of McMullen, Shishikura, Yoccoz, and others. In this course, starting from scratch, we will attempt to go through the necessary background and the construction.