Ma147c: Complex Dynamics 

Instructor: Anton Gorodetski
Office: 282 Sloan
Phone: 395-4350
E-mail: asgor@caltech.edu

Office Hours: after the lectures or by appointment.

       The study of holomorphic maps under iteration was already started at the end of the last century. It goes back to Cayley, who investigated Newton's method for complex mappings, and to Schroder, Koenigs, Leau and Botkher, who analyzed the local behavior of holomorphic maps at fixed points in general. The first who studied the global behavior of iterated rational maps were Fatou and Julia in the early twenties. They observed that by the action of such maps the complex sphere divides into a stable and an instable set, today named the Fatou set and the Julia set, respectively. Fatou and Julia started to classify the components of the stable set for rational maps. Then, with the exception of some substantial progress by Cremer, Siegel and a few other mathematicians, there followed a period of dormancy.
    The second period of interest in the subject began in 1980, and it is not surprising that the revival of complex iteration theory fell in the time of modern computer graphics: what was found by Fatou and Julia could now be seen on a computer display, and the pictures obtained were so fascinating that they forced the further development of the mathematical theory. In particular, the Mandelbrot set became extremely popular, also in the non-mathematical world. Sullivan introduced quasi-conformal techniques into the field of holomorphic dynamics, and his proof of absence of non-wandering domains allowed completion of the classification of components in the stable set. Douady and Hubbard, the pioneers in modern complex quadratic dynamics, started a deep exploration of the Mandelbrot set. In the meantime the investigations were extended to higher degree polynomials, to entire functions and to several variables, but there remained open problems concerning the quadratic case. In particular, the celebrated problem of local connectivity of the Mandelbrot set is still open.