Cornell Math - MATH 628, Fall 2005

MATH 628: Complex Dynamical Systems (Fall 2005)

Instructor: Yulij Ilyashenko

Meeting Time & Room

The course will consist in three parts.

The first one is Complex Dynamics. This name is traditionally reserved for iterates of rational maps of the Riemann sphere. Classical results of Fatou and Julia will be presented, like the estimates of the number of stable periodic orbits, and fractal structure of the Julia sets. The Sullivan theorem about nonwandering components of the Fatou set will be exposed. This theorem produced a revolution in complex dynamics.

The second part is the theory of the nonlinear Stokes phenomena. This is a branch of the theory of normal forms of germs of maps and vector fields. The main result is the description of the functional moduli of analytic classification of these germs. The main tool is almost complex structures used as well in the proof of the Sullivan theorem.

The third part presents the theory of linear equations with complex time. The main result is the solution of the Riemann-Hilbert problem due to Plemelj and Bolibrukh. The dramatic history of the problem will be presented.

All the three parts are unified by the use of the theory of normal forms, whose main results will be presented as well.