MATH 6280: Complex Dynamical Systems (Fall 2010)

Instructor: Yulij Ilyashenko

The course is devoted to three central topics in complex dynamical systems: holomorphic dynamics in one variable; Riemann-Hilbert problem for complex linear systems; polynomial differential equations and complex foliations in the real and complex planes. Fatou and Julia sets, periodic points, structure of Fatou sets, hyperbolicity; regular and irregular singular points of linear systems, necessary and sufficient conditions for solvability of the Riemann-Hilbert problem, Stokes phenomena; limit cycles of planar polynomial vector fields, nonaccumulation of limit cycles to hyperbolic polycycles, density and rigidity properties of the polynomial foliations in the complex plane.

Textbooks:

• John Milnor, Dynamics in One Complex Variable.
• L. Carleson and T. Gamelin, Complex Dynamical Systems, Springer, 1993.
• Yu. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, AMS, 2007.