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.
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.