Cornell Math - MATH 628, Fall 2000

MATH 628 — Fall 2000
Complex Dynamical Systems

Complex dynamical systems is a classical subject going back to Poincaré, Fatou and Julia. In the last three decades this subject is very much in the focus of interest. The course will present the material from basic topics to open problems.

We will focus on planar differential equations and one-dimensional maps. Despite the low dimension of the phase space, the theory has numerous multidimensional applications. The reason is that many problems in higher dimension may be reduced to low-dimensional ones making use of symmetries or invariant manifolds. On the other hand, low dimensional theory has important applications itself; for instance, it is a base for mathematical models in engineering, ecology and so on.

Theory of differential equations is in turn subject of applications of basic mathematical disciplines like algebra, complex analysis and algebraic geometry. The course will present basic results and tools of the low dimensional theory of differential equations and maps.

Part I. Real and complex planar dynamical systems.

Elementary and complex singular points. Desingularization theorems. Linear and nonlinear Stokes Phenomena. Hilbert sixteenth problem: preliminary results.

Part II. Dynamics on the Riemann sphere

Topics in complex analysis: distortion theorems, quasiconformal maps and uniformization. Normal forms. Julia and Fatou sets. Quasiconformal surgery. Quadratic polynomials.

Prerequisites: Basic knowledge of complex analysis and differential equations.

The first part is covered by several research and survey papers. Amidst them, Paper 1 in the book: Yu.Ilyashenko, editor, Nonlinear Stokes Phenomena, AMS, Providence, 1993.

Part II will be mainly covered by the book: L.Carleson, T.Gamelin, Complex dynamics, Springer 1993.