北京雁栖湖应用数学研究院

This course provides an introduction to the theory of dynamical systems generated by the iteration of holomorphic maps. Beginning with rational maps on the Riemann sphere, we study the fundamental dichotomy between the Fatou and Julia sets and develop the basic tools used to analyze their structure. Topics include normal families, Montel’s theorem, periodic points and their classification, critical orbits, and the role of the postcritical set. We will explore key examples such as quadratic polynomials and the Mandelbrot set, emphasizing the interplay between complex analysis, geometry, and dynamical systems. Depending on time and interests, additional topics may include polynomial-like mappings, quasiconformal techniques, and selected connections with Teichmüller theory and moduli spaces. The course is intended for graduate students with a background in complex analysis.