Introduction to Complex Dynamical Systems

An introductory course of lectures is presented on the theory of complex dynamical systems. The main attention is paid to the case of holomorphic mappings of the open unit disk ∆ = {z ∈ C : |z| < 1}. The course starts by presenting detailed information on the simplest holomorphic mappings ∆ → C, which includes the group of homographic maps. Then a description of the theory of holomorphic selfmappings of ∆ based on the Schwarz Lemma is given. It is followed by a detailed discussion of the fixed points of such self-mappings, based on the Denjoy-Wolff theory. Thereafter, a presentation of the most important results in the theory of continuous-time semigroups of holomorphic self-mappings of ∆ is given. It includes the question of the generation of such semigroups and a continuous version of the DenjoyWolff theory.