Mathematical Tools for One-Dimensional Dynamics
The title may be a bit misleading; this book is largely focused on complex dynamics, which is one-dimensional from the complex point of view. There is no doubt that bizarre pictures of Mandelbrot and Julia sets attract attention to this field of mathematics. This is the theme for numerous expositions at a popular level. Here, however, the subject is treated as a precise and deep mathematical theory. Let us mention the main goals of the text. The uniformisation theorem states that a subdomain of the Riemann sphere whose complement contains at least three points can be covered by a holomorphic map from the unit disc. The measurable Riemann mapping theorem (Alfors-Bers version) on the solution of the Beltrami equation with measurable coefficients is used to prove Sullivan's no-wandering-domains theorem on Fatou sets of rational maps. Finally, the Bers-Royden theorem discusses the existence of an extension of a holomorphic motion. This concerns functions of two variables, holomorphic in one variable and quasiconformal in the other. These theorems are presented with full proofs, applications and related developments. The book also contains a chapter on some topics of real dynamics and appendices on Riemann surfaces and Teichmüller theory. Each chapter is equipped with exercises. The book is intended for advanced students and researchers. It is a successful self-contained exposition of an important part of the theory with indications for further studies and discussion of perspectives, including fundamental open problems.