Our main goal in this book is to introduce the reader to some of the most useful tools of modern one-dimensional dynamics. We do not aim at being comprehensive but prefer instead to focus our attention on certain key tools. We believe that the topics covered here are representative of the depth and beauty of the ideas in the subject. For each tool presented in the book, we have selected at least one non-trivial dynamical application to go with it.

Almost all the topics discussed in the text have their source in complex function theory and the related areas of hyperbolic geometry, quasiconformal mappings and Teichmüller theory. This is true even of certain tools, such as the distortion of cross-ratios, that are applied to problems in real one-dimensional dynamics. The main tools include three deep theorems: the uniformization theorem (for domains in the Riemann sphere), the measurable Riemann mapping theorem and the Bers–Royden theorem on holomorphic motions. These are presented, with complete proofs, in Chapters 3, 4 and 5 respectively.

The present book originated in a set of notes for a short course we taught at the 23rd Brazilian Mathematics Colloquium (IMPA, 2001). We have benefited from useful criticism of the original notes from friends and colleagues, especially André de Carvalho, who read through them and found several inaccuracies. We are also grateful to two anonymous referees for their perspicacious remarks and suggestions.

Complex analysis is a vast and very beautiful subject, and the key to its beauty is the harmonious coexistence of analysis, algebra, geometry and topology in its most fundamental entity, the complex plane. We will assume that the reader is already familiar with the basic facts about analytic functions in one complex variable, such as Cauchy's theorem, the Cauchy–Riemann equations, power series expansions, residues and so on. Holomorphic functions in one complex variable enjoy a double life, as they can be viewed both as analytic objects (power series, integral representations) and as geometric objects (conformal mappings). The topics presented in this book exploit freely this dual character of holomorphic functions. Our purpose in this short chapter is to present some well-known or not so well-known analytic and geometric facts that will be necessary later. The reader is warned that what follows is only a brief collection of facts to be used, not a systematic exposition of the theory. For general background reading in complex analysis, see for instance [A2], [An] or [Rud].

Analytic facts

Let us start with some differential calculus of complex-valued functions defined on some domain in the complex plane (by a domain we mean as usual a non-empty, connected, open set).

It is fair to say that the subject known today as complex dynamics – the study of iterations of analytic functions – originated in the pioneering works of P. Fatou and G. Julia early in the twentieth century (see the references [Fat] and [Ju]). In possession of what was then a new tool, Montel's theorem on normal families, Fatou and Julia each investigated the iteration of rational maps of the Riemann sphere and found that these dynamical systems had an extremely rich orbit structure. They observed that each rational map produced a dichotomy of behavior for points on the Riemann sphere. Some points – constituting a totally invariant open set known today as the Fatou set – showed an essentially dissipative or wandering character under iteration by the map. The remaining points formed a totally invariant compact set, today called the Julia set. The dynamics of a rational map on its Julia set showed a very complicated recurrent behavior, with transitive orbits and a dense subset of periodic points. Since the Julia set seemed so difficult to analyse, Fatou turned his attention to its complement (the Fatou set). The components of the Fatou set are mapped to other components, and Fatou observed that these seemed to eventually to fall into a periodic cycle of components. Unable to prove this fact, but able to verify it for many examples, Fatou nevertheless conjectured that rational maps have no wandering domains.