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.