23\(^{\text{o}}\) colóquio Brasileiro de matemática (Q2753379)

This is a very interesting book containing a hot material. It includes topics very useful for conformal dynamics but is self-contained and interesting in itself. It consists of three great themes: the uniformization theorem, the measurable Riemann mapping theorem and quasiconformal mappings and, as the third topic, holomorphic motions. But this is not all. Lots of proofs in the first chapter (the uniformization theorem) rely on heavy use of Poincaré metric and hyperbolic contractions. The proof of Montel's theorem and a non-standard proof of a qualitative version (usually sufficient for dynamical applications) of Koebe's distortion theorem is included. The uniformization theorem itself is stated for (not necessarily simply connected) domains of the Riemann sphere which includes the classical Riemann mapping theorem. NEWLINENEWLINENEWLINEAs the main dynamical application of the methods developed in this chapter, the classification of periodic components of the Fatou set of a rational function is provided. As a main application of quasiconformal methods, the proof of the celebrated Sullivan's no-wandering domain theorem is provided. NEWLINENEWLINENEWLINEIn the last section the \(\lambda\)-lemma is proven and its improvement due to Bers-Royden is stated. As a main application of \(\lambda\)-lemma the density of structural stability is proved and in the last section Yoccoz puzzles are mentioned. NEWLINENEWLINENEWLINEIt is remarkable that such a short book (97 pages of text) contains complete proofs of so many fundamental theorems.