Teichmüller theory and applications to geometry, topology, and dynamics. Volume 2: Surface homeomorphisms and rational functions (Q2812682)

This volume is the second of four volumes devoted to Teichmüller theory and its applications to geometry, topology and dynamics. The final purpose of these volumes is to give complete and self-contained proofs of four theorems by W. Thurston:NEWLINENEWLINENEWLINE(1) The classification of homeomorphisms of a surfaceNEWLINENEWLINENEWLINE(2) The topological classification of rational mapsNEWLINENEWLINENEWLINE(3) The hyperbolization theorem for 3-manifolds that fiber over the circleNEWLINENEWLINENEWLINE(4) The hyperbolization theorem for Haken 3-manifolds NEWLINENEWLINENEWLINEThe Teichmüller theory plays an essential role in the proofs of these theorems. In the first volume the author has given a detailed presentation on the analytic and geometric part of Teichmüller theory.NEWLINENEWLINEThis volume consists of three chapters and an appendix with 9 subsections, which is mainly devoted to prove the first two theorems mentioned above. Chapter 8 contains the proof of the first theorem on the classification of homeomorphisms on a surface, essentially given by \textit{L. Bers} [Acta Math. 141, 73--98 (1978; Zbl 0389.30018)]. Chapter 10 contains the proof of the second theorem on the topological characterization of rational maps on the Riemann sphere, which was given by the author jointly with \textit{A. Douady} [Acta Math. 171, No. 2, 263--297 (1993; Zbl 0806.30027)]. The main part of Chapter 9 is an introduction to the dynamics on the sphere, mainly of polynomials. NEWLINENEWLINENEWLINENEWLINE Appendix C contains more basic definitions and preliminary results on the following topics: NEWLINENEWLINENEWLINENEWLINE C1. The Perron-Frobenius theoremNEWLINENEWLINEC2. The Alexander trickNEWLINENEWLINEC3. Homotopy implies isotopyNEWLINENEWLINEC4. The mapping class group and outer automorphismsNEWLINENEWLINEC5. Totally real stretch factorsNEWLINENEWLINEC6. Linearizing at irrationally indifferent fixed pointsNEWLINENEWLINEC7. Examples of Thurston pullback mapsNEWLINENEWLINEC8. Thurston maps with nonhyperbolic orbifoldsNEWLINENEWLINEC9. Sullivan's dictionary NEWLINENEWLINENEWLINENEWLINE Both the first and the second volume are very useful not only for people who are not very familiar with the Teichmüller theory, but also for experts in the fields of complex analysis, geometry, topology and dynamics.