Lectures on Quasiconformal Mappings

by Ahlfors, Lars V.; Earle, C. J. (CON); Kra, I. (CON); Shishikura, M. (CON); Hubbard, J. H. (CON)

• ISBN13: 9780821836446
• ISBN10: 0821836447
• Edition: 2nd
• Format: Paperback
• Copyright: 2006-08-30
• Publisher: Amer Mathematical Society

Summary

Lars Ahlfors's Lectures on Quasiconformal Mappings, based on a course he gave at Harvard University in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. These lectures develop the theory of quasiconformal mappings from scratch, give a self-contained treatment of the Beltrami equation, and cover the basic properties of Teichmuller spaces, including the Bers embedding and the Teichmuller curve. It isremarkable how Ahlfors goes straight to the heart of the matter, presenting major results with a minimum set of prerequisites. Many graduate students and other mathematicians have learned the foundations of the theories of quasiconformal mappings and Teichmuller spaces from these lecture notes. This editionincludes three new chapters. The first, written by Earle and Kra, describes further developments in the theory of Teichmuller spaces and provides many references to the vast literature on Teichmuller spaces and quasiconformal mappings. The second, by Shishikura, describes how quasiconformal mappings have revitalized the subject of complex dynamics. The third, by Hubbard, illustrates the role of these mappings in Thurston's theory of hyperbolic structures on 3-manifolds. Together, these threenew chapters exhibit the continuing vitality and importance of the theory of quasiconformal mappings. This book is a collection of research and expository papers reflecting the interfacing of two fields: nonlinear dynamics (in the physiological and biological sciences) and statistics. It presents theproceedings of a four-day workshop entitled ''Nonlinear Dynamics and Time Series: Building a Bridge Between the Natural and Statistical Sciences'' held at the Centre de Recherches Mathematiques (CRM) in Montreal in July 1995. The goal of the workshop was to provide an exchange forum and to create a link between two diverse groups with a common interest in the analysis of nonlinear time series data. The editors and peer reviewers of this work have attempted to minimize the problems ofmaintaining communication between the different scientific fields. The result is a collection of interrelated papers that highlight current areas of research in statistics that might have particular applicability to nonlinear dynamics and new methodology and open data analysis problems in nonlinear dynamicsthat might find their way into the toolkits and research interests of statisticians. Features: A survey of state-of-the-art developments in nonlinear dynamics time series analysis with open statistical problems and areas for further research. Contributions by statisticians to understanding and improving modern techniques commonly associated with nonlinear time series analysis, such as surrogate data methods and estimation of local Lyapunov exponents. Starting point for both scientists andstatisticians who want to explore the field. Expositions that are readable to scientists outside the featured fields of specialization. Information for our distributors: Titles in this series are copublished with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario,Canada).

Table of Contents

Preface	vii
The Ahlfors Lectures
Acknowledgments	3	(2)
Differentiable Quasiconformal Mappings	5	(10)
The Problem and Definition of Grotzsch	5	(3)
Solution of Grotzsch's Problem	8	(1)
Composed Mappings	8	(2)
Extremal Length	10	(3)
A Symmetry Principle	13	(1)
Dirichlet Integrals	13	(2)
The General Definition	15	(8)
The Geometric Approach	15	(1)
The Analytic Definition	16	(7)
External Geometric Properties	23	(16)
Three Extremal Problems	23	(2)
Elliptic and Modular Functions	25	(5)
Mori's Theorem	30	(4)
Quadruplets	34	(5)
Boundary Correspondence	39	(12)
The M-condition	39	(3)
The Sufficiency of the M-condition	42	(3)
Quasi-isometry	45	(1)
Quasiconformal Reflection	45	(4)
The Reverse Inequality	49	(2)
The Mapping Theorem	51	(16)
Two Integral Operators	51	(3)
Solution of the Mapping Problem	54	(4)
Dependence on Parameters	58	(4)
The Calderon-Zygmund Inequality	62	(5)
Teichmuller Spaces	67	(16)
Preliminaries	67	(2)
Beltrami Differentials	69	(4)
Δ Is Open	73	(4)
The Infinitesimal Approach	77	(6)
Editors' Notes	83
The Additional Chapters
A Supplement to Ahlfors's Lectures	87	(32)
Clifford J. Earle
Irwin Kra
Complex Dynamics and Quasiconformal Mappings	119	(24)
Mitsuhiro Shishikura
Hyperbolic Structures on Three-Manifolds that Fiber over the Circle	143
John H. Hubbard