## Lectures on Quasiconformal Mappings (Google eBook)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 is remarkable 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 edition includes 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. theory of hyperbolic structures on 3-manifolds. Together, these three new 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 the proceedings 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 of maintaining communication between the different scientific fields. areas of research in statistics that might have particular applicability to nonlinear dynamics and new methodology and open data analysis problems in nonlinear dynamics that might find their way into the toolkits and research interests of statisticians. It features a survey of state-of-the-art developments in nonlinear dynamics time series analysis with open statistical problems and areas for further research. It contains 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. It serves as a starting point for both scientists and statisticians who want to explore the field. Expositions are readable to scientists outside the featured fields of specialization. Titles in this series are copublished with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada). |

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an important book.

### Contents

Differentiable Quasiconformal Mappings | 5 |

The General Definition | 15 |

Extremal Geometric Properties | 23 |

Boundary Correspondence | 39 |

The Mapping Theorem | 51 |

Teichmiiller Spaces | 67 |

Editors Notes | 83 |

116 | |

Hyperbolic Structures on ThreeManifolds that Fiber over the Circle | 140 |

### Common terms and phrases

3-manifolds Acta Math Ahlfors Ahlfors's Amer analytic basepoint basin Beltrami differential Beltrami form Bers biholomorphic boundary bounded Chapter compact support complex conformal mapping conformal structure conjecture conjugacy conjugate converges corresponding curve defined definition denote dilatation dynamics equation equivalent exists extremal mappings F-tree Fatou set finite conformal type finitely generated Kleinian follows Fuchsian Fuchsian group function geodesic geometric hence holomorphic motion homeomorphism homotopic hyperbolic 3-manifolds hyperbolic Riemann surface integral invariant line fields isometry isomorphic Julia set Kleinian groups Kobayashi LEMMA Let F linear transformation M-condition map f measured foliations Mobius modular group norm normalized orbit parabolic parameter periodic point pleated surfaces Poincare Poincare metric polynomials proved pseudo-Anosov pseudo-Anosov mapping q.c. mapping quadratic differentials quasiconformal mappings R-trees rational map real axis renormalizable renormalization Riemann sphere satisfies Siegel disk solution space T(So subgroup subset Teichmuller spaces theory Thurston's topology unique unit disk whole plane zero