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Quasihomographies in the theory of Teichmüller spaces

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Zając J.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction............................................................................................................................5 I. Special functions of quasiconformal theory.....................................................................10 1. Introduction.................................................................................................................10 2. The distortion function $Φ_K$.....................................................................................11 3. Quasisymmetric functions............................................................................................19 4. Functional identities for special functions....................................................................27 5. Applications..................................................................................................................38 II. Quasihomographies of a circle.......................................................................................42 1. Introduction..................................................................................................................42 2. Introduction to quasihomographies..............................................................................42 3. Quasihomographies and quasisymmetric functions on the real line.............................45 4. Quasihomographies and quasisymmetric functions on the unit circle...........................48 5. Quasisymmetric functions as quasihomographies.........................................................51 III. Distortion theorems for quasihomographies....................................................................57 1. Introduction...................................................................................................................57 2. Similarities.....................................................................................................................57 3. Distortion theorems.......................................................................................................60 4. Normal and compact families of quasihomographies.....................................................67 5. Topological characterization of quasihomographies.......................................................69 IV. Quasihomographies of a Jordan curve ...........................................................................72 1. Introduction...................................................................................................................72 2. Harmonic cross-ratio.....................................................................................................72 3. One-dimensional quasiconformal mappings..................................................................76 4. Complete boundary transformations.............................................................................78 5. Quasicircles...................................................................................................................80 V. The universal Teichmüller space.......................................................................................84 1. Introduction....................................................................................................................84 2. The universal Teichmüller space of a circle...................................................................85 3. The universal Teichmüller space of an oriented Jordan curve........................................87 4. The space of normalized quasihomographies................................................................91 5. A linearization formula....................................................................................................94 Acknowledgements...................................................................................................................97 References...............................................................................................................................98

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1 Zając J.

1 1996

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Quasihomographies in the theory of Teichmüller spaces