Quasihomographies in the theory of Teichmüller spaces

Zając, Józef

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Quasihomographies in the theory of Teichmüller spaces

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Zając Józef

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Rozprawy Matematyczne tom/nr w serii: 357 wydano: 1996

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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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Instytut Matematyczny Polskiej Akademii Nauk

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Rozprawy Matematyczne tom/nr w serii: 357

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102

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Dissertationes Mathematicae, Tom CCCLVII

Daty

wydano

1996

otrzymano

1993-06-27

poprawiono

1995-05-25

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autor

Zając Józef

jzajac@zeus.kul.lublin.pl

Institute of Mathematics, The Catholic University of Lublin, P.O. Box 129, Al. Racławickie 14, 20-950 Lublin, Poland
Institute of Mathematics, Polish Academy of Sciences, ul. Narutowicza 56 , 90-136 Łódź, Poland

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1991 Mathematics Subject Classification: 32H10, 42B10, 32A07, 46E22.

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Quasihomographies in the theory of Teichmüller spaces

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