Dynamical aspects of teichmüller theory : SL(2,R)-action on moduli spaces of flat surfaces / Carlos Matheus Silva Santos.
This book is a remarkable contribution to the literature on dynamical systems and geometry. It consists of a selection of work in current research on Teichmüller dynamics, a field that has continued to develop rapidly in the past decades. After a comprehensive introduction, the author investigates t...
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Full Text (via Springer) |
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| Format: | eBook |
| Language: | English |
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Cham : [Amsterdam] :
Springer ; Atlantis Press,
[2018]
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| Series: | Atlantis series in dynamical systems ;
v. 7. |
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Table of Contents:
- Intro; Preface; Acknowledgements; Contents; 1 Introduction; 1.1 Abelian Differentials and Their Moduli Spaces; 1.2 Translation Structures; 1.3 Some Examples of Translation Surfaces; 1.3.1 Abelian Differentials on Complex Torus; 1.3.2 Square-Tiled Surfaces; 1.3.3 Suspensions of Interval Exchange Transformations; 1.3.4 Billiards in Rational Polygons; 1.4 Stratification of Moduli Spaces of Translation Surfaces; 1.5 Period Coordinates; 1.6 Connected Components of Strata; 1.7 GL+(2,mathbbR) Action on mathcalHg; 1.8 SL(2,mathbbR)-Action on mathcalHg.
- 1.9 Teichmüller Flow and Kontsevich-Zorich Cocycle1.10 Teichmüller Curves, Veech Surfaces and Affine Homeomorphisms; 2 Proof of the Eskin-Kontsevich-Zorich Regularity Conjecture; 2.1 Eskin-Kontsevich-Zorich Formula; 2.2 Statement of the Eskin-Kontsevich-Zorich Regularity Conjecture; 2.3 Idea of the Proof of Theorem 9; 2.4 Reduction of Theorem 9 to Propositions 14 and 15; 2.5 Proof of Proposition 14 (Modulo Propositions 16 and 17); 2.6 Proof of Proposition 15 (Modulo Proposition 16); 2.7 Proof of Proposition 16 via Rokhlin's Disintegration Theorem.
- 2.8 Proof of Proposition 17 via Rokhlin's Disintegration Theorem3 Arithmetic Teichmüller Curves with Complementary Series; 3.1 Exponential Mixing of the Teichmüller Flow; 3.2 Teichmüller Curves with Complementary Series; 3.3 Idea of Proof of Theorem 40; 3.4 Quick Review of Representation Theory of SL(2,mathbbR); 3.4.1 Spectrum of Unitary SL(2,mathbbR)-Representations; 3.4.2 Bargmann's Classification; 3.4.3 Hyperbolic Surfaces and Examples of Regular Unitary SL(2,mathbbR)-Representations; 3.4.4 Rates of Mixing and Spectral Gap.
- 5.2 Lyapunov Exponents of Teichmüller Curves and Random Products of Matrices5.3 Galois-Theoretical Criterion for Simplicity of Exponents of Origamis; 5.3.1 Galois-Pinching Matrices; 5.3.2 Twisting with Respect to Galois-Pinching Matrices I: Statements of Results; 5.3.3 Twisting with Respect to Galois-Pinching Matrices II: Proof of Theorem 67; 5.3.4 Two Simplicity Criteria for the Lyapunov Exponents of Origamis; 5.4 A Counterexample to an Informal Conjecture of Forni; 6 An Example of Quaternionic Kontsevich-Zorich Monodromy Group.
- 3.5 Explicit Hyperbolic Surfaces mathbbH/Γ6(2k) with Complementary Series3.6 Arithmetic Teichmüller Curves mathcalS2k Birational to mathbbH/Γ6(2k); 4 Some Finiteness Results for Algebraically Primitive Teichmüller Curves; 4.1 Some Classification Results for the Closures of SL(2,mathbbR)-Orbits in Moduli Spaces; 4.2 Statement of the Main Results; 4.3 Proof of Theorem 48; 4.4 Sketch of Proof of Theorem 49; 5 Simplicity of Lyapunov Exponents of Arithmetic Teichmüller Curves; 5.1 Kontsevich-Zorich Conjecture and Veech's Question.