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E-raamat: Dynamical Aspects of Teichmuller Theory: SL(2,R)-Action on Moduli Spaces of Flat Surfaces

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Dynamical Aspects of Teichmuller Theory: SL(2,R)-Action on Moduli Spaces of Flat SurfacesSuurem pilt
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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 the dynamics of the Teichmüller flow, presenting several self-contained chapters, each addressing a different aspect on the subject. The author includes innovative expositions, all the while solving open problems, constructing examples, and supplementing with illustrations. This book is a rare find in the field with its guidance and support for readers through the complex content of moduli spaces and Teichmüller Theory.

The author is an internationally recognized expert in dynamical systems with a talent to explain topics that is rarely found in the field. He has created a text that would benefit specialists in, not only dynamical systems and geometry, but also Lie theory and number theory.


1 Introduction
1(18)
1.1 Abelian Differentials and Their Moduli Spaces
1(1)
1.2 Translation Structures
2(1)
1.3 Some Examples of Translation Surfaces
3(3)
1.4 Stratification of Moduli Spaces of Translation Surfaces
6(2)
1.5 Period Coordinates
8(1)
1.6 Connected Components of Strata
9(1)
1.7 GL+(2,R) Action on Hg
10(1)
1.8 SL(2, R)-Action on Hg
11(1)
1.9 Teichmuller Flow and Kontsevich-Zorich Cocycle
12(3)
1.10 Teichmuller Curves, Veech Surfaces and Affine Homeomorphisms
15(4)
2 Proof of the Eskin-Kontsevich-Zorich Regularity Conjecture
19(28)
2.1 Eskin-Kontsevich-Zorich Formula
19(1)
2.2 Statement of the Eskin-Kontsevich-Zorich Regularity Conjecture
20(3)
2.3 Idea of the Proof of Theorem 9
23(2)
2.4 Reduction of Theorem 9 to Propositions 14 and 15
25(2)
2.5 Proof of Proposition 14 (Modulo Propositions 16 and 17)
27(3)
2.6 Proof of Proposition 15 (Modulo Proposition 16)
30(4)
2.7 Proof of Proposition 16 via Rokhlin's Disintegration Theorem
34(6)
2.8 Proof of Proposition 17 via Rokhlin's Disintegration Theorem
40(7)
3 Arithmetic Teichmuller Curves with Complementary Series
47(16)
3.1 Exponential Mixing of the Teichmuller Flow
47(1)
3.2 Teichmuller Curves with Complementary Series
48(1)
3.3 Idea of Proof of Theorem 40
49(1)
3.4 Quick Review of Representation Theory of SL(2, R)
49(4)
3.5 Explicit Hyperbolic Surfaces H/Π(,(2k)with Complementary Series
53(5)
3.6 Arithmetic Teichmuller Curves S2k Birational to H/Π6(2k)
58(5)
4 Some Finiteness Results for Algebraically Primitive Teichmuller Curves
63(16)
4.1 Some Classification Results for the Closures of SL(2, R)-Orbits in Moduli Spaces
63(1)
4.2 Statement of the Main Results
64(3)
4.3 Proof of Theorem 48
67(1)
4.4 Sketch of Proof of Theorem 49
68(11)
5 Simplicity of Lyapunov Exponents of Arithmetic Teichmuller Curves
79(26)
5.1 Kontsevich-Zorich Conjecture and Veech's Question
79(1)
5.2 Lyapunov Exponents of Teichmuller Curves and Random Products of Matrices
80(3)
5.3 Galois-Theoretical Criterion for Simplicity of Exponents of Origamis
83(20)
5.4 A Counterexample to an Informal Conjecture of Forni
103(2)
6 An Example of Quaternionic Kontsevich-Zorich Monodromy Group
105(12)
6.1 Filip's Classification of Kontsevich-Zorich Monodromy Groups
105(1)
6.2 Realizability Problem for Kontsevich-Zorich Monodromy Groups
106(1)
6.3 A Quaternionic Cover of a L-Shaped Orgami
107(2)
6.4 Block Decomposition of the KZ Cocycle Over SL(2, R)-L
109(1)
6.5 Some Constraints on Kontsevich-Zorich Monodromy Group of W
110(1)
6.6 Ruling Out SO* (2) × SO* (2) x SO* (2) Monodromy on W
111(1)
6.7 Ruling Out SO* (4) x SO* (2) Monodromy on W
112(5)
References 117(4)
Index 121
Carlos Matheus Silva Santos is a Brazilian mathematician who is an expert in the field of dynamical systems. He currently works at the Centre national de la recherche scientifique (CNRS), while teaching at both École Polytechnique and Université Paris 13. Additionally, he has organized various conferences in Paris and Rio de Janiero on the topic of dynamical systems. 
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