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El. knyga: Ergodic Theory: with a view towards Number Theory

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  • Formatas: PDF+DRM
  • Serija: Graduate Texts in Mathematics 259
  • Išleidimo metai: 11-Sep-2010
  • Leidėjas: Springer London Ltd
  • Kalba: eng
  • ISBN-13: 9780857290212
Kitos knygos pagal šią temą:
Ergodic Theory: with a view towards Number TheoryDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Graduate Texts in Mathematics 259
  • Išleidimo metai: 11-Sep-2010
  • Leidėjas: Springer London Ltd
  • Kalba: eng
  • ISBN-13: 9780857290212
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This text is a rigorous introduction to Ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. It describes some recent applications to number theory, and goes beyond the standard texts in this topic.

This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Recenzijos

From the reviews:

The book is an introduction to ergodic theory and dynamical systems. The book is intended for graduate students and researchers with some background in measure theory and functional analysis. Definitely, it is a book of great interest for researchers in ergodic theory, homogeneous dynamics or number theory. (Antonio Dķaz-Cano Ocańa, The European Mathematical Society, January, 2014)

A book with a wider perspective on ergodic theory, and yet with a focus on the interaction with number theory, remained a glaring need in the overall context of the development of the subject. The book under review goes a long way in fulfilling this need. it covers a good deal of conventional ground in ergodic theory . a very welcome addition and would no doubt inspire interest in the area among researchers as well as students, and cater to it successfully. (S. G. Dani, Ergodic Theory and Dynamical Systems, Vol. 32 (3), June, 2012)

The book under review is an introductory textbook on ergodic theory, written with applications to number theory in mind. it aims both to provide the reader with a solid comprehensive background in the main results of ergodic theory, and of reaching nontrivial applications to number theory. The book should also be very appealing to more advanced readers already conducting research in representation theory or number theory, who are interested in understanding the basis of the recent interaction with ergodic theory. (Barak Weiss, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 114, 2012)

This introductory book, which goes beyond the standard texts and allows the reader to get a glimpse of modern developments, is a timely and welcome addition to the existing and ever-growing ergodic literature. This book is highly recommended to graduate students and indeed to anyone who is interested in acquiring a better understanding of contemporarydevelopments in mathematics. (Vitaly Bergelson, Mathematical Reviews, Issue 2012 d)

The book contains a presentation of the ergodic theory field, focusing mainly on results applicable to number theory. of interest for researchers, specialists, professors and students that work within some other areas than precisely the ergodic theory. Ergodic Theory. With a view toward number theory is now an indispensable reference in the domain and offers important instruments of research for other theoretical fields. (Adrian Atanasiu, Zentralblatt MATH, Vol. 1206, 2011)

1 Motivation
1(12)
1.1 Examples of Ergodic Behavior
1(2)
1.2 Equidistribution for Polynomials
3(1)
1.3 Szemeredi's Theorem
4(1)
1.4 Indefinite Quadratic Forms and Oppenheim's Conjecture
5(2)
1.5 Littlewood's Conjecture
7(1)
1.6 Integral Quadratic Forms
8(1)
1.7 Dynamics on Homogeneous Spaces
9(1)
1.8 An Overview of Ergodic Theory
10(3)
2 Ergodicity, Recurrence and Mixing
13(56)
2.1 Measure-Preserving Transformations
13(8)
2.2 Recurrence
21(2)
2.3 Ergodicity
23(5)
2.4 Associated Unitary Operators
28(4)
2.5 The Mean Ergodic Theorem
32(5)
2.6 Pointwise Ergodic Theorem
37(11)
2.6.1 The Maximal Ergodic Theorem
37(1)
2.6.2 Maximal Ergodic Theorem via Maximal Inequality
38(2)
2.6.3 Maximal Ergodic Theorem via a Covering Lemma
40(4)
2.6.4 The Pointwise Ergodic Theorem
44(1)
2.6.5 Two Proofs of the Pointwise Ergodic Theorem
45(3)
2.7 Strong-Mixing and Weak-Mixing
48(6)
2.8 Proof of Weak-Mixing Equivalences
54(7)
2.8.1 Continuous Spectrum and Weak-Mixing
59(2)
2.9 Induced Transformations
61(8)
3 Continued Fractions
69(28)
3.1 Elementary Properties
69(7)
3.2 The Continued Fraction Map and the Gauss Measure
76(11)
3.3 Badly Approximable Numbers
87(4)
3.3.1 Lagrange's Theorem
88(3)
3.4 Invertible Extension of the Continued Fraction Map
91(6)
4 Invariant Measures for Continuous Maps
97(24)
4.1 Existence of Invariant Measures
98(5)
4.2 Ergodic Decomposition
103(2)
4.3 Unique Ergodicity
105(5)
4.4 Measure Rigidity and Equidistribution
110(11)
4.4.1 Equidistribution on the Interval
110(3)
4.4.2 Equidistribution and Generic Points
113(1)
4.4.3 Equidistribution for Irrational Polynomials
114(7)
5 Conditional Measures and Algebras
121(32)
5.1 Conditional Expectation
121(5)
5.2 Martingales
126(7)
5.3 Conditional Measures
133(12)
5.4 Algebras and Maps
145(8)
6 Factors and Joinings
153(18)
6.1 The Ergodic Theorem and Decomposition Revisited
153(3)
6.2 Invariant Algebras and Factor Maps
156(2)
6.3 The Set of Joinings
158(1)
6.4 Kronecker Systems
159(4)
6.5 Constructing Joinings
163(8)
7 Furstenberg's Proof of Szemeredi's Theorem
171(60)
7.1 Van der Waerden
172(3)
7.2 Multiple Recurrence
175(3)
7.2.1 Reduction to an Invertible System
177(1)
7.2.2 Reduction to Borel Probability Spaces
177(1)
7.2.3 Reduction to an Ergodic System
177(1)
7.3 Furstenberg Correspondence Principle
178(2)
7.4 An Instance of Polynomial Recurrence
180(8)
7.4.1 The van der Corput Lemma
184(4)
7.5 Two Special Cases of Multiple Recurrence
188(4)
7.5.1 Kronecker Systems
188(2)
7.5.2 Weak-Mixing Systems
190(2)
7.6 Roth's Theorem
192(7)
7.6.1 Proof of Theorem 7.14 for a Kronecker System
194(1)
7.6.2 Reducing the General Case to the Kronecker Factor
195(4)
7.7 Definitions
199(2)
7.8 Dichotomy Between Relatively Weak-Mixing and Compact Extensions
201(6)
7.9 SZ for Compact Extensions
207(9)
7.9.1 SZ for Compact Extensions via van der Waerden
210(2)
7.9.2 A Second Proof
212(4)
7.10 Chains of SZ Factors
216(2)
7.11 SZ for Relatively Weak-Mixing Extensions
218(8)
7.12 Concluding the Proof
226(1)
7.13 Further Results in Ergodic Ramsey Theory
227(4)
7.13.1 Other Furstenberg Ergodic Averages
227(4)
8 Actions of Locally Compact Groups
231(46)
8.1 Ergodicity and Mixing
231(4)
8.2 Mixing for Commuting Automorphisms
235(8)
8.2.1 Ledrappier's "Three Dots" Example
236(3)
8.2.2 Mixing Properties of the x2, x3 System
239(4)
8.3 Haar Measure and Regular Representation
243(8)
8.3.1 Measure-Theoretic Transitivity and Uniqueness
245(6)
8.4 Amenable Groups
251(3)
8.4.1 Definition of Amenability and Existence of Invariant Measures
251(3)
8.5 Mean Ergodic Theorem for Amenable Groups
254(3)
8.6 Pointwise Ergodic Theorems and Polynomial Growth
257(9)
8.6.1 Flows
257(2)
8.6.2 Pointwise Ergodic Theorems for a Class of Groups
259(7)
8.7 Ergodic Decomposition for Group Actions
266(6)
8.8 Stationary Measures
272(5)
9 Geodesic Flow on Quotients of the Hyperbolic Plane
277(54)
9.1 The Hyperbolic Plane and the Isometric Action
277(5)
9.2 The Geodesic Flow and the Horocycle Flow
282(6)
9.3 Closed Linear Groups and Left Invariant Riemannian Metric
288(17)
9.3.1 The Exponential Map and the Lie Algebra of a Closed Linear Group
289(6)
9.3.2 The Left-Invariant Riemannian Metric
295(6)
9.3.3 Discrete Subgroups of Closed Linear Groups
301(4)
9.4 Dynamics on Quotients
305(9)
9.4.1 Hyperbolic Area and Fuchsian Groups
306(4)
9.4.2 Dynamics on I\PSL2(R)
310(1)
9.4.3 Lattices in Closed Linear Groups
311(3)
9.5 Hopf's Argument for Ergodicity of the Geodesic Flow
314(3)
9.6 Ergodicity of the Gauss Map
317(10)
9.7 Invariant Measures and the Structure of Orbits
327(4)
9.7.1 Symbolic Coding
327(1)
9.7.2 Measures Coming from Orbits
328(3)
10 Nilrotation
331(16)
10.1 Rotations on the Quotient of the Heisenberg Group
331(2)
10.2 The Nilrotation
333(1)
10.3 First Proof of Theorem 10.1
334(2)
10.4 Second Proof of Theorem 10.1
336(5)
10.4.1 A Commutative Lemma; The Set K
336(1)
10.4.2 Studying Divergence; The Set X1
337(2)
10.4.3 Combining Linear Divergence and the Maximal Ergodic Theorem
339(2)
10.5 A Non-ergodic Nilrotation
341(1)
10.6 The General Nilrotation
342(5)
11 More Dynamics on Quotients of the Hyperbolic Plane
347(56)
11.1 Dirichlet Regions
347(10)
11.2 Examples of Lattices
357(7)
11.2.1 Arithmetic and Congruence Lattices in SL2 (R)
358(1)
11.2.2 A Concrete Principal Congruence Lattice of SL2 (R)
358(3)
11.2.3 Uniform Lattices
361(3)
11.3 Unitary Representations, Mautner Phenomenon, and Ergodicity
364(6)
11.3.1 Three Types of Actions
364(2)
11.3.2 Ergodicity
366(3)
11.3.3 Mautner Phenomenon for SL2(R)
369(1)
11.4 Mixing and the Howe---Moore Theorem
370(8)
11.4.1 First Proof of Theorem 11.22
370(2)
11.4.2 Vanishing of Matrix Coefficients for PSL2(R)
372(1)
11.4.3 Second Proof of Theorem 11.22; Mixing of All Orders
372(6)
11.5 Rigidity of Invariant Measures for the Horocycle Flow
378(10)
11.5.1 Existence of Periodic Orbits; Geometric Characterization
379(4)
11.5.2 Proof of Measure Rigidity for the Horocycle Flow
383(5)
11.6 Non-escape of Mass for Horocycle Orbits
388(11)
11.6.1 The Space of Lattices and the Proof of Theorem 11.32 for X2 = SL2(Z)\SL2(R)
390(5)
11.6.2 Extension to the General Case
395(4)
11.7 Equidistribution of Horocycle Orbits
399(4)
Appendix A Measure Theory
403(14)
A.1 Measure Spaces
403(3)
A.2 Product Spaces
406(1)
A.3 Measurable Functions
407(2)
A.4 Radon-Nikodym Derivatives
409(1)
A.5 Convergence Theorems
410(1)
A.6 Well-Behaved Measure Spaces
411(1)
A.7 Lebesgue Density Theorem
412(1)
A.8 Substitution Rule
413(4)
Appendix B Functional Analysis
417(12)
B.1 Sequence Spaces
417(1)
B.2 Linear Functionals
418(1)
B.3 Linear Operators
419(2)
B.4 Continuous Functions
421(1)
B.5 Measures on Compact Metric Spaces
422(3)
B.6 Measures on Other Spaces
425(1)
B.7 Vector-valued Integration
425(4)
Appendix C Topological Groups
429(12)
C.1 General Definitions
429(2)
C.2 Haar Measure on Locally Compact Groups
431(2)
C.3 Pontryagin Duality
433(8)
Hints for Selected Exercises 441(6)
References 447(16)
Author Index 463(4)
Index of Notation 467(4)
General Index 471
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