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El. knyga: Dynamical Systems and Ergodic Theory

(University of Manchester), (Sapporo University, Japan)
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Dynamical Systems and Ergodic TheoryDidesnis paveikslėlis
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Essentially a self-contained text giving an introduction to topological dynamics and ergodic theory.

This book is an introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. The authors provide a number of applications, principally to number theory and arithmetic progressions (through Van der Waerden's theorem and Szemerdi's theorem). This text is suitable for advanced undergraduate and beginning graduate students.

Recenzijos

' the volume achieves its goals well. It covers a broad range of topics clearly and succinctly There is much material here to interest and stimulate the reader I thoroughly recommend it to anyone of has some knowledge of the subject matter and wants a concise and well presented reference for more advanced concepts.' UK Non-Linear News

Daugiau informacijos

Essentially a self-contained text giving an introduction to topological dynamics and ergodic theory.
Introduction ix(2)
Preliminaries xi
1. Conventions xi(1)
2. Notation xi(1)
3. Pre-requisities in point set topology (Chapters 1-6) xi(1)
4. Pre-requisities in measure theory (Chapters 7-12) xii(1)
5. Subadditive sequences xiii(1)
6. References xiii
Chapter
1. Examples and basic properties
1(10)
1.1. Examples
1(1)
1.2. Transitivity
2(2)
1.3. Other characterizations of transitivity
4(1)
1.4. Transitivity for subshifts of finite type
5(1)
1.5. Minimality and the Birkhoff recurrence theorem
6(2)
1.6. Commuting homeomorphisms
8(1)
1.7. Comments and references
9(2)
Chapter
2. An application of recurrence to arithmetic progressions
11(8)
2.1. Van der Waerden's theorem
11(1)
2.2. A dynamical proof
12(3)
2.3. The proofs of Sulemma 2.2.2 and Sublemma 2.2.3
15(2)
2.4. Comments and references
17(2)
Chapter
3. Topological entropy
19(14)
3.1. Definitions
19(4)
3.2. The Perron-Frobenius theorem and subshifts of finite type
23(3)
3.3. Other definitions and examples
26(4)
3.4. Conjugacy
30(2)
3.5. Comments and references
32(1)
Chapter
4. Interval maps
33(14)
4.1. Fixed points and periodic points
33(4)
4.2. Topological entropy of interval maps
37(2)
4.3. Markov maps
39(5)
4.4. Comments and references
44(3)
Chapter
5. Hyperbolic toral automorphisms
47(10)
5.1. Definitions
47(2)
5.2. Entropy for Hyperbolic Toral Automorphisms
49(3)
5.3. Shadowing and semi-conjugacy
52(3)
5.4. Comments and references
55(2)
Chapter
6. Rotation numbers
57(8)
6.1. Homeomorphisms of the circle and rotation numbers
57(3)
6.2. Denjoy's theorem
60(4)
6.3. Comments and references
64(1)
Chapter
7. Invariant measures
65(8)
7.1. Definitions and characterization of invariant measures
65(1)
7.2. Borel sigma-algebras for compact metric spaces
65(2)
7.3. Examples of invariant measures
67(2)
7.4. Invariant measures for other actions
69(2)
7.5. Comments and references
71(2)
Chapter
8. Measure theoretic entropy
73(18)
8.1. Partitions and conditional expectations
73(3)
8.2. The entropy of a partition
76(3)
8.3. The entropy of a transformation
79(3)
8.4. The increasing martingale theorem
82(2)
8.5. Entropy and sigma algebras
84(2)
8.6. Conditional entropy
86(1)
8.7. Proofs of Lemma 8.7 and Lemma 8.8
87(1)
8.8. Isomorphism
88(1)
8.9. Comments and references
89(2)
Chapter
9. Ergodic measures
91(8)
9.1. Definitions and characterization of ergodic measures
91(1)
9.2. Poincare recurrence and Kac's theorem
91(2)
9.3. Existence of ergodic measures
93(1)
9.4. Some basic constructions in ergodic theory
94(3)
9.4.1. Skew products
95(1)
9.4.2. Induced transformations and Rohlin towers
95(1)
9.4.3. Natural extensions
96(1)
9.5. Comments and references
97(2)
Chapter
10. Ergodic theorems
99(14)
10.1. The Von Neumann ergodic theorem
99(3)
10.2. The Birkhoff theorem (for ergodic measures)
102(4)
10.3. Applications of the ergodic theorems
106(5)
10.4. The Birkhoff theorem (for invariant measures)
111(1)
10.5. Comments and references
112(1)
Chapter
11. Mixing Properties
113(12)
11.1. Weak mixing
113(1)
11.2. A density one convergence characterization of weak mixing
114(2)
11.3. A generalization of the Von Neumann ergodic theorem
116(2)
11.4. The spectral viewpoint
118(2)
11.5. Spectral characterization of weak mixing
120(2)
11.6. Strong mixing
122(1)
11.7. Comments and reference
123(2)
Chapter
12. Statistical properties in ergodic theory
125(14)
12.1. Exact endomorphisms
125(1)
12.2. Statistical properties of piecewise expanding Markov maps
126(7)
12.3. Rohlin's entropy formula
133(1)
12.4. The Shannon-McMillan-Brieman theorem
134(3)
12.5. Comments and references
137(2)
Chapter
13. Fixed points for homeomorphisms of the annulus
139(8)
13.1. Fixed points for the annulus
139(5)
13.2. Outline proof of Brouwer's theorem
144(2)
13.3. Comments and references
146(1)
Chapter
14. The variational principle
147(6)
14.1. The variational principle for entropy
147(1)
14.2. The proof of the variational principle
147(5)
14.3. Comments and reference
152(1)
Chapter
15. Invariant measures for commuting transformations
153(8)
15.1. Furstenberg's conjecture and Rudolph's theorem
153(1)
15.2. The proof of Rudolph's theorem
153(6)
15.3. Comments and references
159(2)
Chapter
16. Multiple recurrence and Szemeredi's theorem
161(16)
16.1. Szemeredi's theorem on arithmetic progressions
161(1)
16.2. An ergodic proof of Szemeredi's theorem
162(1)
16.3. The proof of Theorem 16.2
163(3)
16.3.1. (UMR) for weak-mixing systems, weak-mixing extensions and compact systems
163(2)
16.3.2. The non-weak-mixing case
165(1)
16.3.3. (UMR) for compact extensions
165(1)
16.3.4. The last step
165(1)
16.4. Appendix to section 16.3
166(10)
16.4.1. The proofs of Propositions 16.3 and 16.4
166(5)
16.4.2. The proof of Proposition 16.5
171(1)
16.4.3. The proof of Proposition 16.6
171(1)
16.4.4. The proof of Proposition 16.7
172(1)
16.4.5. The proof of Proposition 16.8
173(2)
16.4.6. The proof of Proposition 16.9
175(1)
16.5. Comments and references
176(1)
Index 177


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