This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon-McMillan-Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included, making this an ideal text for graduate courses. Professionals needing a quick review, or seeking a different perspective on the subject, will also value this book.
Author(s): Steven Kalikow, Randall McCutcheon
Series: Cambridge Studies in Advanced Mathematics 122
Publisher: CUP
Year: 2010
Language: English
Pages: 184
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 9
Introduction......Page 11
1.1 Basic definitions......Page 15
1.2 Carathéodory's theorem, isomorphism, Lebesgue spaces......Page 17
1.3 Properties of Lebesgue spaces, factors......Page 21
1.4 Random variables, integration, (stationary) processes......Page 26
1.5 Conditional expectation......Page 33
2.1 Systems and homomorphisms......Page 36
2.2 Constructing measure-preserving transformations......Page 37
2.3 Types of processes; ergodic, independent and (P,T)......Page 40
2.4 Rohlin tower theorem......Page 42
2.5 Countable generator theorem......Page 47
2.6 Birkhoff ergodic theorem and the strong law......Page 49
2.7 Measure from a monkey sequence......Page 53
2.8 Ergodic decomposition......Page 55
2.9 Ergodic theory on L......Page 58
2.10 Conditional expectation of a measure......Page 60
2.11 Subsequential limits, extended monkey method......Page 62
3.1 Martingales......Page 65
3.2 Coupling; the basics......Page 69
3.3 Applications of coupling......Page 72
3.4 The dbar and variation distances......Page 77
3.5 Preparation for the Shannon–Macmillan–Breiman theorem......Page 79
4.1 The 3-shift is not a factor of the 2-shift......Page 82
4.2 The Shannon–McMillan–Breiman theorem......Page 84
4.3 Entropy of a stationary process......Page 88
4.4 An abstraction: partitions of 1......Page 89
4.5 Measurable partitions, entropy of measure-preserving systems......Page 92
4.6 Krieger finite generator theorem......Page 95
4.7 The induced transformation and fbar......Page 102
5.1 The cast of characters......Page 106
5.2. Step 2: FB ⊂ IC......Page 109
5.3. Step 3: IC ⊂ EX......Page 112
5.4. Step 4: FD ⊂ IC......Page 122
5.5. Step 5: EX ⊂ VWB......Page 125
5.6. Step 6: EX ⊂ FD......Page 131
5.7. Step 7: VWB ⊂ IC......Page 132
6.1 Copying in distribution......Page 134
6.2 Coding......Page 138
6.3 Capturing entropy: preparation......Page 139
6.4 Tweaking a copy to get a better copy......Page 145
6.5 Sinai's theorem......Page 149
6.6 Ornstein isomorphism theorem......Page 153
7.1 The varieties of mixing......Page 156
7.2 Ergodicity vs. weak mixing......Page 157
7.3 Weak mixing vs. mild mixing......Page 160
7.4 Mild mixing vs. strong mixing......Page 163
7.5 Strong mixing vs. Kolmogorov......Page 165
7.6 Kolmogorov vs. Bernoulli......Page 172
Appendix......Page 177
References......Page 180
Index......Page 183
