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Weakly Wandering Sequences in Ergodic Theory 2014 ed. [Kietas viršelis]

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  • Formatas: Hardback, 153 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 15 Illustrations, black and white; XIV, 153 p. 15 illus., 1 Hardback
  • Serija: Springer Monographs in Mathematics
  • Išleidimo metai: 01-Sep-2014
  • Leidėjas: Springer Verlag, Japan
  • ISBN-10: 4431551077
  • ISBN-13: 9784431551072
Kitos knygos pagal šią temą:
Weakly Wandering Sequences in Ergodic Theory 2014 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 153 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 15 Illustrations, black and white; XIV, 153 p. 15 illus., 1 Hardback
  • Serija: Springer Monographs in Mathematics
  • Išleidimo metai: 01-Sep-2014
  • Leidėjas: Springer Verlag, Japan
  • ISBN-10: 4431551077
  • ISBN-13: 9784431551072
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9784431551072.html
Raktažodžiai:

The appearance of weakly wandering (ww) sets and sequences for ergodic transformations over half a century ago was an unexpected and surprising event. In time it was shown that ww and related sequences reflected significant and deep properties of ergodic transformations that preserve an infinite measure.

This monograph studies in a systematic way the role of ww and related sequences in the classification of ergodic transformations preserving an infinite measure. Connections of these sequences to additive number theory and tilings of the integers are also discussed. The material presented is self-contained and accessible to graduate students. A basic knowledge of measure theory is adequate for the reader.

Recenzijos

This is a well-written book that should be the place to go to for someone interested in weakly wandering sequences, their properties and extensions. Most of the work the authors discuss is the result of their research over a number of years. At the same time we would have liked to see discussions of several topics that are connected to the topics of the book, such as inducing, rank-one transformations, and Maharam transformations. (Cesar E. Silva, Mathematical Reviews, May, 2016)

The subject of the book under review is ergodic theory with a stress on WW sequences. The book is interesting, well written and contains a lot of examples. It constitutes a valuable addition to the mathematical pedagogical literature. (Athanase Papadopoulos, zbMATH, 1328.37006, 2016)

1 Existence of Finite Invariant Measure
1(16)
1.1 Recurrent Transformations
2(2)
1.2 Finite Invariant Measure
4(13)
2 Transformations with No Finite Invariant Measure
17(8)
2.1 Measurable Transformations
17(4)
2.2 Ergodic Transformations
21(4)
3 Infinite Ergodic Transformations
25(16)
3.1 General Properties of Infinite Ergodic Transformations
25(1)
3.2 Weakly Wandering Sequences
26(6)
3.3 Recurrent Sequences
32(9)
3.3.1 Transformations with Recurrent Sequences
34(2)
3.3.2 Transformations Without Recurrent Sequences
36(5)
4 Three Basic Examples
41(24)
4.1 First Basic Example
41(7)
4.1.1 Induced Transformations
42(1)
4.1.2 Construction of the First Basic Example
43(5)
4.2 Second Basic Example
48(12)
4.2.1 Non-measure-Preserving Commutators
48(4)
4.2.2 A General Class of Transformations
52(4)
4.2.3 Construction of the Second Basic Example
56(4)
4.3 Third Basic Example
60(5)
4.3.1 Construction of the Third Basic Example
60(2)
4.3.2 Random Walk on the Integers
62(3)
5 Properties of Various Sequences
65(14)
5.1 Properties of ww and Recurrent Sequences
65(5)
5.2 Dissipative Sequences
70(5)
5.3 The Sequences in a Different Setting
75(4)
6 Isomorphism Invariants
79(24)
6.1 Exhaustive Weakly Wandering Sets
79(5)
6.2 α-Type Transformations
84(3)
6.3 Recurrent Sequences as an Isomorphism Invariant
87(6)
6.3.1 Construction of the Transformation Tε
88(3)
6.3.2 The Recurrent Sequences for Tε
91(2)
6.4 Growth Distributions for a Transformation
93(10)
7 Integer Tilings
103(44)
7.1 Infinite Tilings of the Integers
103(5)
7.1.1 Structure of Complementing Pairs in N
104(1)
7.1.2 Complementing Pairs in Z When A or B Is Finite
105(1)
7.1.3 Infinite A, B: No Structure Expected
106(2)
7.2 How Tilings Arise in Ergodic Theory
108(7)
7.2.1 Constructing a Transformation from a Hitting Sequence
110(5)
7.3 Examples of Complementing Pairs
115(9)
7.3.1 A Complementing Set That Is Not a Hitting Sequence
116(1)
7.3.2 A ww Sequence Which Is Not eww for Any Transformation
117(2)
7.3.3 An eww Sequence with a Complementing Set That Does Not Come from a Point
119(5)
7.4 Extending a Finite Set to a Complementing Set
124(6)
7.4.1 Definitions and Notations
124(1)
7.4.2 Extension Theorem
125(5)
7.5 Complementing Sets of A and the 2-Adic Integers
130(6)
7.5.1 The 2-Adic Integers
131(3)
7.5.2 Condition (iv) Is Not Enough to Be Complementing
134(2)
7.6 Examples: Non-isomorphic Transformations
136(5)
7.6.1 Two Non-isomorphic Transformations
136(3)
7.6.2 An Uncountable Family of Non-isomorphic Transformations
139(2)
7.7 An Odometer Construction from M
141(6)
7.7.1 Set Theoretic Construction of X
141(1)
7.7.2 Defining the Sequences A and B Associated to M
142(2)
7.7.3 The Sequence M and Multiple Recurrence
144(3)
References 147(4)
Index 151
Arshag Hajian Professor of Mathematics at Northeastern University, Boston, Massachusetts, U.S.A. Stanley Eigen Professor of Mathematics at Northeastern University, Boston, Massachusetts, U. S. A. Raj. Prasad Professor of Mathematics at University of Massachusetts at Lowell, Lowell, Massachusetts, U.S.A. Yuji Ito Professor Emeritus of Keio University, Yokohama, Japan.