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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.
List of contents
1. Existence of a finite invariant measure 2. Transformations with no Finite Invariant Measure 3. Infinite Ergodic Transformations 4. Three Basic Examples 5. Properties of Various Sequences 6. Isomorphism Invariants 7. Integer Tilings
About the author
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.
Summary
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.
Additional text
“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)
Report
"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)
