Weakly Wandering Sequences in Ergodic Theory

1. Front Matter

   Pages i-xiv

   PDF

2. Existence of Finite Invariant Measure

   • Stanley Eigen, Arshag Hajian, Yuji Ito, Vidhu Prasad

   Pages 1-16

3. Transformations with No Finite Invariant Measure

   • Stanley Eigen, Arshag Hajian, Yuji Ito, Vidhu Prasad

   Pages 17-24

4. Infinite Ergodic Transformations

   • Stanley Eigen, Arshag Hajian, Yuji Ito, Vidhu Prasad

   Pages 25-39

5. Three Basic Examples

   • Stanley Eigen, Arshag Hajian, Yuji Ito, Vidhu Prasad

   Pages 41-63

6. Properties of Various Sequences

   • Stanley Eigen, Arshag Hajian, Yuji Ito, Vidhu Prasad

   Pages 65-77

7. Isomorphism Invariants

   • Stanley Eigen, Arshag Hajian, Yuji Ito, Vidhu Prasad

   Pages 79-102

8. Integer Tilings

   • Stanley Eigen, Arshag Hajian, Yuji Ito, Vidhu Prasad

   Pages 103-146

9. Back Matter

   Pages 147-153

   PDF

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.

• Direct sum decompositions of N and Z
• Infinite ergodic transformations
• Invariant measures for ergodic transformations
• Recurrent and dissipative sequences
• Weakly wandering and exhaustive weakly wandering sequences

“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)

• Department of Mathematics, Northeastern University, Boston, USA

  Stanley Eigen, Arshag Hajian

• Department of Mathematics, Keio University, Yokohama, Japan

  Yuji Ito

• Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, USA

  Vidhu Prasad

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.

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