## Weakly Wandering Sequences in Ergodic TheoryThe 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. |

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### Contents

1 | |

2 Transformations with No Finite Invariant Measure | 17 |

3 Infinite Ergodic Transformations | 25 |

4 Three Basic Examples | 41 |

5 Properties of Various Sequences | 64 |

### Other editions - View all

Weakly Wandering Sequences in Ergodic Theory Stanley Eigen,Arshag Hajian,Yuji Ito,Vidhu Prasad No preview available - 2016 |

Weakly Wandering Sequences in Ergodic Theory Stanley Eigen,Arshag Hajian,Yuji Ito No preview available - 2014 |

Weakly Wandering Sequences in Ergodic Theory Stanley Eigen,Arshag Hajian,Yuji Ito,Vidhu Prasad No preview available - 2014 |

### Common terms and phrases

absolutely continuous assume column C1 complementing pairs complementing set construction coordinates cylinder set deﬁned defined on X;B definition denote direct sum disj dissipative sequence dyadic dyadic rationals Eigen equivalent ergodic measure-preserving transformation ergodic theory ergodic transformation defined eww sequence eww set ﬁnite finite invariant measure finite measure space ﬁrst growth distribution Hajian hitting sequence implies Individual Ergodic Theorem induction inﬁnite infinite ergodic transformation infinite sequence isomorphism invariant Lebesgue measure Lemma lim n!1 m.TnA measurable and nonsingular measure space X;B measure-preserving transformation defined mutually disjoint non-isomorphic nonnegative nonsingular transformation odometer positive integer positive measure possess recurrent sequences preserve a finite Proof Proposition right subcolumn satisfies Second Basic Example sequence fnig sequence of integers set of finite set of integers set X0 skyscraper stack subsets Theorem transformation Q wandering set Weakly Wandering Sequences