Custom cover image
Weakly Wandering Sequences in Ergodic Theory / by Stanley Eigen, Arshag Hajian, Yuji Ito, Vidhu Prasad
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Monographs in Mathematics | SpringerLink BücherPublisher: Tokyo ; s.l. : Springer Japan, 2014Description: Online-Ressource (XIV, 153 p. 15 illus, online resource)ISBN:- 9784431551089
- 515.39
- 515.48
- QA313
Contents:
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 readerPPN: PPN: 1659076161Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Foreword; Preface; Contents; 1 Existence of Finite Invariant Measure; 1.1 Recurrent Transformations; 1.2 Finite Invariant Measure; 2 Transformations with No Finite Invariant Measure; 2.1 Measurable Transformations; 2.2 Ergodic Transformations; 3 Infinite Ergodic Transformations; 3.1 General Properties of Infinite Ergodic Transformations; 3.2 Weakly Wandering Sequences; 3.3 Recurrent Sequences; 3.3.1 Transformations with Recurrent Sequences; 3.3.2 Transformations Without Recurrent Sequences; 4 Three Basic Examples; 4.1 First Basic Example; 4.1.1 Induced Transformations
4.1.2 Construction of the First Basic Example4.2 Second Basic Example; 4.2.1 Non-measure-Preserving Commutators; 4.2.2 A General Class of Transformations; 4.2.3 Construction of the Second Basic Example; 4.3 Third Basic Example; 4.3.1 Construction of the Third Basic Example; 4.3.2 Random Walk on the Integers; 5 Properties of Various Sequences; 5.1 Properties of ww and Recurrent Sequences; 5.2 Dissipative Sequences; 5.3 The Sequences in a Different Setting; 6 Isomorphism Invariants; 6.1 Exhaustive Weakly Wandering Sets; 6.2 α-Type Transformations
6.3 Recurrent Sequences as an Isomorphism Invariant6.3.1 Construction of the Transformation T; 6.3.2 The Recurrent Sequences for T; 6.4 Growth Distributions for a Transformation; 7 Integer Tilings; 7.1 Infinite Tilings of the Integers; 7.1.1 Structure of Complementing Pairs in N; 7.1.2 Complementing Pairs in Z When A or B Is Finite; 7.1.3 Infinite A, B: No Structure Expected; 7.2 How Tilings Arise in Ergodic Theory; 7.2.1 Constructing a Transformation from a Hitting Sequence; 7.3 Examples of Complementing Pairs; 7.3.1 A Complementing Set That Is Not a Hitting Sequence
7.3.2 A ww Sequence Which Is Not eww for AnyTransformation7.3.3 An eww Sequence with a Complementing Set That Does Not Come from a Point; 7.4 Extending a Finite Set to a Complementing Set; 7.4.1 Definitions and Notations; 7.4.2 Extension Theorem; 7.5 Complementing Sets of A and the 2-Adic Integers; 7.5.1 The 2-Adic Integers; 7.5.2 Condition (iv) Is Not Enough to Be Complementing; 7.6 Examples: Non-isomorphic Transformations; 7.6.1 Two Non-isomorphic Transformations; 7.6.2 An Uncountable Family of Non-isomorphicTransformations; 7.7 An Odometer Construction from M
7.7.1 Set Theoretic Construction of X7.7.2 Defining the Sequences A and B Associated to M; 7.7.3 The Sequence M and Multiple Recurrence; References; Index
No physical items for this record