**University Lecture Series**

Volume: 70;
2018;
149 pp;
Softcover

MSC: Primary 37; 20;

Print ISBN: 978-1-4704-4115-9

Product Code: ULECT/70

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

**Electronic ISBN: 978-1-4704-4731-1
Product Code: ULECT/70.E**

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

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#### Supplemental Materials

# Cantor Minimal Systems

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*Ian F. Putnam*

Within the subject of topological dynamics,
there has been considerable recent interest in systems where the
underlying topological space is a Cantor set. Such systems have an
inherently combinatorial nature, and seminal ideas of Anatoly Vershik
allowed for a combinatorial model, called the Bratteli-Vershik model,
for such systems with no non-trivial closed invariant subsets. This
model led to a construction of an ordered abelian group which is an
algebraic invariant of the system providing a complete classification
of such systems up to orbit equivalence.

The goal of this book is to give a statement of this classification
result and to develop ideas and techniques leading to it. Rather than
being a comprehensive treatment of the area, this book is aimed at
students and researchers trying to learn about some surprising
connections between dynamics and algebra. The only background material
needed is a basic course in group theory and a basic course in general
topology.

#### Readership

Undergraduate and graduate students and researchers interested in dynamical systems.

#### Table of Contents

# Table of Contents

## Cantor Minimal Systems

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface ix10
- Chapter 1. An example: A tale of two equivalence relations 116
- Chapter 2. Basics: Cantor sets and orbit equivalence 722
- Chapter 3. Bratteli diagrams: Generalizing the example 1934
- Chapter 4. The Bratteli-Vershik model: Generalizing the example 2944
- Chapter 5. The Bratteli-Vershik model: Completeness 3752
- Chapter 6. Étale equivalence relations: Unifying the examples 4358
- Chapter 7. The 𝐷 invariant 5368
- Chapter 8. The Effros-Handelman-Shen Theorem 7590
- Chapter 9. The Bratteli-Elliott-Krieger Theorem 85100
- Chapter 10. Strong orbit equivalence 91106
- Chapter 11. The 𝐷_{𝑚} invariant 95110
- 1. An innocent’s guide to measure theory 95110
- 2. States on ordered abelian groups 99114
- 3. 𝑅-invariant measures 102117
- 4. 𝑅-invariant measures and the 𝐷 invariant 103118
- 5. The invariant 104119
- 6. The invariant for AF-equivalence relations 109124
- 7. The invariant for \Z-actions 113128
- 8. The classification of odometers 114129

- Chapter 12. The absorption theorem 117132
- Chapter 13. The classification of AF-equivalence relations 129144
- Chapter 14. The classification of \Z-actions 137152
- Appendix A. Examples 139154
- Bibliography 145160
- Index of terminology 147162
- Index of notation 149164
- Back Cover Back Cover1167