**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.

#### Reviews & Endorsements

[This] book is a very nicely written introduction to the orbit equivalence theory of Cantor minimal systems. I highly recommend it for students who would like to conduct research in this area...The book gives the intuition needed to work in this area and the inspiration for further research. The style of writing is very encouraging and leaves a reader with an impression of being at the lecture and listening to the author.

-- Olena Karpel, Mathematical Reviews

#### 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