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# Cantor Minimal Systems by Ian F. Putnam

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## Cantor Minimal Systems Summary

### Cantor Minimal Systems by 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.

## About Ian F. Putnam

Ian F. Putnam, University of Victoria, BC, Canada.

• An example: A tale of two equivalence relations
• Basics: Cantor sets and orbit equivalence
• Bratteli diagrams: Generalizing the example
• The Bratteli-Vershik model: Generalizing the example
• The Bratteli-Vershik model: Completeness
• Etale equivalence relations: Unifying the examples
• The \$D\$ invariant
• The Effros-Handelman-Shen theorem
• The Bratteli-Elliott-Krieger theorem
• Strong orbit equivalence
• The \$D_m\$ invariant
• The absorption theorem
• The classification of AF-equivalence relations
• The classification of \$\mathbb{Z}\$-actions
• Examples
• Bibliography
• Index of terminology
• Index of notation

NGR9781470441159
9781470441159
1470441152
Cantor Minimal Systems by Ian F. Putnam
New
Paperback
American Mathematical Society
2018-04-30
184
N/A
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