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| Softcover ISBN: | 978-1-4704-4115-9 |
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| Softcover ISBN: | 978-1-4704-4115-9 |
| Product Code: | ULECT/70 |
| List Price: | $69.00 |
| MAA Member Price: | $62.10 |
| AMS Member Price: | $55.20 |
| eBook ISBN: | 978-1-4704-4731-1 |
| Product Code: | ULECT/70.E |
| List Price: | $65.00 |
| MAA Member Price: | $58.50 |
| AMS Member Price: | $52.00 |
| Softcover ISBN: | 978-1-4704-4115-9 |
| eBook ISBN: | 978-1-4704-4731-1 |
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| List Price: | $134.00 $101.50 |
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Book DetailsUniversity Lecture SeriesVolume: 70; 2018; 149 ppMSC: Primary 37; 20
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.
ReadershipUndergraduate and graduate students and researchers interested in dynamical systems.
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Table of Contents
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Chapters
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An example: A tale of two equivalence relations
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Basics: Cantor sets and orbit equivalence
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Bratteli diagrams: Generalizing the example
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The Bratteli-Vershik model: Generalizing the example
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The Bratteli-Vershik model: Completeness
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Étale equivalence relations: Unifying the examples
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The $D$ invariant
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The Effros-Handelman-Shen theorem
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The Bratteli-Elliott-Krieger theorem
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Strong orbit equivalence
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The $D_m$ invariant
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The absorption theorem
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The classification of AF-equivalence relations
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The classification of $\mathbb {Z}$-actions
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Examples
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Additional Material
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Reviews
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[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
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RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
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- Reviews
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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.
Undergraduate and graduate students and researchers interested in dynamical systems.
-
Chapters
-
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
-
Étale 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
-
[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
