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Softcover ISBN:  9781470441159 
Product Code:  ULECT/70 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
eBook ISBN:  9781470447311 
Product Code:  ULECT/70.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Softcover ISBN:  9781470441159 
eBook ISBN:  9781470447311 
Product Code:  ULECT/70.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $107.20 $81.20 

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 BratteliVershik model, for such systems with no nontrivial 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.

Table of Contents

Chapters

An example: A tale of two equivalence relations

Basics: Cantor sets and orbit equivalence

Bratteli diagrams: Generalizing the example

The BratteliVershik model: Generalizing the example

The BratteliVershik model: Completeness

Étale equivalence relations: Unifying the examples

The $D$ invariant

The EffrosHandelmanShen theorem

The BratteliElliottKrieger theorem

Strong orbit equivalence

The $D_m$ invariant

The absorption theorem

The classification of AFequivalence relations

The classification of $\mathbb {Z}$actions

Examples


Additional Material

Reviews

[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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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 BratteliVershik model, for such systems with no nontrivial 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 BratteliVershik model: Generalizing the example

The BratteliVershik model: Completeness

Étale equivalence relations: Unifying the examples

The $D$ invariant

The EffrosHandelmanShen theorem

The BratteliElliottKrieger theorem

Strong orbit equivalence

The $D_m$ invariant

The absorption theorem

The classification of AFequivalence 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