Cantor minimal systems
Author(s)
Bibliographic Information
Cantor minimal systems
（University lecture series, v. 70）
American Mathematical Society, c2018
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Includes bibliographical references (p. 145-146) and indexes
Description and Table of Contents
Description
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.
Table of Contents
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
by "Nielsen BookData"