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Cantor Minimal Systems By Ian F. Putnam

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.

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

    Additional information

    Cantor Minimal Systems by Ian F. Putnam
    American Mathematical Society
    Book picture is for illustrative purposes only, actual binding, cover or edition may vary.
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