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BL
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| Record Number
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850294
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| Main Entry
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Putnam, Ian F., (Ian Fraser),1958-
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| Title & Author
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Cantor minimal systems /\ Ian F. Putnam.
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| Publication Statement
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Providence, Rhode Island :: American Mathematical Society,, [2018]
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| Series Statement
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University lecture series ;; volume 70
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| Page. NO
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xiii, 149 pages :: illustrations ;; 26 cm.
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| ISBN
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1470441152
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: 9781470441159
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| Bibliographies/Indexes
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Includes bibliographical references and indexes.
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| Contents
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Machine generated contents note: ch. 1 An example: A tale of two equivalence relations -- ch. 2 Basics: Cantor sets and orbit equivalence -- 1.Cantor sets -- 2.Orbit equivalence -- ch. 3 Bratteli diagrams: Generalizing the example -- ch. 4 The Bratteli-Vershik model: Generalizing the example -- ch. 5 The Bratteli-Vershik model: Completeness -- ch. 6 Etale equivalence relations: Unifying the examples -- 1.Local actions and etale equivalence relations -- 2.Re as an etale equivalence relation -- 3.Rq as an etale equivalence relation -- ch. 7 The D invariant -- 1.The group C(X, Z) -- 2.Ordered abelian groups -- 3.The invariant -- 4.Inductive limits of groups -- 5.The dimension group of a Bratteli diagram -- 6.The invariant for AF-equivalence relations -- 7.The invariant for Z-actions -- ch. 8 The Effros-Handelman-Shen Theorem -- 1.The statement -- 2.The proof -- ch. 9 The Bratteli-Elliott-Krieger Theorem -- ch. 10 Strong orbit equivalence -- 1.Orbit cocycles
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Note continued: 2.Strong orbit equivalence and classification -- ch. 11 The Dm invariant -- 1.An innocent's guide to measure theory -- 2.States on ordered abelian groups -- 3.R-invariant measures -- 4.R-invariant measures and the D invariant -- 5.The invariant -- 6.The invariant for AF-equivalence relations -- 7.The invariant for Z-actions -- 8.The classification of odometers -- ch. 12 The absorption theorem -- 1.The simplest version -- 2.The proof -- 3.Matui's absorption theorem -- ch. 13 The classification of AF-equivalence relations -- 1.An example -- 2.The classification theorem -- ch. 14 The classification of $$-actions.
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| Abstract
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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 and a basic course in general topology.
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| Subject
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Cantor sets.
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Compact spaces.
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Topological spaces.
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| Subject
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Cantor sets.
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| Subject
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Compact spaces.
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| Subject
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Dynamical systems and ergodic theory -- Topological dynamics -- Transformations and group actions with special properties (minimality, distality, proximality, etc.).
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| Subject
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Group theory and generalizations -- Special aspects of infinite or finite groups -- Ordered groups.
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| Subject
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Topological spaces.
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| Dewey Classification
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514/.32
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| LC Classification
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QA611.23.P88 2018
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| NLM classification
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37B0520F60msc
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