Cantor minimal systems

167 Pages·2018·0.835 MB·English

by  Putnam, Ian F

UNIVERSITY LECTURE SERIES VOLUME 70
Cantor Minimal Systems
Ian F. Putnam
Cantor Minimal Systems

UNIVERSITY LECTURE SERIES VOLUME 70
Cantor Minimal Systems
Ian F. Putnam
EDITORIAL COMMITTEE
Jordan S. Ellenberg Robert Guralnick
WilliamP. Minicozzi II (Chair) Tatiana Toro
2010 Mathematics Subject Classification. Primary 37B05; Secondary 20F60.
For additional informationand updates on this book, visit
www.ams.org/bookpages/ulect-70
Library of Congress Cataloging-in-Publication Data
Names: Putnam,IanF.(IanFraser),1958-author.
Title: Cantorminimalsystems/IanF.Putnam.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2018]|Series: Univer-
sitylectureseries;volume70|Includesbibliographicalreferencesandindexes.
Identifiers: LCCN2017057279|ISBN9781470441159(alk. paper)
Subjects: LCSH:Compactspaces. |Cantorsets. |Topologicalspaces. |AMS:Dynamicalsystems
andergodictheory–Topologicaldynamics–Transformationsandgroupactionswithspecial
properties(minimality,distality, proximality, etc.). msc|Grouptheoryandgeneralizations–
Specialaspectsofinfiniteorfinitegroups–Orderedgroups. msc
Classification: LCCQA611.23.P882018|DDC514/.32–dc23
LCrecordavailableathttps://lccn.loc.gov/2017057279
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Contents
Preface ix
Chapter 1. An example: A tale of two equivalence relations 1
Chapter 2. Basics: Cantor sets and orbit equivalence 7
1. Cantor sets 7
2. Orbit equivalence 14
Chapter 3. Bratteli diagrams: Generalizing the example 19
Chapter 4. The Bratteli-Vershik model: Generalizing the example 29
Chapter 5. The Bratteli-Vershik model: Completeness 37
Chapter 6. E´tale equivalence relations: Unifying
the examples 43
1. Local actions and ´etale equivalence relations 44
2. R as an ´etale equivalence relation 48
E
3. R as an ´etale equivalence relation 50
ϕ
Chapter 7. The D invariant 53
1. The group C(X,Z) 53
2. Ordered abelian groups 55
3. The invariant 56
4. Inductive limits of groups 58
5. The dimension group of a Bratteli diagram 61
6. The invariant for AF-equivalence relations 68
7. The invariant for Z-actions 70
Chapter 8. The Effros-Handelman-Shen Theorem 75
1. The statement 75
2. The proof 78
Chapter 9. The Bratteli-Elliott-Krieger Theorem 85
Chapter 10. Strong orbit equivalence 91
1. Orbit cocycles 91
2. Strong orbit equivalence and classification 92
Chapter 11. The D invariant 95
m
vii
viii CONTENTS
1. An innocent’s guide to measure theory 95
2. States on ordered abelian groups 99
3. R-invariant measures 102
4. R-invariant measures and the D invariant 103
5. The invariant 104
6. The invariant for AF-equivalence relations 109
7. The invariant for Z-actions 113
8. The classification of odometers 114
Chapter 12. The absorption theorem 117
1. The simplest version 117
2. The proof 118
3. Matui’s absorption theorem 126
Chapter 13. The classification of AF-equivalence relations 129
1. An example 129
2. The classification theorem 133
Chapter 14. The classification of Z-actions 137
Appendix A. Examples 139
Bibliography 145
Index of terminology 147
Index of notation 149
Preface
In mathematics, dynamical systems is broadly concerned with self-maps
ofspaces. Thespacesunderconsiderationusuallyhavesomekindofgeomet-
ric or measure structure. Here, we will look at compact topological spaces.
For the most part, we will also content ourselves with having a single self-
map of this space, which we assume to be both continuous and invertible.
The main feature of such a self-map is that it can be iterated, as can its
inverse, and this collection of maps is the main object of study.
We willdenotethespacebyX andthe mapbyϕ, so thatforanyinteger
n, ϕn is the composition of this map with itself n times, when n is positive,
and the composition of ϕ−1 with itself |n| times, when n is negative.
There are many applications of these ideas. The space is a model for
the possible configurations of some system and the map is the evolution of
this system in time or space.
Returning to mathematics, such systems display a very wide range of
different phenomena and it is common to restrict attention to a class having
some special features. We will have two specific restrictions and the first
is that our systems will (usually) be minimal: for every point x in X, its
orbit, {ϕn(x) | n ∈ Z}, is dense in X. Intuitively, this suggests that the
system has a certain amount of complexity: every point is moved around by
the iteration under ϕ into every open subset of the space. We will see that
this is equivalent to the condition that there is no proper compact subset
of X which is mapped onto itself by ϕ, other than the trivial case of the
empty set. This means that it is not possible to study the dynamics by
decomposing it into smaller pieces. In a certain sense, it is irreducible.
We should mention that chaotic systems also have a weaker type of
irreduciblity condition, but they also have a wealth of finite orbits and so
are quite far from our minimal case.
The second restriction we place on our systems is that the underlying
space X be totally disconnected: that is, its only non-empty connected sub-
sets are single points.We refer to any compact, totally disconnected metric
spacewithnoisolatedpointsasaCantor set, asCantor’sfamousternaryset
is an example. This may seem slightly less natural. But such spaces have a
nice universal property: every compact metric is the image of such a space
under a continuous function. A version of this even holds if we include the
dynamics, but we will not give a precise statement here. So our systems,
while special, have a universal property among minimal systems.
ix