Automorphisms and Equivalence Relations in Topological Dynamics

287 Pages·2014·1 MB·English

by  Ellis D.B., Ellis R.

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376 Permutationpatterns,S.LINTON,N.RUŠKUC&V.VATTER(eds)
377 AnintroductiontoGaloiscohomologyanditsapplications,G.BERHUY
378 Probabilityandmathematicalgenetics,N.H.BINGHAM&C.M.GOLDIE(eds)
379 Finiteandalgorithmicmodeltheory,J.ESPARZA,C.MICHAUX&C.STEINHORN(eds)
380 Realandcomplexsingularities,M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds)
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382 Forcingwithrandomvariablesandproofcomplexity,J.KRAJÍCEK
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404 Operatormethodsforboundaryvalueproblems,S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds)
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LondonMathematicalSocietyLectureNoteSeries:412
Automorphisms and Equivalence Relations
in Topological Dynamics
DAVID B. ELLIS
BeloitCollege,Wisconsin
ROBERT ELLIS
UniversityofMinnesota
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UniversityPrintingHouse,Cambridgecb28bs,UnitedKingdom
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitofeducation,
learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107633223
©D.B.EllisandR.Ellis
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2013
PrintedandboundintheUnitedKingdomby<NameofPrinter>
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-Publicationdata
Ellis,D.(David),1958–Automorphismsandequivalencerelationsintopologicaldynamics
/DavidB.Ellis,BeloitCollege,Wisconsin,RobertEllis,UniversityofMinnesota.
pages cm.–(LondonMathematicalSocietylecturenoteseries;412)
ISBN978-1-107-63322-3(pbk.)
1. Topologicaldynamics. 2. Algebraictopology. 3. Automorphisms.
I. Ellis,Robert,1926– II. Title.
QA611.5.E3942014
512’.55–dc23
2013043992
ISBN978-1-107-63323-3Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
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To Valerie, Carrie, and Kathleen; in memory of Betty.
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Contents
Introduction pageix
PARTI UNIVERSALCONSTRUCTIONS 1
1 TheStone-cechcompactificationβT 3
2 Flowsandtheirenvelopingsemigroups 19
3 Minimalsetsandminimalrightideals 27
4 Fundamentalnotions 37
5 Quasi-factorsandthecircleoperator 54
PARTII EQUIVALENCERELATIONSAND
AUTOMORPHISMSOFFLOWS 65
6 Quotientspacesandrelativeproducts 67
7 IcersonMandautomorphismsofM 83
8 Regularflows 103
9 Thequasi-relativeproduct 111
PARTIII THEτ-TOPOLOGY 125
10 Theτ-topologyonAut(X) 127
11 Thederivedgroup 138
12 Quasi-factorsandtheτ-topology 149
vii
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viii Contents
PARTIV SUBGROUPSOFGANDTHEDYNAMICS
OFMINIMALFLOWS 155
13 TheproximalrelationandthegroupP 157
14 DistalflowsandthegroupD 167
15 EquicontinuousflowsandthegroupE 179
16 Theregionallyproximalrelation 205
PARTV EXTENSIONSOFMINIMALFLOWS 211
17 Openandhighlyproximalextensions 213
18 Distalextensionsofminimalflows 231
19 Almostperiodicextensionsofminimalflows 245
20 Ataleoffourtheorems 260
References 267
Index 269
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Introduction
To a large extent this book is an updated version of Lectures on Topological
Dynamics by Robert Ellis [Ellis,R ., (1969)]. That book gave an exposition
of what might be called an algebraic theory of minimal sets. Our goal here
is to give a clear, self contained exposition of a new approach to the theory
whichallowsformorestraightforwardproofsanddevelopsaclearerlanguage
for expressing many of the fundamental ideas. We have included a treatment
of many of the results in the aforementioned exposition, in addition to more
recent developments in the theory; we have not attempted, however, to give
a complete or exhaustive treatment of all the known results in the algebraic
theoryofminimalsets.Ourhopeisthatthereaderwillbemotivatedtousethe
languageandtechniquestostudyrelatedtopicsnottouchedonhere.Someof
thesearementionedeitherintheexercisesornotesgivenattheendofvarious
sections. This book should be suitable for a graduate course in topological
dynamicswhoseprerequisitesneedonlyincludesomebackgroundintoplogy.
WeassumethereaderisfamiliarwithcompactHausdorffspaces,convergence
of nets, etc., and perhaps has had some exposure to uniform structures and
pseudometricswhichplayalimitedroleinourexposition.
A flow is a triple (X,T,π) where X is a compact Hausdorff space, T is a
topological group, and π : X ×T → X is a continuous action of T on X,
so that xe = x and (xt)s = x(ts) for all x ∈ X, s,t ∈ T. Here we write
xt = π(x,t) for all x ∈ X and t ∈ T, and e is the identity of the group
T. Usually the symbol π will be omitted and the flow (X,T,π) denoted by
(X,T) or simply by X. In the situations considered here there is no loss of
generalityifT isgiventhediscretetopology.Theassumptionsmadethusfar
donotsufficetoproduceaninterestingtheory.ThegroupT maybetoo“small”
in its action on X. Thus for example, the trivial case where xt = x , a fixed
0
element of X, for all x ∈ X and t ∈ T, is not ruled out. To eliminate such
degenerate behavior it is convenient to assume that the flow (X,T) is point
ix
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