Table Of ContentSpringerBriefs in Mathematics
Anima Nagar · Manpreet Singh
Topological Dynamics
of Enveloping
Semigroups
SpringerBriefs in Mathematics
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·
Anima Nagar Manpreet Singh
Topological Dynamics
of Enveloping Semigroups
AnimaNagar ManpreetSingh
DepartmentofMathematics IndianInstituteofScience
IndianInstituteofTechnologyDelhi Bengaluru,Karnataka,India
NewDelhi,Delhi,India
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN 978-981-19-7876-0 ISBN 978-981-19-7877-7 (eBook)
https://doi.org/10.1007/978-981-19-7877-7
MathematicsSubjectClassification:37B05,37B10,37B20,54B20
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Preface
A compact metric space X and a discrete topological acting group T give a flow
(X,T).RobertEllishadinitiatedthestudyofdynamicalpropertiesoftheflow(X,T)
via the algebraic properties of its “enveloping semigroup” E(X). This concept of
envelopingsemigroupsthathedefinedhasturnedouttobeaveryfundamentaltool
intheabstracttheoryof“topologicaldynamics”.
Theflow(X,T)inducestheflow(2X,T).SuchastudywasfirstinitiatedbyEli
Glasner who studied the properties of this induced flow by defining and using the
notionofa“circleoperator”asanactionofβT on2X,whereβT istheStone–Cˇech
compactificationof T andalsoauniversalenvelopingsemigroup.Weproposethat
the study of properties for the induced flow (2X,T) be made using the algebraic
propertiesofE(2X)onthelinesofEllis’theory,insteadoflookingintotheactionof
βT on2X viathecircleoperatorasdonebyGlasner.Suchastudyrequiresextending
the present theory on the flow (E(X),T). In this book, we take up such a study
giving some subtle relations between the semigroups E(X) and E(2X) and some
interestingassociatedconsequences.
We are indebted to Ethan Akin and Joseph Auslander for all the enlighting
discussionsthroughoutandtheirmanyhelpfulcomments.
NewDelhi,India AnimaNagar
June2022 ManpreetSingh
v
Contents
1 Introduction .................................................... 1
1.1 Introduction ................................................ 1
1.2 Chapters’Overview .......................................... 3
References ...................................................... 4
2 Bird’s-EyeViewonDynamicalSystems ........................... 7
2.1 DefinitionsandElementaryProperties .......................... 7
2.2 InducedSpaces ............................................. 15
2.3 EnvelopingSemigroups ...................................... 16
2.3.1 SomeBasicTheory ................................... 16
2.3.2 FunctionSpaces ...................................... 24
2.4 SymbolicDynamics ......................................... 26
References ...................................................... 30
3 DynamicsofInducedSystems .................................... 33
3.1 For(semi)Cascades .......................................... 34
3.2 For(semi)Flows ............................................. 39
References ...................................................... 43
4 DynamicalPropertiesofEnvelopingSemigroups ................... 45
4.1 RecurrenceinEnvelopingSemigroups .......................... 45
4.2 PeriodicPointsforEnvelopingSemigroups ..................... 46
4.3 FiniteMinimalIdealsinEnvelopingSemigroupandProximal
Relations ................................................... 49
4.4 VariationsofTransitivityforEnvelopingSemigroups ............. 56
4.4.1 TransitiveEnvelopingSemigroups ....................... 57
4.4.2 Stronger Forms of Transitivity for Enveloping
Semigroups .......................................... 59
References ...................................................... 61
vii
viii Contents
5 EnvelopingSemigroupoftheInducedSystems ..................... 63
5.1 InducibleMappings .......................................... 63
5.2 EnvelopingSemigroupof(2Z,σ),ItsSubshiftsandof(22Z,σ∗) .... 71
5.2.1 IsomorphismofβZandβnZ, ∀n ∈ N ................... 76
5.3 SaturatedEnvelopingSemigroups .............................. 78
References ...................................................... 80
About the Authors
Anima Nagar is Associate Professor at the Department of Mathematics, Indian
InstituteofTechnologyDelhi,NewDelhi,India.Shecompletedherdoctoralstudies
at Gujarat University in 1999 and spent her postdoctoral years at the University
of Hyderabad, India. She has mainly held visiting positions at the University of
Maryland,USA,andTechnicalUniversityWeinaswellastheUniversityofWein,
Austria.Herresearchinterestsareintopologicaldynamicsandsymbolicdynamics.
Manpreet Singh is a mathematics instructor at the Indian Institute of Science,
Bengaluru,India.BeforethathewasvisitingscientistattheIndianStatisticalInsti-
tute,Bengaluru,India.HecompletedhisPh.D.degreefromtheIndianInstituteof
Technology Delhi. His broad research area is topological dynamics. He is mainly
interestedintherelationbetweenenvelopingsemigroupofaflowandtheenveloping
semigroupofitsinducedflow.
ix
Chapter 1
Introduction
Abstract Ourgoalistostudy“EnvelopingsemigroupsofInducedSystems”.Inthis
chapter, we briefly introduce the background behind our motivation for this study.
Thisalsoservesasanintroductiontoourassumptionsforbuildingupthistheory.
· ·
Keywords Flows Subflows Envelopingsemigroups
1.1 Introduction
ADynamicalSystemisusuallydenotedasapair(X,T),where X iscalledaphase
space,anddefinedasanactionofsome group(semigroup)T on X.ThesetTx =
{tx :t ∈T} is called the orbit of a point x ∈ X and Topological Dynamics is the
studyoforbitsforallpointsin X.
Let T be a topological group (monoid) acting on a compact Hausdorff space
or a compact metric space X then π(X,T) is called a flow (semiflow), where π is
thecontinuous group (monoid) action of T on X,i.e.,π :T ×X → X defined as
π(t,x)=tx iscontinuousinbothvariables.Thusforeveryt ∈T,onecanconsider
the function πt : X → X, given by πt(x)=π(t,x)=tx. If T =Z or N then the
flow is called a cascade or semicascade. So, we can write π(X,Z) or π(X,N) as
(X, f)andtheactionofZorNwillbedefinedasπ(m,x)=m·x := fm(x),where
f =π1 givesthegeneratoroftheaction.Wedropnamingtheactionπ ofT on X,
andabbreviatethesystemas(X,T).
HerethroughoutweshalltalkofflowsifT isagroup,semiflowsifT isamonoid,
andingeneralwillreferthemtogetheras(semi)flow.Similarly,whenT =ZorNwe
havecascadesandsemicascadeswhichtogetherwillbereferredtoas(semi)cascades.
In our study, X is mostly a compact, infinite metric and T is discrete but not
necessarilyabelian(semi)group,thoughforsomeparticularresultswemayneedT
tobeabelian.Someresultsrequire T =ZorNandinthosecaseswewillbeonly
considering(semi)cascades.WearemainlyinterestedinthecasewhenT isinfinite.
ForafiniteT,manyofourresultsmaynothavethesameformandwesimplyignore
thosecases.
©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2022 1
A.NagarandM.Singh,TopologicalDynamicsofEnvelopingSemigroups,
SpringerBriefsinMathematics,
https://doi.org/10.1007/978-981-19-7877-7_1
