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SpringerBriefs in Mathematics Anima Nagar · Manpreet Singh Topological Dynamics of Enveloping Semigroups SpringerBriefs in Mathematics SeriesEditors NicolaBellomo,Torino,Italy MicheleBenzi,Pisa,Italy PalleJorgensen,Iowa,USA TatsienLi,Shanghai,China RoderickMelnik,Waterloo,Canada OtmarScherzer,Linz,Austria BenjaminSteinberg,NewYork,USA LotharReichel,Kent,USA YuriTschinkel,NewYork,USA GeorgeYin,Detroit,USA PingZhang,Kalamazoo,USA SpringerBriefs present concise summaries of cutting-edge research and practical applicationsacrossawidespectrumoffields.Featuringcompactvolumesof50to125 pages,theseriescoversarangeofcontentfromprofessionaltoacademic.Briefsare characterizedbyfast,globalelectronicdissemination,standardpublishingcontracts, standardized manuscript preparation and formatting guidelines, and expedited productionschedules. 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Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews,andzbMATH. · 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 ©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore 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