SPECTRA OF FINITELY GENERATED BOOLEAN FLOWS

TheoryandApplicationsofCategories,Vol.16,No.17,2006,pp.434–459.SPECTRAOFFINITELYGENERATEDBOOLEANFLOWSJOHNF.KENNISONAbstract.AflowonacompactHausdorffspaceXisgivenbyamapt:X→X.Thegeneralgoalofthispaperistofindthe“cyclicparts”ofsuchaflow.Todothis,weapproximate(X,t)byaflowonaStonespace(thatis,atotallydisconnected,compactHausdorffspace).SuchaflowcanbeexaminedbyanalyzingtheresultingflowontheBooleanalgebraofclopensubsets,usingthespectrumdefinedinourpreviouspaper,ThecyclicspectrumofaBooleanflowTAC10392-419.Inthispaper,wedescribethecyclicspectrumintermsthatdonotrelyontopostheory.WethencomputethecyclicspectrumofanyfinitelygeneratedBooleanflow.WedefinewhenasheafofBooleanflowscanberegardedascyclicandfindnecessaryconditionsforrepresentingaBooleanflowusingtheglobalsectionsofsuchasheaf.Inthefinalsection,wedefineandexplorearelatedspectrumbasedonminimalsubflowsofStonespaces.1.IntroductionThispapercontinuestheresearchstartedin[Kennison,2002].Theunderlyingissueswehopetoaddressareillustratedbyconsidering“flowsincompactHausdorffspaces”ormapst:X→XwhereXissuchaspace.Eachx∈Xhasanorbit{x,t(x),t2(x),...,tn(x),...}andwewanttoknowwhenitisreasonabletosaythatthisorbitis“closetobeingcyclic”.WealsowanttobreakXdownintoits“close-to-cyclic”components.Todothis,weapproximateXbyaStonespace,whichhasanassociatedBooleanalgebratowhichwecanapplythecyclicspectrumdefinedin[Kennison,2002].Insection4,weexaminewaysofcomputingthecyclicspectrumandgiveacompletedescriptionofitforBooleanflowsthatarisefromsymbolicdynamics.Section5discussesnecessaryconditionsforcyclicrepresentations.Section6considersthe“simplespectrum”whichisricherthanthecyclicspectrum.Wehavetriedtopresentthismaterialinawaythatisunderstandabletoexpertsindynamicalsystemswhoarenotspecialistsincategorytheory.(Wedoassumesomebasiccategorytheory,asfoundin[Johnstone,1982,pages15–23].Forfurtherdetails,[MacLane,1971]isagoodreference.)Insection3,wedefinethecyclicspectrumconstructionTheauthorthanksMichaelBarrandMcGillUniversityforprovidingastimulatingresearchat-mosphereduringtheauthor’srecentsabbatical.Theauthoralsothankstherefereeforhelpfulsuggestions,particularywiththeexposition.Receivedbytheeditors2003-11-03and,inrevisedform,2006-08-20.TransmittedbySusanNiefield.Publishedon2006-08-28.2000MathematicsSubjectClassification:06D22,18B99,37B99.Keywordsandphrases:Booleanflow,dynamicalsystems,spectrum,sheaf.cJohnF.Kennison,2006.Permissiontocopyforprivateusegranted.434SPECTRAOFFINITELYGENERATEDBOOLEANFLOWS435withoutusingtopostheory.Inthatsection,wereviewthebasicnotionofasheafoveralocale.Fordetails,seethebookonStonespaces,[Johnstone,1982],whichprovidesareadabletreatmentoftheideasandtechniquesusedinthispaper.Asdiscussedinsection2,theuseofsymbolicdynamicsallowsustorestrictourattentiontoflowst:X→XwhereXisaStonespace,whichmeansitistotallydisconnectedinadditiontobeingcompactandHausdorff.ButifXisaStonespace,thenXisdeterminedbytheBooleanalgebra,Clop(X),ofitsclopensubsets(where“clopen”subsetsarebothclosedandopen).BytheStoneRepresentationTheorem,ClopiscontravariantlyfunctorialandsetsupanequivalencebetweenthecategoryofStonespacesandthedualofthecategoryofBooleanalgebras.Itfollowsthatt:X→XgivesrisetoaBooleanhomomorphismτ:B→BwhereB=Clop(X)andτ=Clop(t)=t−1.Mappingaflowfromonecategorytoanotherissignificantbecausethenotionofacyclicflowdependsontheambientcategory.Werecallthefollowingdefinitionfrom[Kennison,2002].Indoingso,weadopttheusefultermiteratorfrom[Wojtowicz,2004]andotherwiseusethenotationalconventionsadoptedin[Kennison,2002].SoiffandgaremorphismsfromanobjectXtoanobjectY,thenEqu(f,g)istheirequalizer(ifitexists).If{Aα}isafamilyofsubobjectsofX,then
{Aα}istheirsupremum(ifitexists)inthepartiallyorderedsetofsubobjectsofX.1.1.Definition.Thepair(X,t)isaflowinacategoryCifXisanobjectofCandt:X→Xisamorphism,calledtheiterator.If(X,t)and(Y,s)areflowsinC,thenaflowhomomorphismisamaph:X→Yforwhichsh=ht.WeletFlow(C)denotetheresultingcategoryofflowsinC.Wesaythat(X,t)∈Flow(C)iscyclicif
Equ(IdX,tn)existsandisX(thelargestsubobjectofX).Inlistingsomeexamplesfrom[Kennison,2002],itisconvenienttosaythatifSisaset(possiblywithsometopologicaloralgebraicstructure)andift:S→S,thens∈Sisperiodicifthereexistsn∈Nwithtn(s)=s.•Aflow(S,t)inSetsiscyclicifandonlyifeveryelementofSisperiodic.•Aflow(X,t)inthecategoryofStonespacesiscyclicifandonlyiftheperiodicelementsofXaredense.•Aflow(B,τ)inthecategoryofBooleanalgebrasiscyclicifandonlyifeveryelementofBisperiodic.•Aflow(X,t)inStonespacesis“Booleancyclic”(meaningthatClop(X,t)iscyclicinBooleanalgebras)ifandonlyifthegroupofprofiniteintegers,Z,actscontinuouslyonXinamannercompatiblewitht.(ThereisanembeddingN⊆Zandanactionα:Z×X→Xiscompatiblewiththeactionoftifα(n,x)=tn(x)forallx∈Xandalln∈N.SinceNisdenseinZ,thereisatmostonesuchcontinuousactionbyZ.Fordetails,see[Kennison,2002]).436JOHNF.KENNISON•Lett:S→SbegivenwhereSisaset.Then(S,t)isacyclicflowinthedualofthecategoryofSetsifandonlyiftisone-to-one.WeareprimarilyinterestedinBooleanflows,orflows(B,τ),inthecategoryofBooleanalgebras.Wesometimessaythat“BisaBooleanflow”,inwhichcasetheiterator(alwaysdenotedbyτ)isleftimplicit.