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AMERICANMATHEMATICALSOCIETYVolume348,Number8,August1996
THEAUTOMORPHISMGROUPOFACODEDSYSTEM
DORISFIEBIGANDULF-RAINERFIEBIG
Abstract.Wegiveageneralconstructionofcodedsystemswithanauto-morphismgroupisomorphictoZ⊕GwhereGisanypreassignedgroupwhichhasa“continuousblockpresentation”(theisomorphismwillmaptheshiftto(1,eG)).Severalapplicationsaregiven.Inparticular,weobtainautomor-phismgroupsofcodedsystemswhichareabelian,whicharefinitelygeneratedandonewhichcontainsZ[1/2].Weshowthatanygroupwhichoccursasasubgroupoftheautomorphismgroupofsomesubshiftwithperiodicpointsdensealreadyoccursforsomesynchronizedsystem.
Introduction
ForafinitesetAconsiderAZendowedwiththeproducttopologyofthediscretetopologyonAtogetherwiththe(left)shiftmap.Asubshift(S,σ)isaclosedshiftinvariantsubsetSofAZtogetherwiththerestrictedshiftmapσ.Inthefollowingweshallsuppressthemapσinournotation.Theautomorphismgroupaut(S)ofSisthegroupofshiftcommutinghomeomorphismsofS.Itiscountable,sinceautomorphismsaregivenbyslidingblockcodes[H].
TheautomorphismgroupsoffullshiftsS=AZhavebeenstudiedintheearly1960sbyHedlundandhiscoworkers[H].MotivatedbyWilliams’sshiftequivalenceproblem[W],inthe1980stherewasrenewedinterestinthegroupsaut(S)whereSisashiftoffinitetype(SFT);see[AM,DGS,LM]forbasicdefinitions.Theyhavebeenstudiedinvariouspapers,forexample[BK,BLR,BFK,KRW],where[BFK]treatsthedifferentone-sidedcase.Theseautomorphismgroupsareveryrichgroups.Bytheso-calledmarkermethoditwasshownthattheyarenotfinitelygeneratedandhavealargecollectionofsubgroups,asforexamplethedirectsumsofeverycountablecollectionoffinitegroupsandthefreeproductsofanyfinitenumberofcopiesofZ/2Z([BLR]).
Itisanaturalquestionhowthestructureoftheseautomorphismgroupsmaychangeifweenlargetheclassofsubshiftsunderconsideration.ShiftsoffinitetypearePPDsubshifts,wherePPDstandsfor“periodicpointsdense”.ThisimpliesthattheautomorphismgroupsofSFTsareresiduallyfinite([BLR]).Forgeneralsub-shiftsothercountablegroupscanberealizedasautomorphismgroups,forexampleQ[BLR].Itisnotclearwhattheobstructionsinthegeneralsettingare.
WewillconsidertwonaturalsubclassesofPPDsubshifts.Forthatwerecallthenotions(see[BH],[FF])ofsynchronizedsystemsandcodedsystems(anytransitiveSFTissynchronizedandanysynchronizedsystemiscoded)andindicatethedra-maticchangeinthepossiblestructuresofautomorphismgroupswhenpassingfrom
ReceivedbytheeditorsDecember13,1994and,inrevisedform,July17,1995.
1991MathematicsSubjectClassification.Primary58F03,20B27;Secondary20E26.
c1996AmericanMathematicalSociety
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3174DORISFIEBIGANDULF-RAINERFIEBIG
synchronizedtogeneralcodedsystems.Forapoints=(si)i∈Z∈S,Sanarbitrarysubshift,lets[n,m]denotethesubblocksn,...,smofs(−∞≤n≤m≤∞).AnS-blockisafinitesubblockofsomepointinS.AnS-blockwissynchronizingifforallpairsofS-blocksu,vtheimplication“ifuwandwvareS-blocksthenuwvisanS-block”holds.ForaSFTthereisannsuchthatallS-blocksoflengthnaresynchronizing(whichmaybeusedasadefinitionoftheSFT-property).
Asynchronizedsystemisatransitivesubshiftwhichhasasynchronizingblock.Theexistenceofasynchronizingblocksufficestocarryoverthemarkerconstructionofautomorphismsasshownin[BLR],thussynchronizedsystemsalsohaverichautomorphismgroups.WewillshowthatgivenanysubshiftRwithperiodicpointsdense,thereisasynchronizedsystemSsuchthataut(S)containsacopyofaut(R).AcodedsystemTisasubshiftwhichhasacodeX,i.e.XisacountablesetoffiniteblocksandTistheclosureofthepointsobtainedbybi-infiniteconcatena-tionsofblocksfromX.Wewillseethattherearecodedsystemswithverysmallautomorphismgroups,infactitmayconsistonlyofthepowersoftheshiftmap.Startingwithsomegeneralremarks,wegiveanoutlineoftheorganizationandmainresultsofthispaper.Wewillstudythesetofgroupswhichoccurasautomor-phismgroups(orassubgroupsofautomorphismgroups)forcodedsystems.Codedsystemshaveperiodicpointsdense,thustheargumentformTheorem3.1in[BLR]showsthattheautomorphismgroupofacodedsystemisresiduallyfinite[MKS](whichimmediatelyexcludesgroupslikeQorZ(p∞)=Z[1/p]/Z).Noteveryresid-uallyfinitegroupisanautomorphismgroupofacodedsystem,forexampleZ[1/2]cannotberealized(seeLemma2.15).Forsubgroupsnofurtherrestrictionthanresidualfinitenessisknowntous(intheSFTcasethesubgroupalsohastohavesolvablewordproblem[BLR,Proposition2.8],arestrictionthatseemstovanishforcodedsystems).
WewillshowthatforavarietyofgroupsGthereisacodedsystemTwithaut(T)≈σ⊕G,whichmeansthatthereisanisomorphismfromaut(T)tothegroupZ⊕G,thatmapstheshiftσto(1,eG),whereeG∈Gistheidentityelement.Insection1wedescribeageneraltechniquetoobtainsuchacodedsystemforanygroupGwhichhasa“continuousblockpresentation”(seesection1fordefinitions).Thisconditionisofaverycombinatorialnature,butiseasilymetbyanyfinitegroupG,andisclosedundertakingcountablesumsofgroups(Theorems2.1and2.6).Insection2wealsoshowthatagroupGhasacontinuousblockpresentationiffGisisomorphictoaclosed(w.r.t.periodicpointtopology)subgroupofaut(R)forsomesubshiftRwithperiodicpointsdense.ThisgivesasecondapproachtoconstructingcodedsystemsTwithaut(T)≈σ⊕GforpreassignedgroupsG:trytorealizeGasaclosedsubgroupoftheautomorphismgroupofsomesubshiftwithperiodicpointsdense.
Wegetinparticularthefollowingresults.
ForanyfinitegroupGthereisacodedsystemTwithaut(T)≈σ⊕G,inparticularthereisacodedsystemSwithtrivialautomorphismgroupaut(T)=σ(Corollary2.2).ThiscontrastswiththefactthatforSFTstheautomorphismgroupis“large”(seeabove).
Anyinfinite,finitelygeneratedabeliangroupistheautomorphismgroupofsomecodedsystem.ThusthecenteroftheautomorphismgroupofacodedsystemneednottobethepowersoftheshiftasintheSFTcase[R].
THEAUTOMORPHISMGROUPOFACODEDSYSTEM3175
Therearetwocodedsystemswithdistinctzetafunctions(thusnonconjugate)whichhaveisomorphicautomorphismgroups(Corollary2.3)(seealsoQuestion4.1in[BLR]).
Thereareuncountablymanynonisomorphicautomorphismgroupsforcodedsys-tems.
ThereisacodedsystemTwithaut(T)≈σ⊕{a,b;a2=b2=e}.
ThereisacodedsystemTwithaut(T)≈σ⊕Z[1/2](itisnotknownifZ[1/2]canbethesubgroupoftheautomorphismgroupofanSFT[BLR,Problem3.4].ThegroupZ[1/2]itselfcannotbetheautomorphismgroupofacodedsystem(seeLemma2.15)).
LetRbeashiftwithperiodicpointsdense.ThenthereisacodedsystemTsuchthataut(T)≈σ⊕aut(R).
Thisshowsinparticularthatpassingfromcodedsystemstoshiftswithperiodicpointsdensedoesnotincreasethesetofsubgroupsoftheautomorphismgroups.Butthesetofpossibleautomorphismgroupsdoesinfactchange:thereisasubshiftRwithperiodicpointsdenseandaut(R)≈Z[1/2]⊕CwhereCissomedirectsumoffinitecyclicgroups(Theorem2.14).Suchagroupcannotbetheautomorphismgroupofacodedsystem(Theorem2.16).WedonotknowifthereissuchanexamplewhereRisalsotransitive.
Itremainsanopenproblemwhatthescopeofourconstructionis:whichgroupsGadmitacontinuousblockpresentation?
1.Codedsystemswithpreassignedautomorphismgroups.
Generalconstruction
Weconsiderinverselimitsoffinitegroups,forwhichwedefineblockpresentationsandtheresultingsubgroupofcontinuouselements.Weshowbyconstructionthatforanysuchsubgroupofcontinuouselements,sayG,thereisacodedsystemwithaut(T)≈σ⊕G(Theorem1.4).
Let(Gn,πn)beasequenceoffinitegroupsGnandontogrouphomomorphismsπn:Gn+1→Gn,n∈N.Letlim←(Gn,πn)denotetheinducedinverselimitsystem{(g1,g2,...)|gn∈Gn,πn(gn+1)=gnforalln∈N},withcoordinatewisegroupmultiplication.
Definition1.1.AblockpresentationBoflim←(Gn,πn)isachoiceofafinitesetA,asequence2≤k(1) Definition1.2.LetωbeanewsymbolnotinA.TheB-actionofgisaslidingblockcode,ifforsomeL≥0thereisamapφ:(A∪{ω})2L+1→Asuchthatforall (φ)(φ) n∈Nandhn∈GnwehaveBn(gnhn)=Φn(Bn(hn)),whereΦn:Ak(n)→Ak(n) (φ) isthemapgivenby(Φn(a1,...,ak(n)))i:=φ(a−L+i,...,ai+L),1≤i≤k(n),whereontheright-handsideam:=ωifm<1orm>k(n). Definition1.3.Anelementg∈lim←(Gn,πn)iscalledB-continuousiftheB-actionsofgandg−1bothareslidingblockcodes.Thisdefinesthesubgroup 3176DORISFIEBIGANDULF-RAINERFIEBIG ofB-continuouselementsinlim←(Gn,πn).AnabstractgroupGhasacontinu-ousblockpresentationifthereissomeblockpresentationBofsomeinverselimitlim←(Gn,πn)offinitegroupssuchthatGisisomorphicto{g∈lim←(Gn,πn)|gisB-continuous}. IfGhasacontinuousblockpresentationthenGisasubgroupoflim←(Gn,π),thusresiduallyfinite.EveryresiduallyfinitegroupGisasubgroupofaninverselimitoffinitegroups,butwedonotknowifitisalwayspossibletoobtainacontinuousblockpresentation. Theorem1.4.LetGbeagroupwithacontinuousblockpresentation.ThenthereisacodedsystemTwithaut(T)≈σ⊕G. TherestofthissectionisdevotedtoaproofofTheorem1.4.FirstageneralmethodofconstructionwillleadustoacodedsystemT,thenweshowaut(T)≈σ⊕Gbyaseriesoflemmataandpropositions.Wemayassumethat G={g∈lim(Gn,πn)|gisB-continuous} ← wherelim←(Gn,πn)isaninverselimitoffinitegroups,Bablockpresentationoflim←(Gn,πn)givenbyasequenceofinjectivemapsBn:Gn→Ak(n),2≤k(1) Y:={0}∪{0|w|+1y(w)0|w|+1|w∈X}. WestartdefiningthesetX.ThesetXwillconsistofallblocksw∈{0,1,2}m,m∈N,whicharestable,neutral,andhaveaproperskeleton.Nowwedefinetheseterms. (i)Stability:LetI=k∈N{n|22k≤n≤22k+1}(anyinfinitesubsetI⊂Nwith1∈I,andI,Ichavingunboundedgapsworks).Let J={(0,0),(1,1),(1,2),(2,1),(2,2)}⊂{0,1,2}2. Ablockw∈{0,1,2}m,m∈N,isstableif (1)theblocks12and21donotoccurassubblocksofw, (2)ifw=0m+1w0m+1,thenforanypairi wehavea=1if(n,(wi−n,wj+n))∈(I×J)∪(I×J),anda=2otherwise.Remark.By(1)theblockwcanbewrittenasafiniteconcatenationofblocksfrom{0}∪{01n|n∈N}∪{02n|n∈N}.ThedefinitionofJismadesuchthat(wi−n,wj+n)∈Jiffbotharezeroorbotharenotzero.Thusanystableblockwcanberecoveredbytherule(2)fromitslengthand0-skeleton(thesetofindicesiwithwi=0). (ii)Neutrality:Wedefineamapcfromthefiniteblockswithsymbolsin{0,1,2}∪Aintolim←(Gn,πn).ForthatletX1={0n+11n0n+1|n∈I}andfirstfixamapc:X1→lim←(Gn,πn)suchthat[g∈c(X1)⇒g−1∈c(X1)]andc(X1)isdense THEAUTOMORPHISMGROUPOFACODEDSYSTEM3177 (i.e.foreachn∈Nandhn∈Gnitholdsc(w)n=hnforsomew∈X1).Nowextendctoamaponblockswwithsymbolsin{0,1,2}∪Aasfollows.Letc(w)=eifnosubblockofwisinX1.Otherwiselet1≤i1 Forexample,anyblockwwhichhasaproperskeletonwherealltheuiareemptyblocksisstableandneutral,thusbelongstoX. WedefinethefunctionyonX.Forn∈N,gn∈Gn,letC(n,gn)betheblockobtainedfromC(n)byreplacingthesubblock2k(n)withBn(gn).Nowletw∈X.Thenwhasaproperskeletonw=C(n1)u1C(n2)u2···C(nk−1)uk−1C(nk)asin(iii).Nowdefine y(w)=C(n1,gn1)u1C(n2,gn2)u2···C(nk−1,gnk−1)uk−1C(nk,gnk) wherethegniareuniquelydeterminedby gni=c(C(n1)u1C(n2)u2···C(ni−1)ui−1C(ni))ni∈Gni, 1≤i≤k. (Notethatgn1=eandalsognk=esincewisneutral.) ThisfinishesthedefinitionofthecodedsystemT.Aseriesoflemmataandpropositionswillleadustoaut(T)≈σ⊕G. WeshowthatT-pointscanbeapproximatedarbitrarilywellbysingleblocksfromY,whichimpliesafundamentalpropertyofT-points. Letfbethe1-blockfactormaponTwhichmapsthesymbolsinAto2andleavesthesymbols0,1,2fixed.Thusf(y(w))=wforallw∈X. Lemma1.5.(a)AnyfiniteconcatenationofblocksinYisasubblockofsomeY-block. (b)AnyT-blockau0z0vbwithz∈({1,2}∪A)n,|a|=|b|=1,|u|=|v|=n−1,satisfiesthefundamentalproperty,thatis(*) z=1norz=2norz∈An, andz=1niff(n,(f(a),f(b)))∈(I×J)∪(Ic×Jc). (c)GivenaT-block0|u|+3u0|u|+3withablocku=u1u2···unoflengthn∈N,thenthe0-skeletonofu(i.e.thesetofiforwhichui=0)determinesforeach1≤i≤ntowhichofthesets{0},{1},{2},Athevalueofuibelongs. Proof.(a)Givenaconcatenationy1y2···yjofblocksyifromY,choosen0solargethatallC(n)-blocksinf(y1y2···yj)satisfyn≤n0.Thenitisnothardtoseethat0M+1C(n0,e)y1y2···yjC(n0,e)0M+1∈YwhereM=lengthofC(n0,e)y1y2···yjC(n0,e). 3178DORISFIEBIGANDULF-RAINERFIEBIG (b)By(a)thegivenblockiscontainedinasingleblock0|w|+1y(w)0|w|+1fromY.ThedefinitionofY-blocksimpliesimmediatelythatz=1norz=2norz∈An.Bydefinition,wisastableblock,whichimpliesthesecondstatement. (c)Consider1≤i≤nsuchthatui=0.Thereisasubblockau0z0vbasin(b)of0|u|+3u0|u|+3suchthatthisithcoordinatebelongstothesubblockz.Thus,thefundamentalpropertydeterminesfromthe0-skeletonofuifui=1orui∈{2}∪A.Thusweknowf(u).Nowassumeui∈{2}∪A,i.e.f(ui)=2.Theblock0|u|+3u0|u|+3iscontainedinaconcatenationofY-blocks,thusby(a)inoneY-block,say0|w|+1y(w)0|w|+1forsomeblockw∈X.Thus,bydefinitionofy(w),ui∈Aifandonlyifiisacoordinatecontainedinthe2k(n)-blockofsomeX2-blockC(n)inf(0|w|+1y(w)0|w|+1).Butthisisactuallydeterminedbyf(u),sincesuchaC(n)hastobeasubblockof0|u|+3u0|u|+3. NowthefundamentalpropertyimpliesthatanyautomorphismofTcanbecombinedwithauniquepoweroftheshiftsothattheresultingmapleavesthe{0,1,2}-skeletonofeverypointinTfixed. Proposition1.6.LetΦbeanautomorphismofT.LetσbetheshiftonT.Thenthereisani∈ZsuchthatΦ◦σiisaskeletonautomorphism,i.e.Φ◦σi(t)0=t0ifft0∈{0,1,2}andΦ◦σi(t)0∈Aifft0∈A,forallt∈T.Furthermore,anautomorphismofTisaskeletonautomorphismiffitfixesallperiodicpoints(02n1n)∞,n∈I. Proof.Theproofwillbegiveninaseriesofclaims.LetLbeacodinglengthforΦ,i.e.Φ(t)0isdeterminedbyt[−L,L],foreveryt∈T.Claim1.Φ(0)∞=0∞. Proof.NecessarilyΦ(0∞)=c∞forsomec∈{0,1,2}∪A.Assumec=0.Thereisapointt∈TwithΦ(t)=0∞.Thent[−L,L]occursinsomeblocky∈YbyLemma1.5(a).Since0∈YwehavethattN=0∞y0Ny0∞isapointinT.LetN>2L.Sinceyseest[−L,L],andLisacodinglengthforΦ,wehaveΦ(tN)=c∞u0ucN−2Lv0vc∞∈T,whereu,v,u,varefixedblocks,eachoflength≤|y|+2L,anduandvdonotseethesymbol0.ConsiderthenumbersNwith|ucN−2Lv|>max{|u|,|v|}.ChoosesuchanNwith|ucN−2Lv|∈I;thenthefundamentalpropertyofTimpliesc=1.ChoosesuchanNwith|ucN−2Lv|∈I,nowthefundamentalpropertyofTimpliesc=1,acontradiction.Thusc=0.NowlettNbedefinedbytN(−∞,0]=0∞,tN[1,∞)=1N0∞. Claim2.ForN∈IwehavetN∈T.Moreover,thereisauniquei∈ZsuchthatΦ◦σi(tN)[0,∞)beginswith01N0forallN∈IwithN>4L. Proof.FixN∈I.Takew∈X1suchthatc(w)=c(0N+11N0N+1)−1.Then,foreachk≥1, 0M+1C(1,e)0N+k1N0N+kwC(1,e)0M+1∈Y forM=|C(1,e)0N+11N0N+kwC(1,e)|,thustN∈T.WeknowΦ(1∞)=c∞forsomec∈{1,2}∪A(c=0byClaim1).SinceLisacodinglengthforΦ,forallN>4LwithN∈IwehaveΦ(tN)=0∞u0ucN−2Lv0v0∞,whereu,v,u,varefixedblocks,eachoflength≤2L,andu,vdonotseethesymbol0.Moreover,theblock0ucN−2Lv0startsatthesamecoordinateiforallN.ItremainstoshowucN−2Lv=1NforallN.Sincen=N−|ucN−2Lv|isafixednumberandI THEAUTOMORPHISMGROUPOFACODEDSYSTEM3179 containsarbitrarilylongintervals(thecomplementhasunboundedgaps),thereisanN>4LwithN∈Iand|ucN−2Lv|=N−n∈I.ForthisNthefundamentalpropertyofTand0∞u0ucN−2Lv0v0∞∈TimplythatucN−2Lv=1N−n.ThusucN−2Lv=1N−nforallN.Finallyn=0,sinceotherwisewecouldchooseN>4LwithN∈Iand|ucN−2Lv|=N−n∈I(Iisinfinitewithunboundedgaps);then0∞u0ucN−2Lv0v0∞∈Twouldviolatethefundamentalproperty.Claim3.ChooseiasinClaim2.ThenΨ=Φ◦σiisaskeletonautomorphism.Proof.LetLbeacodinglengthforΨ.Firstlett∈Twitht0=0.WeshowthatΨ(t)0=0.ThereisaY-blockw=w−r···w0···wssuchthatw−L···w0···wL=t[−L,L].ChooseanN>2L+|w|+4withN∈I.Foru:=w0020N−s−31N02Nwehavethatf(u)isstable,sincew0=t0=0,andneutral,sincec(u)=c(w)=e.Thus0MC(p,e)uC(p,e)0M∈YforplargeenoughandsuitableM,whichshowsthatuisaT-block.Solett∈Tbesuchthatt(−∞,0]endswithw−r···w0andt[0,∞)beginswithw0···ws0020N−s−31N02N.ThenΨ(t)[N,2N+1]=01N0byClaim2,andΨ(t)3N+1=0byClaim1.FromthefundamentalpropertyitfollowsthatΨ(t)0=0. Nowlett∈Twitht0=0.AsaboveitfollowsthatforsomeY-blockwcontainingt[−L,L]thereareN,K∈IwithN>2L+|w|+4and4L≤K Again,lett∈Twitht0=0.ThereisaY-blockw=w−r···w0···wssuchthatw−L···w0···wL=t[−L,L].Lett∈Twitht[−r−|w|−3,s+|w|+3]=0|w|+3w0|w|+3.Bytheabove,tandΨ(t)havethesame0-skeleton(i.e.,tn= 0iffΨ(t)n=0foralln).Thustheblockst[−r−|w|−3,s+|w|+3]andΨ(t)[−r−|w|−3,s+|w|+3]areofthetypeasinLemma1.5(c),andhavethesame0-skeleton;thusΨ(t)0belongstothesamesetof{1},{2},Aast0does.Since t0=t0andΨ(t)0=Ψ(t)0,thesameholdsfort0andΨ(t)0. ThecharacterizationofskeletonautomorphismsfollowsimmediatelyfromClaim3. SofarwehaveusedthefundamentalpropertyofT.NowweshallusethefollowingdependencystructurebetweenneighboringC(n,gn)-blocks.Lemma1.7.LetBn(gn)wBn(hn)beaT-block.Thenhn=gn·c(w)n.LetC(n+1,hn+1)C(n,hn)beaT-block.Thenhn=πn(hn+1). Proof.Bn(gn)wBn(hn)occursassubblockofablockfromY.Thelemmanowfol-lowsfromthedefinitionofthecodeY.ThesameappliestoC(n+1,hn+1)C(n,hn).Proposition1.8.LetΦbeaskeletonautomorphismofT.Ift,t∈Twitht[0,k(n)+1]=t[0,k(n)+1]=0Bn(gn)0forsomegn∈Gn,thenΦ(t)[1,k(n)]=Φ(t)[1,k(n)]=Bn(hn)forsomehn∈Gn. Proof.LetLbeacodinglengthforΦ.Thereareblocksy,yfromYsuchthatt[−L,k(n)+L]isasubblockofyandt[−L,k(n)+L]isasubblockofy.ThenforlargeenoughNandsuitableM,u=0M+1C(N,e)yyC(N,e)0M+1∈Y.Writey=aBn(gn)bandy=aBn(gn)b.ByLemma1.7,gn=gn·c(ba)n;thusc(ba)n=einGn.SinceΦisaskeletonautomorphism,wewillseesomeblockBn(hn) 3180DORISFIEBIGANDULF-RAINERFIEBIG inΦ(0∞u0∞)atthecoordinatesofBn(gn)iny,andsomeblockBn(hn)atthe ∞ coordinatesofBn(gn)iny.LetwdenotetheblockoccurringinΦ(0u0∞)atthesamecoordinatesastheblockbain0∞u0∞.Thenc(w)=c(ba)sinceΦisaskeletonautomorphism.Again,byLemma1.7,hn=hn·c(w)n=hn.Proposition1.9.LetΦbeaskeletonautomorphismofT.ThenΦinducesinaninjectivehomomorphicwayaB-continuouselementgΦ∈lim←(Gn,πn). Proof.Forgn∈Gnchoosew∈X1withc(w)n=gn.Letu∈X1withc(u)=c(w)−1.Then 0M+1C(n,e)wC(n,gn)uC(n,e)0M+1∈Y for M=|C(n,e)wC(n,gn)uC(n,e)|. ThusC(n,e)wC(n,gn)isaT-block.SinceΦisaskeletonautomorphism,by Proposition1.8ΦmapsthisblocktoC(n,hn)wC(n,hn)forsomehn,hn∈Gn.Thushn=hnc(w)n=hngnbyLemma1.7.ThusΦactsonGnasmultipli-cationfromtheleftbyhn.ForeachnwehavethatC(n+1,e)C(n,e)isaT-blockwhichismappedbyΦtoC(n+1,hn+1)C(n,hn).Thushn=πn(hn+1)byLemma1.7.ThereforegΦ:=(h1,h2,...)∈lim←(Gn,πn).TheB-actionofgΦisaslidingblockcodewithL=thecodinglengthofΦ.Analogously,Φ−1inducesgΦ−1=(gΦ)−1∈lim←(Gn,πn),whoseB-actionisaslidingblockcode.ThusgΦisB-continuous.ClearlythemapΦ→gΦisaninjectivehomomorphism. Proposition1.10.ThemapΦ→gΦdefinesanisomorphismbetweenthegroupofskeletonautomorphismsofTandthesubgroupofB-continuouselementsoflim←(Gn,πn). Proof.ItonlyremainstoshowthatΦ→gΦissurjective.Forthatletg=(g1,g2,...)∈lim←(Gn,πn)beB-continuous.LetDbethesetofpointsinTwhichseethesymbol0infinitelyofteninthepastandinthefuture.Firstde-fineΦgonthesetDasthemapwhichleavesthesymbols0,1,2fixedandmapseachBn(hn)toBn(gnhn).WeshowthatΦg(D)⊂D.ByLemma1.5(a)itsufficestoshowthatΦg(0∞y(w)0∞)∈Tforallw∈X.Sincey(w)isoftheformC(n1,e)u1···uk−1C(nk,e)withn1=nk,wehaveΦg(0∞y(w)0∞)=0∞C(n1,gn1)u1···C(nk−1,gnk−1)uk−1C(n1,gn1)0∞.Letu∈X1withc(u)n1=gn1andu∈X1withc(u)=c(u)−1.Thenforeachn≥|w|+1thereisM=M(n)suchthat0MC(n1,e)u0nC(n1,gn1)···C(n1,gn1)0nuC(n1,e)0M∈Y.Thus,Φg(0∞y(w)0∞)∈T.DefineΦg−1analogously.SincetheB-actionsofgandg−1areslidingblockcodes,ΦgandΦg−1arebothuniformlycontinuousonD.Thusbothmapscanbeuniquelyextendedtocontinuousshift-invariantmappingsdefinedonT,whichwecallagainΦgandΦg−1.Bydefinition,ΦgΦg−1=idonD,thusΦgΦg−1=idonT.ThesameappliestoΦg−1Φg,thusΦgisanautomor-phismofT.AndobviouslyΦg→gΦg=gunderthemapdescribedintheproofofProposition1.9. NowwecanfinishtheproofofTheorem1.4.LetHbethegroupofskeletonautomorphismsofT.Definej:Z⊕H→aut(T)byj((i,φ))=σiφ.Sinceσcommuteswithanyautomorphism,jisahomomorphism.ByProposition1.6anyautomorphismofThasapresentationasσiφwithi∈Z,φ∈H;thusjissurjective.Thispresentationisunique;thusjisinjective.Soj−1isanisomorphism THEAUTOMORPHISMGROUPOFACODEDSYSTEM3181 aut(T)→Z⊕Hwithj−1(σ)=(1,id).Thisshowsaut(T)≈σ⊕H.ByProposition1.10H≈{g∈lim←(Gn,πn)|gisB-continuous}. 2.Codedsystemswithpreassignedautomorphismgroups. Applications ForavarietyofgroupsGweobtaincodedsystemsTwithaut(T)≈σ⊕G.ThiswillbeanapplicationofTheorem1.4:ineachcaseweshowthatGhasacontinuousblockpresentation.Weshowthatthispropertyisclosedundertakingdirectsums.Moreover,agroupGhasacontinuousblockpresentationiffGisisomorphictoaclosedsubgroupofaut(R)forsomesubshiftRwithperiodicpointsdense.WeusethischaracterizationtoobtainacodedsystemTwithaut(T)≈σ⊕Z[1/2]. Recallthatwewriteaut(T)≈σ⊕Gifthereisanisomorphismfromaut(T)toZ⊕Gwhichmapstheshiftσtotheelement(1,e),whereeistheidentityelementinG. AutomorphismgroupsfornontrivialSFTsarenotfinitelygenerated[BLR,The-orem7.8].Incontrasttothat,theautomorphismgroupofacodedsystemcanbefinitelygenerated.Theorem2.1,Theorem2.4andTheorem2.5giveparticularexamplesforthis. Theorem2.1.AnyfinitegroupGhasacontinuousblockpresentation.Inpartic-ular,thereisacodedsystemTsuchthataut(T)≈σ⊕G. Proof.LetA=Gand(Gn,πn)=(G,id)foralln.Letk(n)=n+1andletBn(g)=gg···gbetheblockofn+1repetitionsofthesymbolg.ThenG=lim←(Gn,πn)andtheB-actionofanyg∈Gisgivenbyasliding1-blockcode(i.e.L=0);thusallelementsinlim←(Gn,πn)areB-continuous.ApplyTheorem1.4.Corollary2.2.ThereisacodedsystemTwithaut(T)=σ. ItisnotknownwhethertherearetwoSFT’sSandTwithisomorphicautomor-phismgroupsbutsuchthatneitherSandTnorSandT−1areconjugate[BLR,Question4.1].Forcodedsystemsthiscanbethecase. Corollary2.3.TherearecodedsystemsTandTwhichhavedistinctzetafunc-tionsbutisomorphicautomorphismgroups. Proof.LetTbegivenasinTheorem2.1withG={e},thetrivial1-elementgroup.ObservethatbyconstructionthefixedpointsofTare0∞,1∞,2∞,e∞.TogetTwechooseanothercontinuousblockpresentationforG={e}.Asbeforelet(Gn,πn)=(G,id)andk(n)=n+1foralln.ButnowchooseA={a,b},andforevennletBn(e)=an+1(theblockofn+1repetitionsofthesymbola),foroddn,Bn(e)=bn+1.ThenG≈thegroupofB-continuouselementsinlim←(Gn,πn),sincetheonlyelementoflim←(Gn,πn),theidentity,isofcourseB-continuous.NowletTbethesystemobtainedfromthiscontinuousblockpresentationofG(Theorem1.4).ObservethatThasfivefixpoints:0∞,1∞,2∞,a∞,b∞.ThusT,Thavedifferentzetafunctionsbutaut(T)≈Z≈aut(T). Theorem2.4.ThegroupZhasacontinuousblockpresentation.Inparticular,thereisacodedsystemTsuchthataut(T)≈σ⊕Z. Proof.ThistheoremfollowsalsofromCorollary2.2incombinationwithTheo-rem2.12.Buthereweprefertogiveamoredirectconstruction.Foreachn≥2letGn=Z/2nZandletπn:Gn+1→Gnbethenaturalprojection.WithHn=2nZ 3182DORISFIEBIGANDULF-RAINERFIEBIG wehaveGn={iHn|0≤i<2n}.LetA={(0,0),(0,1),(1,0)},i.e.thesymbolsofouralphabetare2-tuples.Letk(n)=2n−1andfor0≤i<2nletBn(iHn)betheblock(x0,x2n−1)(x1,x2n−2)(x2,x2n−3)···(x2n−1−1,x2n−1)∈Ak(n)withxi=1andxj=0forallj=i.WeclaimthatthegroupofB-continuouselementsinlim←(Gn,πn)isisomorphictoZandthusthesystemTexistsbyTheorem1.4.Toverifytheclaimwewillshowthatg=(i2H2,i3H3,...)∈lim←(Gn,πn)isB-continuousiffthereisanm∈Zsuchthatin=mmod2nforalln.Ifgisofthisform,thengn=inHnmapsBn(iHn)toBn((i+m)mod2nHn)foralln,i.e.xi=1becomesxi+mmod2n=1,amapwhichobviouslycanbedescribedbyaslidingblock-codewithcodelengthL=|m|.Andwithgalsog−1isoftheaboveformwith−m,thusgisB-continuous.Ontheotherhand,ifgisB-continuous,thentheB-actionofgisaslidingblockcode,sayofcodinglengthL.Letn>100Landconsidertheactionofgn=inHnontheblockBn(2n−2Hn)withthesymbol(1,0)inthemiddle.Theimageofthisblockhastobeablockwhichseesatsomeoffsetmwith−L≤m≤Lfromthepositionofthissymbolthesymbol(1,0)or(0,1),wheremisindependentofnsinceweconsideraslidingblockcode.Letkbetheindexsuchthatxk=1intheimage-block;thenin=(k−2n−2)mod2n.Ifthesymbolintheimageblockwouldbe(0,1),theninHnwouldmaptheblockBn((2n−2+1)Hn)toBn((k−1)Hn)(sinceagain(0,1)intheimagewouldoccuratthesameoffsetfromtheoriginal(1,0)-symbol)andin=(k−2−2n−2)mod2nwouldcontradicttheabovevalueofin.Thustheimagesymbolhastobe(1,0).Thusin=mmod2nforalln>100L,andthus,bythestructureoftheinverselimit,foralln. Theorem2.5.ThereisacodedsystemTwithaut(T)≈σ⊕G,whereG={a,b;a2=b2=e}isthefreeproductoftwocopiesofZ/2Z. Proof.Theconstructionisquitesimilartothelastone.Foreachn≥2letHn n−1 bethesubgroupofGgeneratedbytheelement(ab)2.ThenHnisnormal;letGn=G/Hn.TheelementsofGncanbepresentedasbj(ab)kHn,0≤k≤2n−1−1,j∈{0,1}.LetA={(3,3),(3,0),(0,3),(3,1),(1,3)}.Letk(n)=2n−1andletBn(bj(ab)k·Hn)betheblock(x0,x2n−1)(x1,x2n−2)(x2,x2n−3)···(x2n−1−1,x2n−1)∈Ak(n)withxi=3foralli=2k+jandx2k+j=j. WeclaimthatthesubgroupofB-continuouselementsinlim←(Gn,πn)isisomor-phictoG;thenwearedonebyTheorem1.4.Toprovethisclaim,weembedGintolim←(Gn,πn)byamapthatactsonthegeneratorsasb→gb=(bH2,bH3,...)anda→ga=(aH2,aH3,...).LetGbetheisomorphicimageofGunderthisembedding.Weshowthatgbandga,andthusallelementsofG,areB-continuous.SincebHnmapsBn(bj(ab)k·Hn)toBn(bj+1mod2(ab)k·Hn),itsactionontheseblocksisapermutationwhichexchangestheblockswithx2k=3andx2k+1=3forall0≤k≤2n−1−1.Sincex2k=0ifitisnot3,andx2k+1=1ifitisnot3,thispermutationcanbedescribedforallnbyaslidingblockcodeofcod-inglengthL=1.ThesameholdsfortheactionofaHnwhichexchangestheblockswithx2k+1=3andx2k+2mod2n=3forall0≤k≤2n−1−1.Nowletg=(h2H2,h3H3,...)beaB-continuouselementoflim←(Gn,πn).SayitsB-actionhascodinglengthL.AsintheproofofTheorem2.4,letn>100LandconsidertheactionofhnHnontheblockBn((ab)k·Hn)wherek=2n−2,i.e.theblockwiththesymbol(0,3)inthemiddle.Iftheimageblockwouldseeasymbol(3,0)or(3,1),thiswouldcontradictthefactthathnHnactsasleft-multiplication(asintheproofofTheorem2.4).Thustheimageblockseesasymbol(0,3)or(1,3)withinanoffset THEAUTOMORPHISMGROUPOFACODEDSYSTEM3183 of−L≤m≤Lfromthepositionof(0,3)inBn((ab)k·Hn).Sinceweconsiderablockcode,thisoffsetandtheimagesymbolarethesameforalln>100L.Thusthereisaproductg=g(1)g(2)···g(j)withj≤Lofgeneratorsg(i)from{gb,ga}suchthat,writingg=(g2,g3,...)∈lim←(Gn,πn),wehavethatgn=hnHnforalln>100L.Bythestructureofaninverselimitthisholdsalsoforn≤100L;thusg=g=g(1)···g(j)∈G. ItremainsanopenproblemwhetherthereisacodedsystemTwithaut(T)≈σ⊕thefreeproductofthreecopiesofZ/2Z. Ourmethodofconstructingautomorphismgroupsisclosedundertakingdirectsums. Theorem2.6.Foreachn∈NletHnbeacountablegroupwhichhasacontinuousblockpresentation.LetG=Hn.ThenGhasacontinuousblockpresentation.Inparticular,thereisacodedsystemTsuchthataut(T)≈σ⊕G. Proof.FirstobservethatanyblockpresentationwithalphabetAofsomeinverselimitcanbereplacedbyonewiththealphabet{3,4}andthesamecontinuouselements—simplyreplaceanysymbolofAbyoneoftheblocks3i4n−i,1≤i≤card(A),n=card(A)+1. Thus,foreachn∈Nthereisaninverselimit←lim−k(Hn,k,πn,k)offinitegroupsHn,kandhomomorphismsπn,k:Hn,k+1→Hn,k,andablockpresentationBnwithalphabet{3,4}andblocksBn,k(h),h∈Hn,k,suchthatthegroupofBn-continuouselementsin←lim−k(Hn,k,πn,k)isisomorphictoHn.Inthefollowingletp=3andq=5. ForeachnwemodifytheblockpresentationBninthatineachBn,k(h)wereplacea3bytheblock5i35ianda4bytheblock5i45i,wherei=pn.WecalltheobtainedblockagainBn,k(h)andthepresentationBn.Observethatthis“stretching”oftheblocksdidnotchangethegroupofcontinuouselementsinlim←−k(Hn,k,πn,k). DefineGn=H1,n⊕H2,n⊕H3,n⊕···⊕Hn,n.Letπn:Gn+1→Gnbethemapwhichdropsthelastcoordinateandthenappliesthesuitableπn,k’s,i.e.πn(u1,u2,...,un+1)=(π1,n(u1),π2,n(u2),...,πn,n(un)). WedefineablockpresentationBforlim←(Gn,πn)withalphabet{∗,3,4,5}.Letu(n)=∗5···535···5∗5···545···5wherethestringsof5’shavelengthqn.LetBn(gn)=u(n)∗B1,n(h1)∗B2,n(h2)∗···∗Bn,n(hn)∗foralln,gn=(h1,...,hn)∈Gn.ItremainstoshowthatthegroupofB-continuouselementsinlim←(Gn,πn)isisomorphictoHn. Forhn∈Hnletα(hn)denotetheelement(g1,g2,...)inlim←(Gn,πn)wherethenthcoordinateofeachgiwithi≥nequalshnandalltheothersaretheidentity element.Considerform≥nablockBm(gm)=u(m)∗B1,m(h1)∗B2,m(h2)∗···∗ Bm,m(hm)∗,gm=(h1,...,hn)∈Gn.ThenundertheB-actionofα(hn)this blockgetsmappedtoablockwherejustBn,m(hn)isreplacedbyBn,m(hnhn)and 2i allothercoordinatesremainfixed.SinceonlyBn,m(hn)containsblocksa5b,withi=pnanda,b∈{3,4},itcanberecognizedbyaslidingblockcode,andthustheblockcodedescribingtheBn-actionofhncanbeextendedtoablockcoden).ObservethatαextendstoanembeddingofdescribingtheB-actionofα(h Hnintolim←(Gn,πn).Thusα(Hn)iscontainedintheB-continuouselementsoflim←(Gn,πn). Nowletg=(g1,g2,...)∈lim←(Gn,πn)beB-continuous.LetLbeacodinglengthoftheblockcodeφdescribingtheB-actionofg.Considernwithqn>L. 3184DORISFIEBIGANDULF-RAINERFIEBIG BygntheblockBn(e)getsmappedtoBn(gn).ButBn(e)andBn(gn)bothbeginwithu(n)=∗5···535···5∗5···545···5.Thustheblockcodeφfixesalltheblocks5···535···5and5···545···5,wherethestringsof5’sexceedacertainlength.ThisimpliesthatφfixesalltheblocksBn,k(h),whenevernislargeenough.ItfollowsthatthereisanNsuchthatforalln≥Ngn=(h1,h2,...,hN,e,e,...,e),i.e. g=α(h1,h2,...,hN)andthusisanelementofα(Hn).ThelaststatementfollowsbyTheorem1.4. Corollary2.7.Foranyinfinite,finitelygeneratedabeliangroupHthereisacodedsystemTwithaut(T)≈H. Proof.AnysuchHcanbewrittenasZ⊕GwithGadirectsumofcopiesofZandfinitecyclicgroups.ByTheorem2.1,Theorem2.4,andTheorem2.6thereisacodedsystemTwithaut(T)≈σ⊕G. Corollary2.8.IfGisasubgroupofQ/ZthenthereisacodedsystemTwithaut(T)≈σ⊕GiffGisresiduallyfinite(whichholdsifitsp-torsionsubgroupisfiniteforeveryprime). Proof.Ifaut(T)≈σ⊕GforacodedsystemT,thenG≈{e}⊕Ghastoberesiduallyfinite.GinQ/ZisOntheotherhand,eachresiduallyfinitesubgroupisomorphictoHpwitheachHpafinitesubgroupofZ(p∞),pprime(seetheproofofProposition3.6in[BLR]).ApplyTheorem2.1andTheorem2.6.SincethereareonlycountablemanySFTs,thereareonlycountablemanyauto-morphismgroupsofSFTs. Corollary2.9.Thereareuncountablymanynonisomorphicautomorphismgroupsforcodedsystems. Proof.LetJ⊂Nbeasetofprimes.ThenthereisacodedsystemT(J)withaut(T(J))≈σ⊕G(J),whereG(J)isthedirectsumofallcyclicgroupsoforderpwithp∈J,byTheorem2.1andTheorem2.6.DistinctsetsJyieldnonisomorphicgroupsG(J),andthereareuncountablymanysuchsetsJ. Aself-containedargumentshowsthatshiftswithperiodicpointsdenseandsyn-chronizedsystemshavethesameautomorphismssubgroups. Theorem2.10.LetRbeasubshiftwithperiodicpointsdense.ThenthereisasynchronizedsystemSsuchthataut(S)containsacopyofaut(R). Proof.WehaveR⊂AZforsomefinitealphabetA.Foraperiodicpointx∈Rofleastperiod,sayk,defineB(x)=(x1,xk)(x2,xk−1)···(xk,x1)∈(A×A)k.LetBR={B(x)|xisaperiodicpointinR}.Let∗beasymbolnotinA×A.LetSbethecodedsystemgivenbythecodeY={∗B(x)|xisaperiodicpointinR}.Then∗isasynchronizingsymbolforS.AnyautomorphismΦofRinducesabijectionπΦ:BR→BRbyB(x)→B(Φ(x)).SinceΦiscontinuousand|πΦ(B(x))|=|B(x)|andbecauseofthespecialstructureoftheB(x)-blocks,πΦextendstoanautomorphismofSwhichleavesthesymbol∗fixed,alsodenotedπΦ.Thenα:Φ∈aut(R)→πΦ∈aut(S)isagrouphomomorphism,whichisinjective,sinceα(Φ)=idimpliesthatΦ=idonperiodicpoints,whicharedenseinR,thusΦ=id. Definition2.11.LetRbeasubshift.ForanautomorphismΦ∈aut(R)letαk(Φ)denotetherestrictionofΦtotheperiodicpointswithleastperiod≤k.Asubgroup THEAUTOMORPHISMGROUPOFACODEDSYSTEM3185 G⊂aut(R)isclosedifwheneverΦn∈G,n∈R,Φn→Φ∈aut(R)inperiodicpointtopology(i.e.foreachkthereisannksuchthatαk(Φn)=αk(Φ)foralln≥nk),thenΦ∈G. Theorem2.12.AnabstractgroupGhasacontinuousblockpresentationiffGisisomorphictoaclosedsubgroupofanautomorphismgroupofsomesubshiftRwithperiodicpointsdense. ThusanapplicationofTheorem1.4yieldsimmediately Theorem2.13.LetRbeasubshiftwithperiodicpointsdense.LetG⊂aut(R)beaclosedsubgroup.ThenthereisacodedsystemTsuchthataut(T)≈σ⊕G.Inparticular,thereisacodedsystemwithaut(T)≈σ⊕aut(R). ProofofTheorem2.12.Agroupwithacontinuousblockpresentationisisomor-phictoaclosedsubgroupoftheautomorphismgroupofacodedsystembyPropo-sitions1.6and1.10. NowassumethatGisaclosedsubgroup.LetPn(R)denotethesetofperiodicpointsinRwithperiodicatmostn.Foranyautomorphismg∈Gletαn(g)denotetherestrictionofgtothepointsinPn(R).LetGn={αn(g)|g∈G}bethegroupofbijectionsonPn(R)obtainedinthisway.SincePn+1(R)⊃Pn(R),therearenaturalgrouphomomorphismsπn:Gn+1→Gn.PeriodicpointsaredenseinR;thusforanyautomorphismg=idwehaveαn(g)=idforsomen.Soα(g):=(α1(g),α2(g),...)definesanembeddingofGintolim←(Gn,πn).WewilldefineablockpresentationBoflim←(Gn,πn)suchthatα(G)isthegroupofB-continuouselementsoflim←(Gn,πn). WehaveR⊂(A)ZforsomeinfinitealphabetA.ThealphabetAfortheblockpresentationBwillbeA:={∗}∪A∪A×A,where∗isanewsym-bol.Forx∈Pn(R)defineB(x)=B(x)x[1,n!]B(x)∈A3n!whereB(x)=(x1,xn!)(x2,xn!−1)···(xn!,x1)∈(A×A)n!.FixanenumerationB(n,1),B(n,2),...,B(n,s)oftheblocksB(x),x∈Pn(R).Anyelementgn∈Gninducesaper-mutationoftheseblocks,andthusof{1,2,...,s},sayτgn.Werepresentgn∈GnbyBn(gn):=∗B(n,τgn(1))∗B(n,τgn(2))∗···∗B(n,τgn(s))∗. Nowletg∈G,andletLbeacodinglengthforg.Then,bydefinition,theB-actionofα(g)isthat,foralln,αn(g)mapsBn(hn)toBn(αn(g)hn).ButthiscanbedescribedbyaslidingblockcodewiththesamecodinglengthLsincethesymbol∗,whichremainsfixed,separatesthedifferentB(n,τgn(i))-blocksandbecauseofthespecialstructureoftheB(x)-blocksattheleftandrightendofaB(n,τgn(i))-block.Thesameholdsforg−1;thusα(g)isB-continuous. Nowleth=(h1,h2,...)∈lim←(Gn,πn)beB-continuous.Thus,forsomeL≥0thereisamapφ:(A∩{ω})2L+1→Asuchthatforalln∈Z,gn∈Gnwe (φ) haveBn(hngn)=Φn(Bn(gn)).DefineashiftcommutingmapΨ:R→AZbyΨ(t)0=φ(t−L,...,tL).Letxbeaperiodicpoint.Choosensuchthatn!≥2L+nandx∈Pn(R).ThenalsoΨ(x)isaperiodicpointinAZwithperiodatmostn. (φ) WeshowthatΨ(x)∈R.Now,ΦnmapsBn(e)=∗B(n,1)∗B(n,2)∗···∗B(n,s)∗toBn(hn)=∗B(n,τhn(1))∗B(n,τhn(2))∗···∗B(n,τhn(s))∗.Theblockx[1,n!]occursasthemiddlepartofB(n,i),say.Lety∈Pn(R)suchthatthemiddlepartofB(n,τhn(i))equalsy[1,n!].ThusΨ(x)[1+L,n!−L]=y[1+L,n!−L].ThelengthoftheseblocksislargerthantheperiodsofΨ(x)andy,thusΨ(x)=y.ThisprovesΨ:R→R,sinceperiodicpointsaredenseinR.Thesameargumentsappliedtoh−1leadtoanendomorphismΨ1:R→R.Consideringagainthesetof 3186DORISFIEBIGANDULF-RAINERFIEBIG periodicpointsshowsthatΨΨ1=Ψ1Ψ=idonperiodicpoints,whicharedense;thusΨ∈aut(R).Observethatα(Ψ)=hbydefinitionofΨ.ThusitonlyremainstoshowthatΨ∈G.BydefinitionofGn,foreachnthereisag(n)∈Gwithαn(g(n))=hn.Thusg(n)actsontheperiodicpointswithperiodlessthanorequaltoninthesamewayasΨ.Thus,Ψ=limg(n)onperiodicpoints.SinceGisaclosedsubgroup,thisimpliesΨ∈G. Asanapplication,fromTheorem2.13wegetacodedsystemwhichhasZ[1/2]asasubgroupofitsautomorphismgroup.Inparticular,thisshowsthatanau-tomorphismofacodedsystemofinfiniteordercanhaventhrootsforinfinitelymanyn.ItisnotknownwhetheranSFTcanhavesuchanautomorphism([BLR,Problem3.5]). Theorem2.14.(a)ThereisasubshiftRwithperiodicpointsdensewithaut(R)≈ n+2 Z[1/2]⊕C,whereCisthedirectsumofthefinitecyclicgroupsoforders2(2)−1,n≥1,andZ[1/2]correspondstoaclosedsubgroupofaut(R).(b)ThereisacodedsystemTwithaut(T)≈σ⊕Z[1/2]. (c)ThereisasynchronizedsystemSsuchthatZ[1/2]isasubgroupofaut(S).Remark.Astraightforwardmodificationofthefollowngproof(whichislefttothereader)showsthatforanyprimepthereisacodedsystemTwithaut(T)≈σ⊕Z[1/p].Fornotationalconveniencewegivetheproofonlyforp=2.Proof.(a)implies(b)and(c)viaTheorem2.13andTheorem2.10,respectively.Weshallprove(a).Thefollowingconstructionwasinspiredby[BLR,Example3.9]. LetA={0,a,b}.WestartbydefiningasubshiftR⊂AZastheclosureofcertainorbitsB(n)∞forcertainblocksB(n),n∈N.ForthatwebegintodefineinductivelyafamilyofblocksB(n,k),1≤k≤2n,n≥1.LetB(1,1)=a,B(1,2)=bandL(1)=1=|B(1,1)|.Thenforn≥1 B(n+1,2i−1):=B(n,i)0L(n)B(n,i), B(n+1,2i):=B(n,i)0L(n)B(n,i+1),B(n+1,2n+1):=B(n,2n)0L(n)−1B(n,1)0, L(n+1):=|B(n+1,1)|. ThenbyinductionL(n)=3n−1=|B(n,k)|,1≤k≤2n.Let B(n):=B(n,1)0kB(n,2)0k···B(n,2n−1)0kB(n,2n)0k−1, wherek=k(n)≥|B(n,1)|ischosen(uniquelymod2f(n)−1)suchthat|B(n)|=2f(n)−1,f(n)=2n+2,n≥1.LetRbetheclosureoftheunionofallorbitsB(n)∞.LetσbetheshiftonR.Byinductiononnitiseasytoseethat•eachB(n,k)beginswithasymbol=0,•eachB(n,1)endswiththesymbol“a”, •foreachB(n,k)oneofthelasttwosymbolsis=0,•theblocksB(n,k),1≤k≤2n,arepairwisedistinct. Thus,inparticularnoblockB(n,k)contains0L(n)−1asasubblock. Forn≥1wedefineamapψ(n)ontheunionofallorbitsB(m)∞,m∈N,asfollows: Ifx∈B(k)∞forsome1≤k 1≤i<2n, THEAUTOMORPHISMGROUPOFACODEDSYSTEM3187 m≥n,wedefineψ(n)byablockexchangerule:replaceB(n,i)byB(n,i+1)for1≤i<2nand0B(n,2n)byB(n,1)0,andletnootheractiontakeplace.Thisgivesawell-definedmaponB(m)∞-orbits,sinceanytwosuchblocksinB(m)∞areondisjointindexsetsbyconstruction.Thenwehavethatψ(n)(x)iisdeterminedbyx[−2|B(n)|+i,i+2|B(n)|]forallpointsxofsomeB(m)∞-orbitandalli∈Z,andthusextendstoacontinuousmaponRcommutingwiththeshift. Wenowshowthatψ(n)mapseachB(m)∞intoitself.Thenψ(n)extendstoanendomorphismR→R,whichwecallagainψ(n).Claim1.Foralln≥0,m≥0,andx∈B(n+m)∞ ψ(n)(x)=σα(n+m,n)(x) whereα(k,n)=2f(k)−n. Proof.Inductiononm. Form=0andarbitraryn≥0,x∈B(n)∞wehaveψ(n)(x)=σα(n,n)(x)byinspection.Nowassumetheclaimholdsforallm≤m+1,n≥0,x∈B(n+m)∞.UsingthedefinitionoftheblocksB(n+1,k)andthedefinitionofthemapψ(n)showsthatψ(n)mapstheblock0B(n+1,i)to0B(n+1,i+2)for1≤i<2n+1−1,theblock0B(n+1,2n+1−1)toB(n+1,1)0,and0B(n+1,2n+1)toB(n+1,2)0.Thusψ(n)actsontheblocks0B(n+1,k)asψ(n+1)2does.Nowletx∈B(n+m+1)∞.Thenψ(n)(x)=ψ(n+1)2(x)=σ2α(n+1+m,n+1)(x)byinductionhypothesis,andbythedefinitionofαthisequalsσα(n+1+m,n)(x).Thusforallnwehavethatψ(n):R→Risontoandcontinuous.Claim2.ψ(n+1)2=ψ(n)foralln,andψ(1)2=σ. Proof.Forx∈B(n+1+k)∞forsomek≥0wehaveψ(n+1)2(x)=ψ(n)(x)fromthelastclaim.IfxliesintheorbitB(k)∞forsome1≤k Thusψ(n)2=σforalln,whichprovesthateachψ(n)isanautomorphismofR.Thisprovesthatσandtheautomorphismsψ(n),n≥1,generateasubgroupGofaut(R)whichisisomorphictoZ[1/2].FinallywehavetoshowthatthesubgroupGisclosed. Claim3.TheperiodicorbitsofRare0∞andtheorbitsB(n)∞,andnoothers.Proof.Ofcourse,0∞∈R.Letx∈RbeperiodicwithperiodMandx0=0.Letm=|B(1)|·|B(2)|·····|B(M)|.SinceRistheclosureoftheperiodicorbitsB(n)∞,theblockx[0,mM]occursasasubblockofsomeB(n)∞.Choosetheleastsuchn.Assumen>M.Bytheminimalityofn,x[0,mM]isnotcontainedinaB(n−1,j)-block.Thusx[0,mM]contains0L(n−1)or0k(n).Butk(n),L(n−1)≥n−1≥M,contradictingthefactthatxdoesnotsee0M.Thusn≤M,andthereforemisamultipleof|B(n)|,whichprovesx∈B(n)∞.Thisprovestheclaim. 3188DORISFIEBIGANDULF-RAINERFIEBIG Claim4.ForanyΦ∈aut(R)thereissomeψ∈GsuchthatΦ=ψonallbutfinitelymanyperiodicorbits. Proof.LetΦ∈aut(R)withcodinglength,say,L.ThenΦ(0)∞=0∞,sinceRhasonlyonefixedpoint.IfxliesintheorbitB(n)∞thensodoesΦ(x)∞,sincethereisonlyoneorbitwithleastperiod|B(n)|.Choosensolargethatk(n)−2L>|B(n,1)|,wherek(n)isfromthedefinitionoftheblockB(n).WehaveΦ(x)=σM(x)forallxintheorbitB(n)∞forsomeM.WewanttoshowthatM=j·2f(n)−n+mforsome−L≤m≤Land0≤j<2n.ForthatconsiderB(n)=B(n,1)0kB(n,2)0k···B(n,2n−1)0kB(n,2n)0k−1.SinceΦ(0)∞=0∞andLisacodinglength,Φmaps0kB(n)toablockoftheform 0k−LC(n,1)0k−2LC(n,2)0k−2LC(n,3)···C(n,2n)0k−1−L where|C(n,i)|=|B(n,i)|+2L.SinceΦactsonxassomepoweroftheshift,andsincek−2L>|B(n,1)|,wehavethattheblockC(n,1)containsexactlyoneoftheblocksB(n,i).ThisdeterminestheaboveformofM.ThusΦ=ψ(n)j◦σmontheorbitB(n)∞.ButnowconsiderB(n+1).Wehave B(n+1)=B(n+1,1)0kB(n+1,2)0k···B(n+1,2n−1)0kB(n+1,2n)0k−1,where B(n+1,1)=B(n,1)0pB(n,1), p=|B(n,1)|≥n. ThusΦmaps0LB(n+1,1)0Ltoσm(B(n,j)0pB(n,j)).Fromthisitfollowsthat Φ(x)=σM(x)forallxintheorbitB(n+1)∞,whereM=2j·2f(n+1)−n−1+m.ThusΦ=ψ(n+1)2j◦σm=ψ(n)j◦σmontheorbitB(n+1)∞.InductivelyitfollowsthatΦ=ψ(n)j◦σmonalltheorbitsB(n+k)∞,withk≥0. Claim5.aut(R)≈Z[1/2]⊕C,whereCisthedirectsumofthefinitecyclicgroups n+2 oforders|B(n)|=2(2)−1,n≥1.Proof.ThisfollowsfromClaim3andClaim4Claim6.G≈Z[1/2]isaclosedsubgroupofaut(R). Proof.Forn∈Nandf(n)=2n+2letGn=Z/(2f(n)−1).Since(2f(n+1)−1)=(2f(n)−1)(2f(n)+1),theordersofthegroupsdivideeachotherandthusthemapsπn:Gn+1→Gndefinedbyπn(a)=amod2f(n)−1areontogrouphomomorphisms. LetGbethegroupgeneratedbyσandtheψ(n),n≥1.Form≥nletα(m,n)=2f(m)−n.Observethat πm(α(m+1,n))=2f(m+1)−nmod2f(m)−1=2f(m)+f(m)−nmod2f(m)−1 =2f(m)2f(m)−nmod2f(m)−1=α(m,n)mod2f(m)−1. Form ByClaim4wemayassumethatΦistheidentityonallorbitsB(n)∞,n≥N,forsomeN.ChooseksolargethatΦandgkcoincideonallorbitsB(n)∞,n≤N.Inparticular,gkistheidentityonB(N)∞.ThusαN(gk)=0,sinceαNdescribestheshiftofgkonB(N)∞.Thusαn(gk)=0foralln≤N,andsoΦ=gkistheidentityonallB(n)∞,n≤N.SoΦistheidentityonallorbitsB(n)∞,n∈N,whicharedenseinR;thusΦistheidentityonR,andsoΦ∈G.ThisprovesthatGisaclosedsubgroupofaut(R). Claims5and6provepart(a)ofthetheorem.ThefollowinglemmashowsinparticularthatanautomorphismofanSFTwhichhasaninfinitechainofpthrootscannotbetopologicallyconjugatetoacodedsystem(whichgeneralizesaremarkin[BLR,p.79]). Lemma2.15.LetSbeanonfinitesubshift,i=0,andpprime.Assumethatthereareφ(n)∈aut(S),n≥0,suchthatφ(0)=σi(whereσistheshiftonS),andφ(n)=φ(n+1)pforalln≥0.ThenSisnotcoded. Proof.AssumethatSiscoded.Wewanttousethefollowingobservation:ifg(n),n≥0,isasequenceofelementsinsomefinitegroupGwithg(n)=g(n+1)pforalln≥0,thentheorderofg(0)isnotdivisiblebyp.(Fixn,m>0suchthatg(n)= nnmm g(n+m).Theng(m)=g(n+m)p=g(n)p=g(0);thusg(0)p=g(m)p=g(0), m andsog(0)p−1=id.) Thusallwehavetoshowtoderiveacontradictionisthatforsomek∈NthesetPpki(S)ofS-pointswithleastperiodpkiisnonempty,sincethentherestrictionsg(n)oftheautomorphismsφ(n)toPpki(S)areelementsofthesymmetricgrouponPpki(S)andsatisfyg(n)=g(n+1)pforalln≥0,buttheorderofg(0)=(σirestrictedtoPpki(S))wouldbepkandthusdivisiblebyp. LetXbeacodeforS.IfforanyfiniteconcatenationxofblocksfromXtheorbitx∞wouldbethesame,thenSwouldcoincidewiththisorbit,thuswouldbefinite.SothereareconcatenationsofX-blocks,sayxandy,suchthatx∞,y∞aredifferentorbits.Nowreplacexbyx|x|andybyy|x|toobtainblocksofthesamelength.Then,foralln>3,(xyn)∞isaperiodicorbitwithleastperiod(n+1)|x|.Thuswemaychoosensuchthat(n+1)/iisamultipleofp;thus(n+1)|x|=pkiforsomek. Theorem2.16.ThereisasubshiftRwithperiodicpointsdensesuchthataut(R)isnotisomorphictotheautomorphismgroupofacodedsystem. Remark.Itremainsanopenproblemwhetheratransitivesubshiftwithperiodicpointsdensecanhaveanautomorphismgroupwhichisnotthatofacodedsystem.Proof.ByTheorem2.14(a)thereisasubshiftRwithperiodicpointsdenseandaut(R)=Z[1/2]⊕C,whereCisthedirectsumofthefinitecyclicgroups.LetTbeasubshiftsuchthataut(T)isisomorphictoZ[1/2]⊕C.Letg∈Z[1/2]⊕CbetheimageoftheshiftσofT.Thenthereisani>0suchthatgi=(h,e),whereeistheidentityelementinC.Thereisasequenceofsquarerootsh(n)∈Z[1/2]withh(0)=handh(n)=h(n+1)2foralln≥0.Thusthepreimagesφ(n)∈aut(T)of(h(n),e),n∈N,satisfyφ(0)=σiandφ(n)=φ(n+1)2foralln≥0.ThenTisnotcodedbyLemma2.15. Theorem2.4in[BLR]showsthatthefreeproductoffinitelymanycopiesofZ/2ZisasubgroupoftheautomorphismgroupofanySFT.Thisgeneralizesto 3190DORISFIEBIGANDULF-RAINERFIEBIG Theorem2.17.TheautomorphismgroupofanynonfinitesynchronizedsystemScontainsacopyofthefreeproductofallfinitegroups. Proof.Fixanenumerationofallfinitegroups,sayH1,H2,....LetendenotetheidentityelementofHn.LetGdenotethefreeproductofH1,H2,....LetAbethedisjointunionofthesetsHn−{en}.ElementsinGaretheidentityeorcanbeuniquelypresentedinreducedform,i.e.byfiniteblocksw1···wkwithsymbolsinAsuchthatwi∈Hnimplieswi+1∈Hn. LetmbeasynchronizingblocksuchthatmmisanS-block.ThesetofallS-blocksoftheformmwm,wherewisablock,isacodeforS.NowcopytheargumentfromtheproofofLemma2.15toseethatthereareblocksu=mamandv=mbmsuchthattheorbitsu∞andv∞aredistinctandofthesamelength|u|.Foreachnfixaninjectivemap Hn−{en}→{v2uvn+3u|Hn|−i+1vui+1v2|1≤i≤|Hn|−1}. Wedenotetheimageofh∈Hn−{en}byB(n,h).LetB(n,en)=vL(n)/|v|.LetL(n)=|B(n,h)|,ifh∈Hn.ObservethateachB(n,h)isanS-blockandthatdistinctblocksB(n,h),B(m,g)canoverlapatmost2|v|. Foreacha∈HnwedefineamapΨ(a)onSasfollows:LetCn={C|CisanS-blockoflengthL(n)andbeginswithv2uv,C=B(n,h)forallh∈Hn−{en}}.ThenΨ(a)istheblockexchangemapwhichforeachC∈Cn,h∈HnreplacesB(n,en)B(n,h)CbyB(n,en)B(n,ah)Candletsnootheractiontakeplace.Ψ(a)iswelldefined,sincetheblocksinthedefinitionofΨ(a)canoverlapatmostL(n)−2|v|andthereplacementschangeonlythemiddlepartoflength≤L(n).SinceEandallblocksB(n,h)beginandendwiththesynchronizingblockv,Ψ(a)isanendomorphismofS.ClearlyΨ(a)◦Ψ(a−1)=id=Ψ(a−1)◦Ψ(a).ThusΨ(a)∈aut(S). Fora,b∈HnwehaveΨ(a)◦Ψ(b)=Ψ(ab)byinspection.NowletGbethesubgroupofaut(S)generatedbyΨ(a),a∈A.Letw1···wk∈Gbeinreducedform.ThemapΨ:w1···wk→Ψ(w1)◦···◦Ψ(wk)∈Gdefinesagroupho-momorphismG→G,whichisontobydefinitionofG.Nowletw1···wk=einGbegiveninreducedformandletnibesuchthatwi∈Hni−{e}.ThenΨ(w1)◦···◦Ψ(wk)(v∞uv∞)=v∞B(n1,w1)···B(nk,wk)v2uv∞=v∞uv∞.ThusΨ(w1···wk)=id,i.e.Ψisinjective.ThusGisisomorphictoG. 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