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INTERMEDIATE β-SHIFTS OF FINITE TYPE BINGLI,TUOMASSAHLSTEN,ANDTONYSAMUEL Abstract. Anaimofthisarticleistohighlightdynamicaldifferencesbetweenthegreedy, andhencethelazy,β-shift(transformation)andanintermediateβ-shift(transformation), 5 forafixedβ∈(1,2).Specifically,aclassificationintermsofthekneadinginvariantsofthe 1 linearmapsTβ,α: x(cid:55)→βx+αmod1forwhichthecorrespondingintermediateβ-shiftis offinitetypeisgiven.Thischaracterisationisthenemployedtoconstructaclassofpairs 0 2 (β,α)suchthattheintermediateβ-shiftassociatedwithTβ,αisasubshiftoffinitetype.Itis alsoprovedthatthesemapsTβ,αarenottransitive.Thisisincontrasttothesituationforthe r correspondinggreedyandlazyβ-shiftsandβ-transformations,forwhichbothofthetwo a propertiesdonothold. M 1. Introduction,motivationandstatementofmainresults 5 1.1. Introductionandmotivation. Foragivenrealnumberβ (1,2)andarealnumber ] ∈ S x [0,1/(β 1)],aninfiniteword(ωn)n Noverthealphabet 0,1 iscalledaβ-expansionof D th∈epointxi−f ∈ { } . h x= ∞ ω β k. t k − a k=1 (cid:88) m If β is a natural number, then the β-expansions of a point x correspond to the β-adic [ expansionsofx. Inthiscase,almostallpositiverealnumbershaveauniqueβ-expansion. On the other hand, in [34] it has been shown that if β is not a natural number, then, 3 v forLebesguealmostall x, thecardinalityofthesetofβ-expansionsof xisequaltothe 7 cardinalityofthecontinuum. 2 Thetheoryofβ-expansionsoriginateswiththeworksofReńyi[32]andParry[29,30], 0 where an important link to symbolic dynamics has been established. Indeed, through 7 iterating the greedy β-transformation G : x βxmod1 and the lazy β-transformation . β 1 (cid:55)→ N L : x β(x 1)+2mod1oneobtainssubsetsof 0,1 whoseclosuresareknownasthe β 0 (cid:55)→ − { } greedyand(normalised)lazyβ-shifts,respectively. Eachpointω+ofthegreedyβ-shiftisa 4 1 β-expansion,andcorrespondstoauniquepointin[0,1];andeachpointω−ofthelazyβ- : shiftisaβ-expansion,andcorrespondstoauniquepointin[(2 β)/(β 1),1/(β 1)]. (Note v − − − thatinthecasewhen(2 β)/(β 1) 1,ifω+andω areβ-expansionsofthesamepoint, Xi thenω+ andω donotn−ecessa−rilyh≤avetobeequal−,see[23].) Throughthisconnection − r weobserveoneofthemostappealingfeaturesofthetheoryofβ-expansions,namelythat a itlinkssymbolicdynamicstonumbertheory. Inparticular,onecanaskquestionsofthe form, for what class of numbers the greedy and lazy β-shifts have given properties and viceversa. Infact,althoughthearithmetical,Diophantineandergodicpropertiesofthe greedyandlazyβ-shiftshavebeenextensivelystudied(see[6,8,33]andreferencestherein), there are many open problems of this form. Further, applications of this theory to the efficiencyofanalog-to-digitalconversionhavebeenexploredin[12]. Moreover,through understandingβ-expansionsofrealnumbers,advanceshavebeenmadeinunderstanding Bernoulliconvolutions. Forliteratureinthisdirection,wereferthereaderto[9,10,11]and referencetherein. ThefirstauthorwassupportedbyNSFC(no.11201155and11371148)andGuangdongNaturalScience Foundation2014A030313230.ThesecondauthorwassupportedbytheERCgrantno.306494.Thethirdauthor thankstheUniversita¨tBremenandtheSCUTfortheirsupport. 1 2 BINGLI,TUOMASSAHLSTEN,ANDTONYSAMUEL There are many ways, other than using the greedy and lazy β-shifts, to generate a β-expansionofapositiverealnumber. Forinstancetheintermediateβ-shiftsΩ which β,α arisefromtheintermediateβ-transformationsT : [0,1](cid:9). Thesetransformationsare β±,α definedfor(β,α) ∆,with ∈ ∆(cid:66) (β,α) R2: β (1,2)and0 α 2 β , { ∈ ∈ ≤ ≤ − } asfollows(alsoseeFigure1). Letting p=p (cid:66)(1 α)/βweset β,α − T+ (p)(cid:66)0 and T+ (x)(cid:66)βx+αmod1, β,α β,α forallx [0,1] p . Similarly,wedefine ∈ \{ } T (p)(cid:66)1 and T (x)(cid:66)βx+αmod1, β−,α β−,α forallx [0,1] p .IndeedwehavethatthemapsT areequaleverywhereexceptatthe ∈ \{ } β±,α point pandthat T (x)=1 T+ (1 x), β−,α − β,2 β α − − − for all x [0,1]. Observe that, when α=0, the maps G and T+ coincide, and when ∈ β β,α α = 2 β, the maps L and T coincide. Here we make the observation that for all − β β−,α (β,α) ∆,thesymbolicspaceΩ associatedtoT isalwaysasubshift,meaningthatitis ∈ β,α β±,α closedandinvariantundertheleftshiftmap(seeCorollary2). ThemapsT aresometimesalsocalledlinearLorenzmapsandarisenaturallyfrom β±,α thePoincare´ mapsofthegeometricmodelofLorenzdifferentialequations. Wereferthe interestedreaderto[13,27,35,37]forfurtherdetails. 1.6L(x) 1T.β+,α 1.G(x) 1.4 0.8 0.8 1.2 0.6 0.6 1. 0.4 0.4 0.8 0.2 0.2 0.8 1. 1.2 1.4 1.6 x 0.2 0.4 0.6 0.8 1. x 0.2 0.4 0.6 0.8 1. x (a)PlotofLβ. (b)PlotofTβ+,α. (c)PlotofGβ. Figure 1. Plot of T+ for β=(√5+1)/2 and α=1 0.474β, and the β,α − correspondinglazyandgreedyβ-transformations.(Theheightofthefilled incircledeterminesthevalueofthemapatthepointofdiscontinuity.) Inthisarticletheintermediateβ-transformationsandβ-shiftsarethemaintopicofstudy, and thus, to illustrate their importance we recall some of the known results in this area. Parry [31] proved that any topological mixing interval map with a single discontinuity is topologically conjugate to a map on the form T where (β,α) ∆. In [17, 20] it is β±,α ∈ shownthatatopologicallyexpansivepiecewisecontinuousmapT canbedescribedupto topologicalconjugacybythekneadinginvariantsofthepointsofdiscontinuityofT. (Inthe casethatT =T ,thekneadinginvariantsofthepointofdiscontinuity parepreciselygiven β±,α bythepointsintheassociatedintermediateβ-shiftwhichareβ-expansionof p+α/(β 1).) − Forsuchmaps,assumingthatthereexistsasinglediscontinuity,theauthorsof[3,20]gave asimpleconditiononpairsofinfinitewordsinthealphabet 0,1 whichissatisfiedifand { } onlyifthatpairofsequencesarethekneadinginvariantsofthepointofdiscontinuityofT. Ourmainresults,Theorems1.1and1.3andCorollary1,contributetotheongoingefforts indeterminingthedynamicalpropertiesoftheintermediateβ-shiftsandexaminingwhether thesepropertiesalsoholdforthecounterpartgreedyandlazyβ-shifts. Wedemonstratethat INTERMEDIATEβ-SHIFTSOFFINITETYPE 3 itispossibletoconstructpairs(β,α) ∆suchthattheassociatedintermediateβ-shiftis ∈ asubshiftoffinitetypebutforwhichthemapsT arenottransitive. Incontrasttothis, β±,α the corresponding greedy and lazy β-shifts are not subshifts of finite type (neither sofic shifts,namelyafactorofasubshiftoffinitetype)and,moreover,themapsG andL are β β transitive. RecallthatanintervalmapT: [0,1](cid:9)iscalledtransitiveifandonlyifforall opensubintervalsJof[0,1]thereexistsm Nsuchthat m Ti(J)=(0,1). ∈ i=1 ToprovethetransitivitypartofourresultweapplyaresultofPalmer[28]andGlendin- (cid:83) ning[17],whereitisshownthatforany1<β<2themapsG andL aretransitive. Infact, β β theygiveacompleteclassificationofthesetofpoints(β,α) ∆forwhichthemapsT are ∈ β±,α nottransitive. (ForcompletenesswerestatetheirclassificationinSection4.) Beforestatingourmainresultsletusemphasistheimportanceofsubshiftsoffinitetype. ThesesymbolicspacesgiveasimplerepresentationofdynamicalsystemswithfiniteMarkov partitions. Therearemanyapplicationsofsubshiftsoffinitetype,forinstanceincoding theory,transmissionandstorageofdataortilings. Wereferthereaderto[7,22,26,35]and referencesthereinformoreonsubshiftsoffinitetypeandtheirapplications. 1.2. Main results. Throughout this article, following convention, we let the symbol N denotethesetofnaturalnumbersandN denotethesetofnon-negativeintegers. Recall, 0 from for instance [8, Example 3.3.4], that the multinacci number of order n N 1 ∈ \{ } is the real number γ (1,2) which is the unique positive real solution of the equation n ∈ 1=x 1+x 2+ +x n. Thesmallestmultinaccinumberisthemultinaccinumberoforder − − − ··· 2 and is equal to the golden mean (1+ √5)/2. We also observe that γ >γ , for all n+1 n n N. Further,forn,k N withn 2,wedefinethealgebraicintegersβ andα by 0 n,k n,k ∈ ∈ ≥ (β )2k =γ andα =1 β /2. n,k n n,k n,k − Theorem1.1. Foralln,k Nwithn 2,wehavethefollowing. ∈ ≥ (i) Theintermediateβ -transformationsT arenottransitive. n,k β±n,k,αn,k (ii) Theintermediateβ -shiftΩ isasubshiftoffinitetype. n,k βn,k,αn,k (iii) Thegreedyandlazyβ -transformationsaretransitive. n,k (iv) Thegreedyandlazyβ -shifts,Ω andΩ respectively,arenotsofic, andhencenotasubshinf,tkoffinitetyβpn,ek,.0 βn,k,2−βn,k Part(iii)iswellknownandholdsforallgreedy,andhencelazy,β-transformations(seefor instance[17,28]). Itisincludedherebothforcompletenessandtoemphasisethedynamical differenceswhichcanoccurbetweenthegreedy,andhencethelazy,β-transformation(shift) andanassociatedintermediateβ-transformation(shift). To verify Theorem 1.1(ii) we apply Theorem 1.3 given below. This latter result is a generalisationofthefollowingresultofParry[29]. Hereandinthesequel,τ (p)denotes ±β,α thepointsintheassociatedintermediateβ-shiftswhichareaβ-expansionof p+α/(β 1), − where p=p . β,α Theorem1.2([29]). Forβ (1,2),wehave ∈ (i) thegreedyβ-shiftisasubshiftoffinitetypeifandonlyifτ (p)isperiodicand −β,0 (ii) thelazyβ-shiftisasubshiftoffinitetypeifandonlyifτ+ (p)isperiodic. β,2 β − Theorem1.3. Letβ (1,2)andα (0,2 β). Theintermediateβ-shiftΩ isasubshiftof β,α ∈ ∈ − finitetypeifandonlyifbothτ (p)areperiodic. ±β,α Thefollowingexampledemonstratesthatitispossibletohavethatoneofthesequences τ (p)isperiodicandthattheotherisnotperiodic. ±β,α Example 1. Let β=γ , α=1/β4 and p=(1 α)/β=1/β3. Clearly α (0,2 β). An 2 − ∈ − elementary calculation will show that τ (p) is the infinite periodic word with period −β,α (0,1,1,0)andτ+ (p)istheconcatenationofthefiniteword(1,0,0)withtheinfiniteperiodic β,α wordwithperiod(1,0). 4 BINGLI,TUOMASSAHLSTEN,ANDTONYSAMUEL Forβ>2andα [0,1]theforwardimplicationsofTheorems1.2and1.3canbefound ∈ in,forinstance,[36,Theorem6.3]. Further,forβ>1,necessaryandsufficientconditions whenΩ issoficshiftisgivenin[22,Theorem2.14]. Thislatterresult,infact,givesthat β,α ifβ (1,2)isaPisotnumber,thenalongthefiber∆(β)(cid:66) (b,α) ∆: b=β thereexistsa ∈ { ∈ } densesetofpoints(β,α)suchthatΩ isasoficshift. β,α Thisleadsustothefinalproblemofstudyinghowagreedy(orlazy)β-shiftbeinga subshiftoffinitetypeisrelatedtothecorrespondingintermediateβ-shiftsbeingasubshift offinitetype. Wealreadyknowthatitispossibletofindintermediateβ-shiftsoffinitetype inthefibres∆(β)eventhoughthecorrespondinggreedyandlazyβ-shiftsarenotsubshiftof finitetype. Giventhis,anaturalquestiontoaskis,canonedeterminewhennointermediate β-shiftisasubshiftoffinitetype,foragivenfixedβ. Thisispreciselywhatweaddress inthefollowingresultwhichisanalmostimmediateapplicationofthecharacterisation providedinTheorems1.2and1.3. Corollary1. Ifβ (1,2)isnotthesolutionofanypolynomialoffinitedegreewithcoeffi- ∈ cientsin 1,0,1 ,thenforallα [0,2 β]theintermediateβ-shiftΩ isnotasubshiftof β,α {− } ∈ − finitetype. We observe that the values β , which are considered in Theorem 1.1, are indeed a n,k solutionofapolynomialwithcoefficientsintheset 1,0,1 andthusdonotsatisfythe {− } conditionsofCorollary1;thisisverifiedintheproofofTheorem1.1(iv). ThephenomenonthatthemapT isnottransitivebutwheretheassociatedshiftspace β,α Ω isasubshiftoffinitetypealsoappearinthesettingof( β)-transformations. Indeed β,α − recentlyItoandSadahiro[21]studiedtheexpansionsofanumberinthebase( β),with − β>1. Theseexpansionsarerelatedtoaspecificpiecewiselinearmapwithconstantslope definedontheinterval[ β/(β+1),1/(β+1)]by x βx βx+β/(β+1) . (Herefor a R,wedenoteby a t−helargestintegernsothatn(cid:55)→−a.) S−u(cid:98)c−hamapiscalle(cid:99)danegative ∈ (cid:98) (cid:99) ≤ β-transformation. Thecorrespondingsymbolicspacespaceiscalleda( β)-shift which N − isasubsetof 0,1 whenβ<2. ByaresultofLiaoandSteiner[24],incontrasttothe { } lazyorgreedyβ-shifts,forallβ<γ ,thenegativeβ-transformationsarenottransitive. On 2 theotherhand, aresultofFrougnyandLai[16]showsthata( β)-shiftisoffinitetype − ifandonlyiftherepresentationof β/(β+1)inthe( β)-shiftisperiodic(see(1)forthe − − definitionofaperiodicword). Thusonecanconstructexamplesofβ (1,2)sothatthe ∈ negativeβ-transformationisnottransitivebutwherethe( β)-shiftisasubshiftoffinite − type. Forinstance,ifwetaketheperiodicsequence (1,0,0,...,0,1)=(1,0,0,...,0,1,1,0,0,...,0,1,1,0,0,...,0,1,...), (cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32) 2k times 2k times 2k times 2k times − − − − withk 1,asth(cid:124)e((cid:123)(cid:122)β)-e(cid:125)xpansion(cid:124)of (cid:123)β(cid:122)/(β(cid:125)+1),t(cid:124)hen(cid:123)(cid:122)we(cid:125)obtain(cid:124)a((cid:123)β(cid:122))-t(cid:125)ransformationwhich ≥ − − − isnottransitiveandwhoseassociated( β)-shiftisasubshiftoffinitetype. Aninteresting − furtherproblemwouldbetoexaminethecorrespondingquestionsforintermediatenegative β-transformationsandintermediate( β)-shifts. Wereferthereaderto[18]andreferences − their in for further details on intermediate negative β-transformations and intermediate ( β)-shifts. − 1.3. Outline. InSection2wepresentbasicdefinitions,preliminariesandauxiliaryresults requiredtoproveTheorems1.1and1.3andCorollary1. TheproofsofTheorem1.3and Corollary1arepresentedinSection3andtheproofofTheorem1.1isgiveninSection4. INTERMEDIATEβ-SHIFTSOFFINITETYPE 5 2. Definitionsandauxiliaryresults N 2.1. Subshifts. We equipped the space 0,1 of infinite sequences with the topology inducedbythewordmetricd: 0,1 N 0{,1 N} Rwhichisgivenby { } ×{ } → 0 ifω=ν, d(ω,ν)(cid:66) 2 ω ν+1 otherwise.  −| ∧ |  Here ω ν(cid:66)min n N: ωn(cid:44)νn ,forallω=(ω1,ω2,...),ν=(ν1ν2,...) 0,1 N with ω(cid:44)ν.| W∧el|etσden{ot∈etheleft-shift}mapwhichisdefinedontheset ∈{ } 0,1 N ∞ 0,1 n, { } ∪{∅}∪ { } n=1 (cid:91) of finite and infinite words over the alphabet 0,1 by σ(ω)(cid:66) , if ω 0,1 , and otherwisewesetσ(ω ,ω ,...)(cid:66)(ω ,ω ,...).A{shif}tinvariantsub∅spacei∈sa{set}Ω∪{∅}0,1 N 1 2 2 3 ⊆{ } suchthatσ(Ω) Ωandasubshiftisaclosedshiftinvariantsubspace. Givenasubs⊆hiftΩandn Nset ∈ Ω (cid:66) (ω ,...,ω ) 0,1 n: thereexists(ξ ,...) Ωwith(ξ ,...,ξ )=(ω ,...,ω ) n 1 n 1 1 n 1 n | ∈{ } ∈ (cid:26) (cid:27) andwriteΩ (cid:66) Ω forthecollectionofallfinitewords. Asubshif|∗tΩis∞ka=1call|kedasubshiftoffinitetype(SFT)ifthereexistsanM Nsuchthat, forallω=(ω ,ω(cid:83),...,ω ),ξ=(ξ ,ξ ,...,ξ ) Ω withn,m Nandn M,∈ 1 2 n 1 2 m ∗ ∈ | ∈ ≥ (ω ,ω ,...,ω ,ξ ,ξ ,...,ξ ) Ω n M+1 n M+2 n 1 2 m ∗ − − ∈ | ifandonlyif (ω ,ω ,...,ω ,ξ ,ξ ,...,ξ ) Ω . 1 2 n 1 2 m ∗ ∈ | The following result gives an equivalent condition for when a subshift is a subshift of finite type (see, for instance, [26, Theorem 2.1.8] for a proof of the equivalence). For this we require the following notation. For n N and ω=(ω ,ω ,...) 0,1 N, we set 1 2 ω (cid:66)(ω ,ω ,...,ω ). Also,forafinitewordξ∈ 0,1 n,forsomen N ,∈w{elet}ξ denote n 1 2 n 0 | ∈{ } ∈ | | thelengthofξ,namelythevaluen. Theorem2.1([26]). ThespaceΩisaSFTifandonlyifthereexistsafinitesetF offinite wordsinthealphabet 0,1 suchthatifξ F,thenσm(ω) (cid:44)ξ,forallω Ωandm N . ξ 0 { } ∈ || | ∈ ∈ ThesetF inTheorem2.1isreferredtoasthesetofforbiddenwords. Further,asubshift ΩiscalledsoficifandonlyifitisafactorofaSFT. Forν=(ν ,ν ,...,ν ) 0,1 nandξ=(ξ ,ξ ,...,ξ ) 0,1 m,denoteby(ν,ξ)thecon- 1 2 n 1 2 m catenation(ν ,ν ,...,ν ,ξ∈,{ξ ,.}..,ξ ) 0,1 n+m,forn,m∈{ N.}Weusethesamenotation 1 2 n 1 2 m N ∈{ } ∈ whenξ 0,1 . Anin∈fi{nite}sequenceω=(ω ,ω ,...) 0,1 Niscalledperiodicifthereexistsann N 1 2 ∈{ } ∈ suchthat (ω ,ω ,...,ω )=(ω ,ω ,...,ω ), (1) 1 2 n (m 1)n+1 (m 1)n+2 mn − − forallm N. Wecalltheleastsuchntheperiodofωandwewriteω=(ω ,ω ,...,ω ). 1 2 n Similarly,∈ω=(ω ,ω ,...) 0,1 Niscalledeventuallyperiodicifthereexistsann,k N 1 2 ∈{ } ∈ suchthat (ω ,...,ω )=(ω ,ω ,...,ω ), k+1 k+n k+(m 1)n+1 k+(m 1)n+2 k+mn − − forallm N,andwewriteω=(ω ,ω ,...,ω ,ω ,ω ,...,ω ). 1 2 k 1 k k+1 k+n ∈ − 6 BINGLI,TUOMASSAHLSTEN,ANDTONYSAMUEL 2.2. Intermediateβ-shiftsandexpansions. Wenowgivethedefinitionoftheintermedi- ateβ-shift. Throughoutthissectionwelet(β,α) ∆befixedandlet pdenotethevalue ∈ p =(1 α)/β. β,α − TheT+ -expansionτ+ (x)of x [0,1]andtheT -expansionτ (x)of x [0,1]are β,α β,α ∈ β−,α −β,α ∈ respectively defined to be ω+ =(ω+,ω+,...,) 0,1 N and ω =(ω ,ω ,...,) 0,1 N, 1 2 ∈{ } − −1 −2 ∈{ } where 0, if (T )n 1(x) p, 0, if(T+ )n 1(x)<p, ω (cid:66) β−,α − ≤ and ω+(cid:66) β,α − −n n 1, otherwise, 1, otherwise, foralln N.Wewilldenotetheimagesoftheunitintervalunderτ byΩ ,respectively, ∈   ±β,α ±β,α andwriteΩ fortheunionΩ+ Ω . TheupperandlowerkneadinginvariantsofT β,α β,α∪ −β,α β±,α aredefinedtobetheinfinitewordsτ (p),respectively,andwillturnouttobeofgreat ±β,α importance. Remark 1. Let ω =(ω ,ω ,...,) denote the infinite words τ (p), respectively. By ± ±1 ±2 ±β,α definition,ω =ω+=0andω+=ω =1. Furthermore,(ω+,ω+ ,...,)=(0,0,0,...)ifand −1 2 1 −2 k k+1 onlyifk=2andα=0;and(ω ,ω ,...,)=(1,1,1,...)ifandonlyifk=2andα=2 β. −k −k+1 − TheconnectionbetweenthemapsT andtheβ-expansionsofrealnumbersisgiven β±,α N throughtheT -expansionsofapointandtheprojectionmapπ : 0,1 [0,1],which β±,α β,α { } → wewillshortlydefine. Theprojectionmapπ islinkedtotheunderlying(overlapping) β,α iteratedfunctionsystem(IFS),namely([0,1];f : x x/β,f : x x/β+1 1/β),where 0 1 → → − β (1,2). (Wereferthereaderto[14]forthedefinitionofandfurtherdetailsonIFSs.) The ∈ projectionmapπ isdefinedby β,α π (ω ,ω ,...)(cid:66)(β 1) 1(lim f f ([0,1]) α)=α(1 β) 1+ ∞ ω β k. β,α 1 2 − − n ω1◦···◦ ωn − − − k − →∞ k=1 (cid:88) Animportantpropertyoftheprojectionmapisthatthefollowingdiagramcommutes. σ Ω Ω ±β,α −→ ±β,α πβ,α πβ,α (2) ↓ ↓ [0,1] [0,1] −T→β±,α (Thisresultisreadilyverifiablefromthedefinitionsofthemapsinvolvedandasketchof theproofofthisresultcanbefoundin[4,Section5].) Fromthis,weconcludethat,foreach x [α/(β 1),1+α/(β 1)],aβ-expansionofxistheinfinitewordτ (x α(β 1) 1). ∈ − − ±β,α − − − 2.3. Theextendedmodel. InourproofofTheorem1.3, wewillconsidertheextended model given as follows. For (β,α) ∆, we let p= p =(1 α)/β and we define the β,α ∈ − transformationT : [ α/(β 1),(1 α)/(β 1)](cid:9)by β±,α − − − − βx+α ifx<p, βx+α ifx p, Tβ+,α(x)= βx(cid:101)+α 1 otherwise, and Tβ−,α(x)= βx+α 1 othe≤rwise    −  − (se(cid:101)eFigure2). NotethatTβ±,α|[0,1]=Tβ±,αandthat,for(cid:101)allpointsxchosenfromtheinterval ( α/(β 1),0) (1,(1 α)/(1 β)),thereexistsaminimaln=n(x) Nsuchthatwehave − − ∪ − − ∈ (T )m(x) [0,1],forallm(cid:101) n. β±,α ∈ ≥ TheT -expansionsτ (x)ofpoints x [ α/(β 1),(1 α)/(β 1)]isdefinedinthe (cid:101) β±,α ±β,α ∈ − − − − samewayasτ exceptonereplacesT byT . Welet ±β,α β±,α β±,α (cid:101) (cid:101) Ω (cid:66)τ ([ α/(β 1),(1 α)/(β 1)]) ±β,α ±β,α − −(cid:101) − − (cid:101) (cid:101) INTERMEDIATEβ-SHIFTSOFFINITETYPE 7 Tβ+,α Tβ−,α e e 1.0 1.0 0.5 0.5 x x 0.5 1.0 0.5 1.0 (a)PlotofT+ . (b)PlotofT . β,α β−,α Figure2. Plotof(cid:101)T forβ=(√5+1)/2andα=1 0.474β. ((cid:101)Theheight β±,α − of the filled in circle determines the value of the map at the point of discontinuity.) (cid:101) andwesetΩ =Ω+ Ω . TheupperandlowerkneadinginvariantofT aredefined β,α β,α∪ −β,α β±,α tobetheinfinitewordsτ (p),respectively. ±β,α (cid:101) (cid:101) (cid:101) (cid:101) 2.4. StructureofΩ andΩ . Thefollowingtheorem,onthestructureofthespaces ±β,α(cid:101) ±β,α Ω andΩ ,willplayacrucialroleintheproofofTheorem1.3. Apartialversionofthis ±β,α ±β,α (cid:101) resultcanbefoundin[19,Lemma1]. Tothebestofourknowledge,thisfirstappearedin [20,Theo(cid:101)rem1],andlaterin[1,2,4,22]. Moreover,aconverseofTheorem2.2holdstrue andwasfirstgivenin[20,Theorem1]andlaterin[3,Theorem1]. Theorem2.2([1,2,3,4,19,20,22]). For(β,α) ∆,thespacesΩ andΩ arecom- ∈ ±β,α ±β,α pletelydeterminedbyupperandlowerkneadinginvariantsofT andT ,respectively: β±,α β±,α (cid:101) Ω+ = ω 0,1 N: n 0,τ+ (0) σn(ω) τ (p)orτ+ (p) σn(ω) τ (1) β,α ∈{ } ∀ ≥ β,α (cid:22) ≺ −β,α β,α (cid:22) (cid:101)(cid:22) −β,α Ω =(cid:110)ω 0,1 N: n 0,τ+ (0) σn(ω) τ (p)orτ+ (p) σn(ω) τ (1)(cid:111) −β,α ∈{ } ∀ ≥ β,α (cid:22) (cid:22) −β,α β,α ≺ (cid:22) −β,α Ω+ =(cid:110)ω 0,1 N: n 0,τ+ α σn(ω) τ (p)orτ+ (p) σn(ω) τ+ (cid:111)1 α β,α ∈{ } ∀ ≥ β,α β−1 (cid:22) ≺ −β,α β,α (cid:22) (cid:22) β,α β−1 − − Ω =(cid:110)ω 0,1 N: n 0,τ+ (cid:16) α(cid:17) σn(ω) τ (p)orτ+ (p) σn(ω) τ+ (cid:16)1 α(cid:17)(cid:111). (cid:101)−β,α ∈{ } ∀ ≥ (cid:101)β,α β−−1 (cid:22) (cid:22)(cid:101)−β,α (cid:101)β,α ≺ (cid:22)(cid:101)β,α β−−1 (Hereth(cid:110)esymbols , , and (cid:16)deno(cid:17)tethelexicographicorderingson 0,1 N.) M(cid:16)oreov(cid:17)(cid:111)er, (cid:101) ≺ (cid:22) (cid:31) (cid:101)(cid:23) (cid:101) (cid:101) { } (cid:101) Ω =Ω+ Ω and Ω =Ω+ Ω . (3) β,α β,α∪ −β,α β,α β,α∪ −β,α Remark2. Bythecommutativityofthediagramin(2),wehavethatτ+ (0)=σ(τ+ (p)) (cid:101) (cid:101) (cid:101) β,α β,α andτ (1)=σ(τ (p)). Additionally,fromthedefinitionoftheT -expansionofapoint −β,α −β,α β±,α τ+ α =τ α =(0) and τ+ 1 α =τ 1 α =(1). β,α β−1 −β,α β−1 β,α β−1 −β,α(cid:101)β−1 − − − − Further,sinceT (cid:16) (cid:17)=T (cid:16),itfo(cid:17)llowsthatτ (p)(cid:16)=τ (cid:17)(p)an(cid:16)dthu(cid:17)s,fromnowonwe (cid:101) β±,α|[0,1] (cid:101)β±,α ±β,α(cid:101) ±β,α (cid:101) willwriteτ+ (p)forthecommonvalueτ+ (p)=τ+ (p)andτ (p)forthecommonvalue β,α β,α β,α −β,α τ (p)=τ (p(cid:101)). (cid:101) −β,α −β,α (cid:101) Corollary2. For(β,α) ∆,wehavethatΩ andΩ areshiftinvariantsubspacesand (cid:101) ∈ ±β,α ±β,α thatΩ andΩ aresubshifts. Moreover, β,α β,α (cid:101) σ(Ω ) Ω , σ(Ω ) Ω , σ(Ω )=Ω and σ(Ω )=Ω . β,α∗ β,α∗ β,α∗ β,α∗ β,α β,α β,α β,α | ⊆(cid:101) | | ⊆ | (cid:101) (cid:101) (cid:101) (cid:101) 8 BINGLI,TUOMASSAHLSTEN,ANDTONYSAMUEL Proof. ThisisadirectconsequenceofTheorem2.2andthecommutativityofthediagram givenin(2). (cid:3) AsonecanseethecodespacestructureofΩ issimplerthanthecodespacestructure ±β,α ofΩ . Infact,toproveTheorem1.3,wewillshowthatitisnecessaryandsufficientto ±β,α (cid:101) showthatΩ isaSFT. β,α 2.5. Perio(cid:101)dicity and zero/one-full words. We now give a sufficient condition for the kneading invariants of T (and hence, by Remark 2, of T ) to be periodic. For this, β±,α β±,α wewillrequirethefollowingnotationsanddefinitions. Indeedbyconsideringzeroand one-fullness(Definition(cid:101)2.3)oftheinfinitewordsσ(τ (p))weobtainacloselinktothe ±β,α definitionofSFT(seeLemma2.4andProposition1(i)). ForagivensubsetEofN,wewrite E forthecardinalityofE. | | N Definition2.3. Let(β,α) ∆. Aninfinitewordω=(ω ,ω ,...) 0,1 iscalled 1 2 ∈ ∈{ } (a) zero-fullwithrespectto(β,α)if N (ω)(cid:66) k>2: (ω ,ω ,...,ω ,1) Ω andω =0 , 0 1 2 k−1 ∈ −β,α|k k isnotfinite,and(cid:12)(cid:12)(cid:110) (cid:111)(cid:12)(cid:12) (cid:12) (cid:101) (cid:12) (cid:12) (cid:12) (b) one-fullwithrespectto(β,α)if N (ω)(cid:66) k>2: (ω ,ω ,...,ω ,0) Ω+ andω =1 , 1 1 2 k−1 ∈ β,α|k k isnotfinite. (cid:12)(cid:12)(cid:110) (cid:111)(cid:12)(cid:12) (cid:12) (cid:101) (cid:12) (cid:12) (cid:12) Remark3. Theconceptofaninfinitewordbeingzero-fullismotivatedfromthefactthata cylindersetI(ω ,...,ω ,0) [0,1)correspondingtothefiniteword(ω ,...,ω ,0)isa 1 n 1 1 n 1 − ⊂ − fullintervalif(ω ,...,ω ,1)isadmissibleforthegreedyβ-expansionwith1<β<2(see 1 n 1 − forinstance[15,Lemma3.2(1)]). Herewerecallfrom[15]that (i) thecylindersetI(ω ,...,ω ,0) [0,1)isdefinedtobethehalfopenintervalof 1 n 1 [0,1]suchthatforanyx I−(ω ,..⊂.,ω ,0)wehaveτ (x) =(ω ,...,ω ,0), ∈ 1 n−1 −β,0 |n 1 n−1 (ii) admissibleforthegreedyβ-expansionmeansthatσk(ω ,...,ω ,1) τ (1) , 1 n−1 (cid:22) −β,0 |n−1−k forallk 0,1,...,n 2 ,and ∈{ − } (iii) full means that the length of the cylinder set I(ω ,...,ω ,0) is equal to β n, 1 n 1 − whichisequivalentto(T )n(I(ω ,...,ω ,0))=[0,1). − β−,0 1 n 1 − Thenotionofaninfinitewordbeingone-fullismotivatedasaboveexceptfromtheprospec- tiveofthelazyβ-expansioninsteadofthegreedyβ-expansion. Beforestatingournextresultweremarkthat p =1 1/βifandonlyifα=2 βand β,α − − p =1/βifandonlyifα=0. β,α Lemma2.4. Let(β,α) ∆andset p=p . β,α ∈ (i) Ifτ+ (p)isnotperiodicandα(cid:44)2 β,thenσ(τ+ (p))isone-full. β,α − β,α (ii) Ifτ (p)isnotperiodicandα(cid:44)0,thenσ(τ (p))iszero-full. −β,α −β,α Proof. Astheproofsfor(i)and(ii)areessentiallythesame,weonlyincludeaproofof(ii). Foreaseofnotationletν=(ν ,ν ,...)denotetheinfinitewordσ(τ (p))and,notethat, 1 2 −β,α sinceτ (p)isnotperiodic,σk(τ (p))(cid:44)τ (p),forallk N. −β,α −β,α −β,α ∈ Definethenon-constantmonotonicsequence(nk)k Nofnaturalnumbersasfollows. Let n Nbetheleastnaturalnumbersuchthatν =∈1andν =0and, forallk>1, let n1∈Nbesuchthat n1−1 n1 k ∈ σnk−1−1(ν)|nk−nk−1=τ−β,α(p)|nk−nk−1 and σnk−1−1(ν)|nk−nk−1+1≺τ−β,α(p)|nk−nk−1+1. (4) INTERMEDIATEβ-SHIFTSOFFINITETYPE 9 SuchasequenceexistsbyRemark1andbecauseτ (p)isassumedtobenotperiodic. We −β,α willnowshowthatν =0andthat nk (ν ,ν ,...,ν ,1) Ω , 1 2 nk−1 ∈ −β,α|nk forallintegersk>1. Bytheorderingsgivenin(4)itimmediatelyfollowsthatν =0. In (cid:101) nk ordertoconcludetheproof,sinceν Ω andsinceτ (p)isnotperiodic,byTheorem2.2, ∈ −β,α −β,α itissufficienttoshow,forallintegersk>1andm 0,1,2,...,n 2,n 1 ,that k k ∈{ − − } (cid:101) τ (p) ifν =0, σm(θ) (cid:22) −β,α m+1 τ+ (p) otherwise,  (cid:31) β,α wheSruepθpo(cid:66)se(νth1a,tν2l,...1,,ν2n,k.−.1.,,1k,νannk−dnmk−1+1,nνnk−,nnk−1++2,1..,..)...,n 2 ,wheren (cid:66)0. Ifν =1, l 1 l 1 l 0 m+1 ∈{ } ∈{ − − − } then σm(θ)=(ν ,ν ,...,ν ,ν ,ν ,...,ν ,1,ν ,ν ,...) m+1 m+2 nl−1 nl nl+1 nk−1 nk−nk−1+1 nk−nk−1+2 (ν ,ν ,...,ν ,ν ,ν ,...,ν ,0,ν ,ν ,...) (cid:31) m+1 m+2 nl−1 nl nl+1 nk−1 nk+1 nk+2 =σm(ν) τ+ (p). (cid:31) β,α Nowsupposev =0. By(4),wehave(ν ,ν ,...,ν ,1)=τ (p) ,and so, m+1 nl−1 nl−1+1 nl−1 −β,α |nl−nl−1+1 σm−nl−1+1(τ−β,α(p))=(νm+1,νm+2,...,νnl−1,1,νnl−nl−1+1,νnl−nl−1+2,...). Letusfirstconsiderthecasel=k.Sincebyassumptionv =0,andsince,byTheorem2.2, m+1 wehavethatσm−nl−1+1(τ−β,α(p))(cid:22)τ−β,α(p),itfollowsthat σm(θ)=(ν ,ν ,...,ν ,1,ν ,ν ,...) τ (p). m+1 m+2 nl−1 nl−nl−1+1 nl−nl−1+2 (cid:22) −β,α Therefore,inthiscase,namelyl=k,theresultfollows. Nowassumethatl<k. Inthiscase wehavethat σm(θ)=(ν ,ν ,...,ν ,ν ,ν ,...,ν ,1,ν ,ν ,...) m+1 m+2 nl−1 nl nl+1 nk−1 nk−nk−1+1 nk−nk−1+2 (ν ,ν ,...,ν ,1,ν ,ν ,...) ≺ m+1 m+2 nl−1 nl−nl−1+1 nl−nl−1+2 τ (p). (cid:22) −β,α Ifl 1,2,...,k 2 andm=n 1,thenbytheorderingsgivenin(4), l ∈{ − } − σm(θ)=( ν ,...,ν ,...,ν ,1,ν ,ν ,...) τ (p). n(cid:32)(cid:32)l(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)n(cid:32)(cid:32)l(cid:32)+(cid:32)1 nk−1 nk−nk−1+1 nk−nk−1+2 ≺ −β,α ≺τ−β,α(p)|nl+1−nl+1 (cid:124) (cid:123)(cid:122) (cid:125) Ifm=n 1,thenbytheorderingsgivenin(4), k 1 − − σm(θ)=(ν ,...,ν ,1,ν ν ,...)=τ (p). n(cid:32)(cid:32)(cid:32)k(cid:32)−(cid:32)(cid:32)(cid:32)1(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)n(cid:32)(cid:32)k(cid:32)(cid:32)−(cid:32)(cid:32)(cid:32)(cid:32)1(cid:32)(cid:32)(cid:32)(cid:32) n(cid:32)(cid:32)k(cid:32)(cid:32)−(cid:32)(cid:32)(cid:32)(cid:32)n(cid:32)(cid:32)k(cid:32)(cid:32)−(cid:32)(cid:32)(cid:32)1(cid:32)(cid:32)+(cid:32)(cid:32)(cid:32)1(cid:32)(cid:32)(cid:32) nk−(cid:32)(cid:32)(cid:32)n(cid:32)(cid:32)(cid:32)k(cid:32)(cid:32)−(cid:32)(cid:32)1(cid:32)(cid:32)(cid:32)+(cid:32)(cid:32)(cid:32)2(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) −β,α =τ−β,α(p)|nk−nk−1+1 =σnk−nk−1+1(τ−β,α(p)) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) Finally, if m=n 1, then either ν =1, in which case the result follows from Remark1orelsekν− =0inwhnki−cnhk−c1a+s1e nk−nk−1+1 σm(θ)=(1,νnk−nk−1+1νnk−nk−1+2,...)=σnk−nk−1(τ−β,α(p))(cid:31)τ+β,α(p). Thisconcludestheproof. (cid:3) 10 BINGLI,TUOMASSAHLSTEN,ANDTONYSAMUEL 2.6. Symmetricintermediateβ-shifts. Thefollowingremarksandresultonsymmetric intermediateβ-shiftswillassistusinshorteningtheproofofTheorem1.3andtheproof ofTheorem1.1(ii). Forthis,letusrecallthattwomapsR: X(cid:9)andS: Y (cid:9)definedon compactmetricspacesarecalledtopologicallyconjugateifthereexistsahomeomorphism h: X Y suchthatS h=h R. → ◦ ◦ ThemapsT andT+ aretopologicallyconjugateandthemapsT andT+ β−,α β,2 α β β−,α β,2 α β aretopologicallyconjugat−e,−wheretheconjugatingmapisgivenbyh¯: x 1 x. Th−is−is (cid:55)→ − immediatefromthedefinitionofthemapsinvolved. Similarly,themapsT+ andT β,α β−,2 α β are topologically conjugate and the maps T and T+ are topologically conjug−a−te, β−,α β,2 α β − −N (cid:101) (cid:101) wheretheconjugatingmapisalsoh¯. Whenliftedto 0,1 themappingh¯ becomesthe symmetricmap : 0,1 N(cid:9)definedby, (cid:101) (cid:101){ } ∗ { } (ω ,ω ,...)=(ω +1mod2,ω +1mod2,...), 1 2 1 2 ∗ N for all ω 0,1 . From this we conclude that the following couples are topologically ∈{ } conjugatewheretheconjugatingmapisthesymmetricmap : ∗ σ|Ω−β,αandσ|Ω+β,2−α−β, σ|Ω+β,αandσ|Ω−β,2−α−β, σ|Ω−β,αandσ|Ω+β,2−α−β and σ|Ω+β,αandσ|Ω−β,2−α−β. Theorem2.5. Forall(β,α) ∆wehavethatΩ(cid:101) =Ω (cid:101) andΩ (cid:101)=Ω (cid:101). More- ∈ ±β,α ∓β,2 β α ±β,α ∓β,2 β α over,givenβ (1,2),thereexistsauniquepointα [0,2 −β]−,namelyα=1 β/−2,−suchthat ∈ ∈ − − τ+ (p)= (τ (p)),where p=p =(1 α)/β. (cid:101) (cid:101) β,α ∗ −β,α β,α − Proof. The first part of the result follows immediately from the above comments and Theorem2.2. Thesecondpartoftheresultfollowsfromanapplicationof[2,Theorem2] togetherwiththefactthatthemapsT aretopologicallyconjugatetoasub-classofmaps β±,α consideredin[2],wheretheconjugatingmaph¯ : [0,1] [ α/(β 1),(1 α)/(β 1)]is β,α givenbyh¯β,α(x)=(x α)/(β 1). (cid:101) → − − − − (cid:3) − − 3. ProofsofTheorem1.3andCorollary1 LetusfirstproveananalogousresultfortheextendedmodelΩ . Throughoutthissection, β,α foragiven(β,α) ∆,weset p=p . β,α ∈ (cid:101) Proposition1. Letβ (1,2). ∈ (i) Ifα (0,2 β),thenΩ isaSFTifandonlyifbothτ (p)areperiodic. ∈ − β,α ±β,α (ii) Ifα=2 β,thenΩ isaSFTifandonlyifτ (p)isperiodic. − β,α(cid:101) −β,α (iii) Ifα=0,thenΩ isaSFTifandonlyifτ+ (p)isperiodic. β,α β,α (cid:101) ProofofProposition1((cid:101)i). Wefirstprovethatτ−β,α(p)isperiodicifΩβ,αisaSFT.Assumeby wayofcontradictionthatΩ isaSFT,butthatτ (p)isnotperiodic. ThenbyLemma2.4 β,α −β,α (cid:101) theinfinitewordν=σ(τ (p))iszero-full,namely −β,α (cid:101) N (ν)= k>2: (ν ,ν ,...,ν ,1) Ω andν =0 0 1 2 k−1 ∈ −β,α|k k insumnobtefirsnsituec.hTthhauts, there(cid:12)(cid:12)(cid:12)(cid:12)(cid:110)exists a non-constant mono(cid:101)tonic sequence(cid:111)((cid:12)(cid:12)(cid:12)(cid:12)nk)k∈N of natural (ν ,ν ,...,ν 1) Ω and (0,ν ,ν ,...,ν ,0)=τ (p) . 1 2 nk−1 ∈ −β,α|∗ 1 2 nk−1 −β,α |nk+1 ThistogetherwithCorollary2impliesthatΩ isnotaSFT,contradictingourassumption. (cid:101) β,α The statement that if Ω is a SFT, then τ+ (p) is periodic, follows in an identical β,α β,α mannertotheabove,whereweuseone-ful(cid:101)lnessinsteadofzero-fullness. Wewillnowprovethe(cid:101)conversestatement: for(β,α) ∆,ifboththekneadinginvariants ∈ ofT areperiodic,thenΩ isaSFT.Withoutlossofgeneralitywemayassumethatboth β±,α β,α (cid:101) (cid:101)