Intermediate \beta-shifts of finite type

0.32 MB·

by  Bing Li

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-expansionsoriginateswiththeworksofRe´nyi[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)