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The syntactic graph of a sofic shift
Marie-Pierre Béal, Francesca Fiorenzi, Dominique Perrin
To cite this version:
Marie-Pierre Béal, Francesca Fiorenzi, Dominique Perrin. The syntactic graph of a sofic shift. 21st
International Symposium on Theoretical Aspects of Computer Science (STACS 2004), Mar 2004,
Montpellier, France. pp.282-293. �hal-00619853�
Marie-PierreBeal,FranesaFiorenzi,andDominiquePerrin
InstitutGaspard-Monge,UniversitedeMarne-la-Vallee
77454 Marne-la-ValleeCedex2,Frane
fbeal,perringuniv-mlv.fr, fiorenzimat.uniroma1.it
Abstrat. Wedene a new invariant for the onjugay of irreduible
so shifts. This invariant, that we all the syntati graph of a so
shift, is the direted ayli graph of harateristi groups of the non
nullregularD-lassesofthesyntatisemigroupoftheshift.
Keywords:Automataandformallanguages,symbolidynamis.
1 Introdution
Soshifts[17℄aresetsofbi-innitelabelsinalabeledgraph.Ifthegraphanbe
hosenstronglyonneted, thesoshiftis saidtobeirreduible.A partiular
sublassofsoshiftsisthelassofshiftsofnitetype,denedbyanitesetof
forbiddenbloks.Twoso shiftsX andY are onjugateifthereis abijetive
blokmap from X onto Y. It is an open questionto deidewhether two so
shiftsareonjugate,eveninthepartiularaseofirreduibleshiftsofnitetype.
Therearemanyinvariantsforonjugayofsubshifts,algebraiorombinato-
rial,see[13,Chapter7℄,[6℄,[12℄,[3℄.Forinstanetheentropyisaombinatorial
invariantwhihgivestheomplexityofallowedbloksin ashift.Thezetafun-
tionisanotherinvariantwhihountsthenumberofperiodiorbitsin ashift.
Inthispaper,wedeneanewinvariantforirreduiblesoshifts.Thisinvari-
antisbasedonthestrutureofthesyntatisemigroupofthelanguageofnite
bloksoftheshift.Irreduiblesoshiftshaveaunique(up toisomorphismsof
automata)minimal deterministipresentation,alled therightFisheroverof
theshift.ThesyntatisemigroupSofanirreduiblesoshiftisthetransition
semigroupofitsrightFisherover.
Ingeneral,thestrutureof anite semigroupisdeterminedby theGreen's
relations(denotedR,L,H ;D;J)[16℄.Ourinvariantistheaylidiretedgraph
whosenodesaretheharateristigroupsofthenonnullregularD-lassesofS.
Theedgesorrespondtothepartialorder
J
betweentheseD-lasses.Weall
itthesyntatigraphofthesoshift. Theresultanbeextended tothease
ofreduiblesoshifts.
TheproofoftheinvariantisbasedonNasu'sClassiationTheoremforso
shifts[15℄thatextendsWilliam'soneforshiftsofnitetype.Thistheoremsays
that twoirreduible so shiftsX;Y are onjugateifand onlyif there isase-
queneoftransitionmatriesofrightFisheroversA = A
0
; A
1
;:::;A
l 1
;A
l
=
B,suhthatA ;A areelementarystrongshiftequivalentfor1il,where
spetively.This meansthatthere aretransitionmatriesU
i
;V
i
suh that,after
reodingthealphabetsofA
i 1 andA
i
,wehaveA
i 1
=U
i V
i andA
i
=V
i U
i .A
bipartiteshiftisassoiatedinanaturalwaytoapairofelementarystrongshift
equivalentandirreduiblesoshifts[15℄.
The key point in our invariant is the fat that an elementary strong shift
equivalene relation between transition matries implies someonjugay rela-
tionsbetweentheidempotentsinthesyntatisemigroupofthebipartiteshift.
Weshowthatpartiularlassesofirreduiblesoshiftsanbeharaterized
withthissyntatiinvariant:thelassofirreduibleshiftsofnitetypeandthe
lassofirreduibleaperiodisoshifts.
Basidenitions relatedto symbolidynamis aregiven in Setion2.1. We
refer to [13℄ or [9℄ for more details. See also [10℄, [11℄, [4℄ about so shifts.
Basidenitionsandpropertiesrelatedtonite semigroupsandtheirstruture
are given Setion 2.2. We refer to [16, Chapter 3℄ for a more omprehensive
expository. Nasu'sClassiation Theorem is realledin Setion 2.3. We dene
andproveourinvariantinSetion3.Aomparisonofthissyntatiinvariantto
somewell knownotherones isgivenin Setion4.ProofsofPropositions1and
2areomitted.Theextensiontotheaseofreduibleso shiftsisdisussed at
theendofSetion3.
2 Denitions and bakground
2.1 So shifts and their presentations
LetAbeanitealphabet,i.e.anitesetofsymbols.Theshiftmap:A Z
!A Z
isdenedby((a
i )
i2Z )=(a
i+1 )
i2Z ,for(a
i )
i2Z 2A
Z
.IfA Z
isendowedwiththe
produttopologyofthedisretetopologyonA,asubshiftisalosed-invariant
subsetofA Z
.
IfX isasubshiftof A Z
andnapositiveinteger,thenthhigherpower ofX
isthesubshiftof(A n
) Z
denedbyX n
=f(a
in
;:::;a
in+n 1 )
i2Z j(a
i )
i2Z 2Xg.
A nite automaton is a nite multigraph labeledon A. It is denoted A =
(Q;E), where Qis anite set of states,and E anite set of edgeslabeledon
A.Itisequivalentto asymboliadjaeny (QQ)-matrix A,whereA
pq isthe
nite formal sumofthe labelsof alltheedges from pto q. A so shift isthe
setofthelabelsofallthebi-innitepathsonaniteautomaton.IfAisanite
automaton,wedenotebyX
A
thesoshiftdenedbytheautomatonA.Several
automata andenethe sameso shift. Theyarealsoalled presentations or
overs of theso shift.Wewillassumethat allpresentationsareessential:all
states haveat least one outgoingedge and one inoming edge. An automaton
is deterministi if for any given state and any given symbol, there is at most
oneoutgoingedgelabeledwiththisgivensymbol.Asoshiftisirreduible ifit
hasapresentationwithastronglyonnetedgraph.Irreduiblesoshiftshave
aunique(up toisomorphismsof automata)minimaldeterministipresentation
phabet A. Eah nite word w of A
denes a partial funtion from Q to Q.
This funtion sends thestate pto thestate q, ifw is thelabel of apathform
pto q. Thesemigroup generated byall these funtions is alled the transition
semigroup of theautomaton. WhenX
A
is notthefull shift, thesemigrouphas
anullelement,denoted 0,whih orrespondsto wordswhih arenotfatorsof
anybi-innitewordofX
A
. Thesyntati semigroup of anirreduible soshift
isdened asthetransitionsemigroupofitsrightFisherover.
Example 1. ThesoshiftpresentedbytheautomatonofFigure1isalledthe
evenshift.Itssyntatisemigroupisdenedbythetableintherightpartofthe
gure.
1 2
b
b a
1 2
a 1
b 2 1
ab 2
ba 1
bb 1 2
bab 2
aba :
Fig.1.TherightFisheroveroftheevenshiftanditssyntatisemigroup.Sineaa
andadene thesamepartialfuntionfromQtoQ,wewriteaa=ainthesyntati
semigroup. Wealsohaveaba=0,or ab 2k +1
a=0for anynonnegativeintegerk.The
wordbbistheidentityinthissemigroup.
2.2 Struture ofnite semigroups
Wereferto [16℄formoredetails aboutthenotionsdened inthis setion.
GivenasemigroupS,wedenotebyS 1
thefollowingmonoid:ifSisamonoid,
S 1
=S.IfS is notamonoid, S 1
=S[f1gtogetherwith thelawdened by
xy=xyifx;y2S and1x=x1=x foreahx2S 1
.
WerealltheGreen'srelationswhiharefundamentalsequivalenerelations
denedinasemigroupS.ThefourequivalenerelationsR,L,H ,J aredened
asfollows.Letx;y2S,
xRy,xS 1
=yS 1
;
xLy,S 1
x=S 1
y;
xJy,S 1
xS 1
=S 1
yS 1
;
xDy,9z2S xRz andzLy:
InanitesemigroupJ =D.Wereallthedenition ofthequasi-order
J :
x
J y,S
1
xS 1
S 1
yS 1
:
AnR-lassisanequivalenelassforarelationR(similarnotationsholdforthe
otherGreen'srelations).Anidempotent isanelemente2S suhthatee=e.A
regularlassisalassontaininganidempotent.InaregularD-lass,anyH -lass
ontaining an idempotentis a maximal subgroup of thesemigroup. Moreover,
tworegularH -lassesontainedinasameD-lassareisomorphi(asgroups),see
for instane [16, Proposition 1.8℄.This groupis alled the harateristi group
oftheregularD-lass.Thequasi-order
J
induesapartialorderbetweenthe
D-lasses(stilldenoted
J
).ThestrutureofthetransitionsemigroupSisoften
desribedbythesoalled\egg-box"pituresoftheD-lasses.
Wesaythattwoelementsx;y2S areonjugateifthereareelementsu;v2
S 1
suh that x = uv and y =vu. Twoidempotents belong to a sameregular
D-lassifandonlyiftheyareonjugate,seeforinstane [16,Proposition1.12℄.
LetSbeatransitionsemigroupofanautomatonA=(Q;E)andx2S.The
rank of x is the ardinalof theimage of x asa partial funtion from Q to Q.
Thekernel ofx isthepartition induedbytheequivalenerelationoverthe
domainof xwherepqifandonlyp;qhavethesameimage byx.Thekernel
of x is thus apartition of the domain of x. We desribe the egg-boxpitures
withExample1ontinuedinFigure2.
12
1=2 b
b 2
1 2
1
a ab
2ba
bab
0
Fig.2.Thesyntatisemigroupofthe evenshift ofExample1is omposed ofthree
D-lassesD1,D2,D3,ofrank2,1and0,respetively,representedbytheabovetables
fromlefttoright.EahsquareinatablerepresentsanH-lass.Eahrowrepresentsan
R-lassandeaholumnanL-lass.Theommonkerneloftheelementsineahrowis
writtenontheleftofeahrow.Theommonimageoftheelementsineaholumnis
writtenaboveeaholumn.Idempotentsaremarkedwiththesymbol.EahD-lass
of this semigroupis regular. Theharateristi groups of D1,D2,D3 are Z=2Z,the
trivialgroupZ=ZandZ=Z,respetively.
LetX beanirreduiblesoshiftandS itssyntatisemigroup.Itisknown
that S has a uniqueD-lass of rank1 whih is regular (see [4℄ or[5℄, see also
[8℄).
Wedeneanitediretedayligraph(DAG)assoiatedwithXasfollows.
rankoftheD-lassanditsharateristigroup.Thereisanedgefromthevertex
assoiatedwithaD-lassDtothevertexassoiatedwithaD-lassD 0
ifandonly
if D 0
J
D. We all this ayligraphthesyntati graph of X (see Figure3
for an example). Note that the regular D-lass of nullrank, if there is one, is
nottakenintoaountinasyntatigraph.Thisislinkedtothefatthatafull
shift(i.e. theset of allbi-innite wordsonanite alphabet)anbeonjugate
to anonfullshift.
rank2,Z=2Z
rank1,Z=Z
Fig.3.ThesyntatigraphoftheevenshiftofExample1.WehaveD2J D1 sine,
forinstane,S 1
abS 1
S 1
bS 1
.
2.3 Nasu's ClassiationTheorem for soshifts
Inthis setion,wereallNasu's ClassiationTheorem forsoshifts[15℄ (see
also [13, p. 232℄),whih extends William's ClassiationTheorem for shiftsof
nitetype(see[13,p. 229℄).
LetX A Z
;Y B Z
betwosubshifts andm;abenonnegativeintegers.A
map:X !Y isa(m;a)-blokmap (or(m;a)-fatormap)ifthereisamapÆ:
A m+a+1
!Bsuhthat((a
i )
i2Z )=(b
i )
i2Z
whereÆ(a
i m :::a
i 1 a
i a
i+1 :::a
i+a )
=b
i
.Ablokmap isa(m;a)-blokmap forsomenonnegativeintegersm;a.The
wellknowntheoremofCurtis,Hedlundand Lyndon[7℄assertsthatontinuous
andshift-ommutingmapsareexatlyblokmaps.Aonjugay isaone-to-one
andontoblokmap(then,beingashiftompat,itsinverseisalsoablokmap).
Let A be a symboli adjaeny (QQ)-matrix of an automaton A with
entriesinanitealphabetA.LetBbeanitealphabetandf aone-to-onemap
from AtoB.Themap f isextendedto amorphismfrom niteformalsumsof
elementsofAto niteformalsumsof elementsofB.Wesaythatf transforms
Ainto anadjaeny(QQ)-matrixB ifB
pq
=f(A
pq ).
Wenowdene thenotionof strongshift equivalenebetweentwosymboli
adjaenymatries.
LetAandB betwonitealphabets. Wedenote byAB theset ofwordsab
witha2Aandb2B.
TwosymboliadjaenymatriesA,withentriesinA,andB,withentriesin
thereisaone-to-onemapfromAtoUVwhihtransformsAintoUV,andthere
isaone-to-onemapfromBto VU whih transformsB into VU.
TwosymboliadjaenymatriesAandB arestrongshiftequivalentwithin
rightFisheroversifthereisasequeneofsymboliadjaenymatriesofright
Fisherovers
A=A
0
;A
1
;:::;A
l 1
;A
l
=B
suh that for1i lthe matriesA
i 1
tand A
i
areelementary strongshift
equivalent.
Theorem1 (Nasu). Let X and Y be irreduible so shifts and let A and
B be the symboli adjaeny matries of the right Fisher oversof X and Y,
respetively.ThenX andY areonjugateifandonlyifAandB arestrongshift
equivalent withinright Fisherovers.
Example 2. Letusonsiderthetwo(onjugate)irreduiblesoshiftsX andY
dened bytherightFisheroversA=(Q;E)and B=(Q 0
;E 0
)inFigure4.
1 2
b
b a
2 0
3 0 1
0 a
0
b 0
b 0
d 0
0
Fig.4.TwoonjugateshiftsX andY.
Thesymboliadjaenymatriesoftheseautomataarerespetively
A=
ab
b 0
; B= 2
4 a
0
0d 0
0
0 b 0
0 b 0
0 3
5
:
ThenAandB are elementarystrongshiftequivalentwith
U =
u
1 0 u
2
0 u
2 0
; V = 2
4 v
1 0
v
2 0
0 v 3
5
:
UV =
u
1 v
1 u
2 v
2
u
2 v
2 0
; VU = 2
4 v
1 u
1 0 v
1 u
2
v
2 u
1 0 v
2 u
2
0 v
2 u
2 0
3
5
:
Theone-to-onemaps fromA=fa;bgto UV andfrom B=fa 0
;b 0
; 0
;d 0
gto VU
aredesribedin thetables below.
au
1 v
1
b u
2 v
2
; a
0
v
1 u
1
b 0
v
2 u
2
0
v
2 u
1
d 0
v
1 u
2 :
Anelementarystrongshiftequivaleneenablestheonstrutionofanirreduible
soshiftZ onthealphabetU[Vasfollows.ThesoshiftZisdenedbythe
automatonC=(Q[Q 0
;F),wherethesymboli adjaenymatrixC ofCis
QQ 0
Q
Q 0
0 U
V 0
:
TheshiftZ isalled thebipartite shift denedbyU;V (seeFigure 5).An edge
ofClabeledonU goesfromastateinQtoastateinQ 0
.An edgeofClabeled
on V goesfrom astate in Q 0
to astate in Q. Remark that the seond higher
powerofZ isthedisjointunionofX andY.Notealsothat CisarightFisher
over(i.e.isminimal).
1
1 0
2 3
0
2 0
u2 v2
u2 v2
u1 v1
Fig.5.ThebipartiteshiftZ.
3 A syntati invariant
In thissetion, wedene asyntatiinvariant forthe onjugayof irreduible
soshifts.
Theorem2. LetX andY betwo irreduible so shifts. If X andY areon-
LetX (respetivelyY)beanirreduiblesoshiftwhosesymboliadjaeny
matrix of its right Fisher overis a(QQ)-matrix (respetively (Q 0
Q 0
)-
matrix)denotedbyA(respetivelybyB).WeassumethatAandBareelemen-
tarystrongshiftequivalentthroughapairofmatries(U;V).Theorresponding
alphabetsare denoted A, B, U, and V asbefore. Wedenote byf aone-to-one
map from A to UV whih transforms A into UV and by g a one-to-one map
from B toVU whihtransformsB intoVU. LetZ bethebipartiteirreduible
so shiftassoiatedtoU;V.WedenotebyS (respetivelyT,R )thesyntati
semigroupofX (respetivelyY,Z).
Letw 2 R . If w is non null,the bipartite nature of Z implies that w is a
funtionfromQ[Q 0
toQ[Q 0
whosedomainisinludedeitherinQorinQ 0
,and
whose imageisinluded eitherin Qorin Q 0
.Ifw6=0with adomaininluded
in P andanimage inludedin P 0
, wesaythat whasthetype(P;P 0
). Remark
that w hastype(Q;Q)if andonly ifw6=0andw2(f(A))
, and whastype
(Q 0
;Q 0
)ifandonlyifw6=0andw2(g(B))
.
Lemma1. Elements ofR inasame nonnullH -lasshave the sametype.
Proof We show the property for the (Q;Q)-type. Let w 2 H and w of type
(Q;Q). If w = w 0
v with w 0
;v 2 R , then w 0
has type (Q;). If w = zw 0
with
z;w 0
2R ,thenw 0
hastype(;Q).Thus,wH w 0
impliesthatw 0
hastype(Q;Q).
TheH -lassesofRontainingelementsoftype(Q;Q)(respetively(Q 0
;Q 0
))
arealled(Q;Q)-H -lasses(respetively(Q 0
;Q 0
)-H -lasses).
Letw =a
1 :::a
n
bean element ofS,wedene theelementf(w) asf(a
1 )
::: f(a
n
).Notethat this denitionis onsistentsineifa
1 :::a
n
=a 0
1 :::a
0
m in
S, then f(a
1
):::f(a
n
)=f(a 0
1
):::f(a 0
m
) in R .Similarly we dene an element
g(w)foranyelementwofT.
Conversely, letw bean elementof R belonging to f(A)
((UV)
). Then
w = f(a
1
):::f(a
n
), with a
i
2 A. We dene f 1
(w) asa
1 :::a
n
. Similarly we
deneg 1
(w).Againthesedenitionsandnotationsareonsistent.Thusf isa
semigroupisomorphismfromStothesubsemigroupofRoftransitionfuntions
dened by thewordsin (f(A))
. Notie that f(0) =0 if 02 S. Analogously,
g is a semigroup isomorphism from T to the subsemigroup of R of transition
funtions denedbythewordsin (g(B))
.
Lemma2. Let w;w 0
2 R of type (Q;Q). Then wH w 0
in R if and only if
f 1
(w)H f 1
(w 0
)inS.
Proof Let w =f(a
1
):::f(a
n
) and w 0
=f(a 0
1
):::f(a 0
m
),with a
i
;a 0
j
2A. We
have w = w 0
v with v 2 R if and only if v = f( a
1
):::f( a
r
) with a
i
2 A and
f(a
1
):::f(a
n )=f(a
0
1
):::f(a 0
m )f( a
1
):::f( a
r
).Thisisequivalenttoa
1 :::a
n
=
a 0
1 :::a
0
m a
1 :::a
r
, that is f 1
(w)R 1
f 1
(w 0
)R 1
. Analogously, wehave w 0
=
0 0 1 0 1 1 1 0
in R if and onlyiff (w)Rf (w)in S. In thesameway, oneanprovethe
samestatementfortherelationL andhenefortherelationH .
Asimilarstatementholdsfor(Q 0
;Q 0
)-H -lasses.
Lemma3. Let w;w 0
2 R of type (Q;Q). Then w
J w
0
in R if and only if
f 1
(w)
J f
1
(w 0
) in S.This implies that wJw 0
in R if and only if f 1
(w)
J f 1
(w 0
) inS.
Proof Therststatementanbeproovedasinthepreviouslemma.
Similar resultshold betweenT and R . As aonsequene we get the following
lemma.
Lemma4. Thebijetionf betweenSandtheelementsofRbelongingto(f(A))
,
induesabijetionbetweenthenonnullH -lassesofS andthe(Q;Q)-H -lasses
of R .Moreoverthis bijetion keepsthe relationsJ,
J
andthe rankof the H -
lasses.
Asimilarstatementholdsforthebijetiong.
We now ome to the main lemma, whih shows the link between the ele-
mentary strong shift equivalene of the symboli adjaeny matries and the
onjugay ofsomeidempotents in thesemigroup. Thislink is the keypoint of
theinvariant.
Lemma5. Let H be a regular (Q;Q)-H -lass of R . Then there is a regular
(Q 0
;Q 0
)-H -lassinthe sameD-lassasH.
Proof Lete2Rbeanidempotentelementoftype(Q;Q).Letu
1 v
1 :::u
n v
n in
(UV)
suhthate=u
1 v
1 :::u
n v
n
.Wedenee=v
1 :::u
n v
n u
1
.Thuseu
1
=u
1 e
inR .Remarkthatedependsonthehoieofthewordu
1 v
1 :::u
n v
n
representing
einR .
Ifw denotes v
1 :::u
n v
n
and v denotes u
1
, wehave e=vw ande=wv. It
followsthat eandeareonjugate,thuse 2
=eande 2
areonjugate.Moreover
e 3
=wvwvwv =weev=wev=wvwv=e 2
:
Thus e 2
is an idempotent onjugate to the idempotent e. As aonsequene e
and e 2
belong to asame D-lass of R (see Setion 2), and e 2
6= 0.The result
followssinee 2
isof type(Q 0
;Q 0
).
Note that the number of regular (Q;Q)-H -lasses and the number of regular
(Q 0
;Q 0
)-H -lassesinasameD-lassofR ,maybedierentin general.
WenowproveTheorem 2.
Proof[ofTheorem 2℄ByNasu's Theorem [15℄ weanassume, withoutloss of
generality, that the symboliadjaeny matries of theright Fisheroversof
respetively.
Let D be a non null regular D-lass of S. Let H be a regular H -lass of
S ontained in D. Let H 00
= f(H). By Lemma 4, the groups H and H 00
are
isomorphi. Let D 00
theD-lass of R ontainingH 00
. By Lemma 5, there is at
least one regular (Q 0
;Q 0
)-H -lass K 00
in D 00
, whih is isomorphi to H 00
. Let
H 0
=g 1
(K 00
)andletD 0
betheD-lassof T ontainingH 0
.ByLemma4,the
groupsH 0
andK 00
areisomorphi.HenethegroupsH andH 0
areisomorphi.
ByLemmas4and5,wehavethattheaboveonstrutionofD 0
fromDisa
bijetivefuntion 'from thenon nullregularD-lassesofS onto thenonnull
regularD-lassesofT.Moreovertheharateristigroupof D isisomorphito
theharateristigroupof'(D)and,byLemma4,therankofDisequaltothe
rankof'(D).
WenowonsidertwononnullregularD-lassesD
1 andD
2
ofS.ByLemma4
and Lemma 5, D
1
J D
2
if andonly if '(D
1 )
J '(D
2
). It follows that the
syntatigraphsofS andT areisomorphithroughthebijetion'.
Nasu'sClassiation Theorem holds for reduible so shifts bythe use of
right Krieger overs instead of right Fisher overs [15℄. This enables the ex-
tension ofour resultto the aseof reduible so shifts. This extensionis not
desribedin thisshortversionofthepaper.
4 How dynami is this invariant?
We briey omparethe syntati onjugayinvariant withother lassial on-
jugay invariants. We refer to [13℄ for the denitions and properties of these
lassialinvariants.
First, on an remark that the syntatiinvariant does not apture all the
dynami. Two so shifts an have the same syntati graph and a dierent
entropy,seetheexamplegiveninFigure 6.
1 2
b
b a
1 2
a
b
b
Fig.6.ThetwoabovesoshiftsX;Y havethesamesyntatigraphandadierent
entropy.Indeed, we have b= inthe syntatisemigroupof Y.Henethe shiftsX
zetafuntionofashiftX is(X)=exp P
n1 p
n z
n
n
,wherep
n
isthenumberof
bi-innite words x 2X suhthat n
(x)=x. Wegive in Figure 7anexample
of two irreduible so shifts whih havethe same zetafuntion and dierent
syntatigraphs.
Irreduible shiftsof nite type anbeharaterizedwith this syntatiin-
variant.Other equivalentharaterizationsofnite typeshiftsanbefoundin
[14℄andin [8℄.
Proposition1. An irreduible so shifts isof nite type if andonly its syn-
tatigraphisreduedtoonenode ofrank1representingthe trivialgroup.
Anotherinterestinglassofirreduiblesoshiftsanbeharaterizedwith
thesyntatiinvariant.Itisthelassofaperiodisoshifts[1℄.
Let x 2 X, we denote by period(x) the least positive integer n suh that
n
(x)=x ifsuhanintegerexists.Itisequalto1otherwise.
LetX;Y betwosubshifts andlet :X !Y bea blok map. Themap is
said aperiodi if period(x) = period((x)) for any x 2 X. Roughly speaking,
suhafatormapdoesnotmakeperiodsderease.
A so shiftX ifaperiodi if it is theimage of ashift of nite typeby an
aperiodi blok map. A haraterization of irreduible aperiodi so shifts is
thefollowing.
Proposition2. Anirreduiblesoshiftisaperiodiifandonlyifitssyntati
graphontainsonlytrivial groups.
Shutzenberger's haraterization of aperiodi languages(see for instane [16,
Theorem2.1℄)assertsthatthesetofbloksofanaperiodisoshiftisaregular
starfreelanguage.
1 2
a
a
b b
x
y
1 2
a
d b
x
y
Fig.7. Two so shifts X;Y whih have the same zeta funtion 1
1 4z+z 2
(see for
instane [13, Theorem 6.4.8℄, or [2℄ for the omputation of the zeta funtion of a
so shift), and dierent syntati invariants. Indeed the syntati graph of X is
(rank2 ;Z=2Z) ! (rank1 ;Z=Z) while the syntati graph of Y has only one node
(rank1 ;Z=Z).Thustheyarenotonjugate.NotiethatY isashiftofnitetype.
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