On half-synchronized systems

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by  Somayyeh Jangjooye Shaldehi

ON HALF-SYNCHRONIZED SYSTEMS
6 S.JANGJOOYESHALDEHI
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Abstract. Asubclassofcodedsystemscontainingsynchronizedsystems
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isthefamilyofhalf-synchronizedsystems. Inthisnote, wewillconsider
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them and show that the property ‘half-synchronized’ lifts under hyper-
a
bolicmaps. Thisenablesustodefineanequivalencerelationonthesetof
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half-synchronized systems. Also, when the domain is half-synchronized,
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weshowthatright-closinga.e.,1-1a.e. factormapshaveastrongdecod-
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ingproperty.

]
S
D
. 1. Introduction
h
at CodedsystemsweredefinedbyBlanchardandHansel[1]asageneralization
m of sofic shifts. Amongst subshifts, a well-known subclass of coded systems is
[ the family of synchronized systems. The purpose of this paper is to generalize

parts of the synchronized theory to certain subclass of coded systems, the so-
1
v calledhalf-synchronizedsystems. Thehalf-synchronizedsystemsoftenserveas
2 alandmark within the codedsystems, showingthe difficulties in extending the
0
class of synchronized systems and keeping a satisfactory analogon of the sofic
2
4 shifts theory.
0 The study of conjugacy is of interest in symbolic dynamics. But there is no
.
1 generalalgorithm for deciding whether two shift spaces are conjugate. Thus it
0 makes sense to ask if there is an equivalence relation, weaker than conjugacy,
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onsomesubclassesofsubshifts. Theinvestigationfortheexistanceofcommon
1
: extensions where the factor maps have certain properties in common is of in-
v
terestincodedsystemsandhasalonghistory. Amongsttheproperties,weare
i
X
interestedinhyperbolicity. Commonextensionswithhyperbolicmapsleadtoa
r versionof almostconjugacy for codedsystems. Fiebig in[3] showsthathaving
a
acommoncoded(orsynchronized)hyperbolicextensionisanequivalencerela-
tiononthe setofcoded(orsynchronized)systems. InSection3,firstwe prove
that ‘being half-synchronized’ is an invariant for hyperbolic maps (Theorem
3.3). Then, We showthat a commonhalf-synchronizedhyperbolic extensionis
anequivalence relationonthe set of half-synchronizedsystems (Theorem 3.4).
Closing maps are important in the coding theory [6, 7]. They also have a
naturaldescriptioninhyperbolicdynamics[2]: right-closing(resp. left-closing)
2010 Mathematics Subject Classification. 37B10,37-XX.
Key words and phrases. synchronized,coded, hyperbolic,right-closing,decoder block.
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2 S.JANGJOOYESHALDEHI
maps are injective on unstable (resp. stable) sets. When the domain is a syn-
chronizedsystem,right-closinga.e.,1-1a.e. factormapshaveastrongdecoding
property[4]. Thispropertygivesfinitary regularisomorphismbetweengeneral
shiftspaces. Insection4,wegeneralizethisresulttohalf-synchronizedsystems
(Theorem 4.2).
2. Background and Notations
Z
LetAbeanon-emptyfiniteset. ThefullA-shift denotedbyA ,isthesetof
allbi-infinite sequences of symbols fromA. A block overA is a finite sequence
ofsymbolsfromA. The shift map onAZ is the mapσ whereσ({x })={y }is
i i
definedbyy =x . Thepair(AZ, σ)isthefull shift andanyclosedinvariant
i i+1
subset of that is called a shift space.
DenotebyB (X)thesetofalladmissiblen-blocksandletB(X)= ∞ B (X)
n n=0 n
be the language of X. For u ∈ B(X), let the cylinder [u] be the setS{x ∈ X :
x =u}.
[l,l+|u|−1]
SupposeX isasubshiftoveralphabetA. Form, n∈Zwith−m≤n,define
the (m+n+1)-block map ϕ:B (X)→D by
m+n+1
(2.1) y =ϕ(x x ...x )=ϕ(x )
i i−m i−m+1 i+n [i−m,i+n]
wherey isasymbolinalphabetD. Themapϕ=ϕ[−m,n] :X →DZ definedby
i ∞
y =ϕ(x) with y given by 2.1 is called the sliding block code (or code) induced
i
by ϕ. So there is N ≥ 0 such that ϕ(x) is determined by x . We call
0 [−N,N]
2N+1acodinglengthforϕ. Ifm=n=0,thenϕis1-blockcode andϕ=ϕ .
∞
An onto (resp. invertible) code is called a factor map (resp. conjugacy).
Apointx∈X isdoubly transitive ifeveryblockinX appearsinxinfinitely
oftento the left andto the right. Let ϕ:X →Y be a factor map. If there is a
positiveintegerdsuchthateverydoublytransitivepointofY hasdpre-images,
then we call d the degree of ϕ.
LetGbeadirectedgraphandV (resp. E)thesetofitsvertices(resp. edges)
which is supposed to be countable. An edge shift, denoted by X , is a shift
G
space which consist of all bi-infinite sequences of edges from E.
A labeled graph G is a pair (G,L) where G is a graph and L : E → A its
labeling. Associated to G, a space
X =closure{L (ξ):ξ ∈X }=L (X )
G ∞ G ∞ G
is defined and G is called a cover of X . When G is a finite graph, X =
G G
L (X ) is a sofic shift.
∞ G
A block v ∈ B(X) is called synchronizing if whenever uv, vw ∈ B(X), we
haveuvw∈B(X). An irreduciblesubshift X is a synchronized system if it has
a synchronizing block.
A coded system is the closure of the set of sequences obtained by freely
concatenating the blocks in a list of blocks.
ON HALF-SYNCHRONIZED SYSTEMS 3
AlabeledgraphG =(G,L)iscalledright-resolving ifforeachvertexI ofG,
the edges starting at I carry different labels. A minimal right-resolving cover
ofasoficshift X is a right-resolvingcoverhavingthe fewestverticesamongall
right-resolvingcoversofX. Itisuniqueuptoisomorphism[7,Theorem3.3.18]
and is called the Fischer cover of X.
For x∈B(X), call x− =(xi)i<0 (resp. x+ =(xi)i∈Z+) the left (resp. right)
infinite X-ray and let X+ = {x : x ∈ X}. The follower set of x is defined
+ −
as ω (x )={x ∈X+ : x x is a point in X}.
+ − + − +
3. One equivalence relation for half-synchronized systems
We prove that hyperbolic maps lift the property half-synchronized. We use
this to show that common half-synchronized hyperbolic extensions define an
equivalence relation on the set of half-synchronized systems.
Definition3.1. AtransitivesubshiftX ishalf-synchronized, ifthereisablock
m ∈ B(X) and a left-transitive point x ∈ X such that x = m and
[−|m|+1,0]
ω (x )=ω (m). Then, m is called a half-synchronizing block for X.
+ (−∞,0] +
Dyck shift and synchronized systems are half-synchronized. Now we review
the concept of the Fischer cover for a half-synchronized system [5]. Let the
collection of all follower sets ω (x ) be the set of vertices of a graph X+.
+ −
There is an edge from I to I labeled a if and only if there is a X-ray x
1 2 −
suchthatx a is a X-rayandI =ω (x ), I =ω (x a). This labeledgraph
− 1 + − 2 + −
is called the Krieger cover for X. If X is a half-synchronized system with
half-synchronizing block α, the irreducible component of the Krieger cover
containing the vertex ω (α) is called the Fischer cover of X.
+
Thefollowingdefinitionismotivatedbythenotionofhyperbolichomeomor-
phism between the sets of doubly transitive points introduced in [8].
Definition3.2. LetX andY betransitivesubshifts. Thefactormapϕ:X →
Y is hyperbolic if there is a d∈N and a block w and d blocks
m(1),m(2),...,m(d) ∈B (X),
2k+1
such that
(1) if y ∈Y such that y =w, then
[−n,n]
ϕ−1(y) ={m(1),m(2),··· ,m(d)}.
[−k,k]
(2) if w′ ∈ B(Y) beginning and ending with w, then for each 1 ≤ i ≤ d
there is a unique block ai ∈ B(X), such that for any x ∈ X with
ϕ(x) =w′ and x =m(i), it holds x =ai.
[−n,n+p] [−k,k] [−k,k+p]
Thenexttheoremsaythat[3,Theorem4.2]thatwerestatedforsynchronized
systemsareactuallyvalidforhalf-synchronizedsystemsaswell. Infact,itsays
that the property ‘half-synchronized’ is weak enough to lift under hyperbolic
maps.
4 S.JANGJOOYESHALDEHI
Theorem3.3. LetX andY betransitivesubshiftsandϕ:X →Y ahyperbolic
factor map. Then, X is half-synchronized if and only if Y is half-synchronized.
Proof. FirstsupposeX ishalf-synchronizedandG =(G,L)istheFischercover
of X. By [5, Theorem 1.4], G has residual image in the one-sided shift. Since
ϕ = Φ has a degree [3, Theorem 3.2], (G,Φ◦L) has residual image in Y+.
∞
So Y is half-synchronized.
Now let Y be half-synchronized. Since ϕ is hyperbolic, it has a degree ℓ
[3, Theorem 3.2]. Hyperbolicity of ϕ implies that the transitive left ray y
−
does not have less than ℓ preimages. On the other hand, if it has more than
ℓ preimages, then a prolongation of y gives a point in Y with more than ℓ
−
preimages. So y has ℓ preimages.
−
Let w ∈ B(Y) and m(1),m(2),··· ,m(d) ∈ B(X) satisfying Definition 3.2.
Without loss of generality, we may assume that ϕ is 1-block and w is half-
synchronizing block. So there is a transitive left ray y terminating at w
−
such that ω (w) = ω (y ). Let m ∈ B (X) such that Φ(m) = w and
+ + − 2n+1
x ,x ,··· ,x be ℓ preimages for y . Then, since ϕ has a degree, x is
1− 2− ℓ− − 1−
transitive[3,Theorem3.2]. Wemayassumethatm=m(1) andx terminates
1−
at m. Now we show that m is a half-synchronizing block for X.
Suppose mu ∈ B(X) and prolongate mu such that muvm ∈ B(X). Then,
wΦ(uv)w ∈ B(Y) and since w is half-synchronizing, Φ(uv)w ∈ ω (y ). Now
+ −
y Φ(uv)walsohasℓpreimagesx u for1≤i≤ℓ. (2)ofDefinition3.2implies
− i− i
that u =uv. So u∈ω (x ).
1 + 1−
(cid:3)
Theorem3.3enablesustointroduceanequivalencerelationusinghyperbolic
factor codes on the set of half-synchronized systems.
Theorem 3.4. Having a common hyperbolic extension is an equivalence rela-
tion on the half-synchronized systems.
Proof. LetX andY (resp. Y andZ)haveacommonhalf-synchronizedhyper-
bolic extension (V,ϕ ,ϕ ) (resp. (W,ϕ′ ,ϕ′ )). Suppose (Σ,ψ ,ψ′ ) is the
X Y Y Z V W
fiberproductof(ϕ ,ϕ′ )andΓisanirreduciblecomponentofΣsuchthatthe
Y Y
restriction of ψ and ψ′ to Γ are onto. Since ϕ is hyperbolic, ψ′ :Γ→W
V W Y W
is hyperbolic and so by Theorem 3.3, Γ is half-synchronized. The result then
followsfromthe fact thatcompositions ofhyperbolic maps arehyperbolic. (cid:3)
4. The existence of decoder block
A factor code ϕ:X →Y is 1-1 a.e. if any doubly transitive point in Y has
exactly one preimage. It is right-closing almost everywhere if there is n ∈ N
such that for any two left-transitive points x,y ∈ X with x = y
(−∞,0] (−∞,0]
and ϕ(x) =ϕ(y) , it holds x =y . Then, ϕ is called n-step right-
(−∞,n] (−∞,n] 1 1
closing a.e..
ON HALF-SYNCHRONIZED SYSTEMS 5
Definition 4.1. Let ϕ : X → Y be a factor map between arbitrary subshifts
X and Y. A block w ∈ B(Y) is a decoder block for ϕ if there is k ∈ N,
the anticipation of w, such that for all n ∈ N and all points x,y ∈ X with
ϕ(x) = ϕ(y) and ϕ(x) = ϕ(y) , it holds that
[−|w|+1,0] [−|w|+1,0] [1,n+k] [1,n+k]
x =y .
[1,n] [1,n]
Remind that if ϕ : X → Y is a factor map with a decoder block and µ an
ergodic measure on X with full support, then ϕ : (X,µ) → (Y,ν) is a finitary
regular isomorphism, where ν =µ◦ϕ−1 [4].
Fiebig shows that when the domain is a synchronized system, right-closing
a.e., 1-1 a.e. factor maps have a strong decoding property [4]. We generalize
this property to half-synchronized systems. The ingredients for the proof is
almost similar to [4, Theorem 2.3].
Theorem 4.2. Let X be a half-synchronized system and ϕ : X → Y a factor
map. Then, ϕis right-closing a.e., 1-1 a.e. ifand onlyif ithas adecoder block.
Proof. Supposeϕisk-stepright-closinga.e. andm∈B(X)isahalf-synchronizing
block with length greater than a coding length for ϕ. Let z ∈ X be a doubly
transitive point satisfying Definition 3.1. By compactness combined with the
fact that ϕ is 1-1 a.e., there is p ∈ N such that for x ∈ X with ϕ(x) =
[−p,p]
ϕ(z) it holds that x = m. We claim that w := ϕ(z) is a
[−p,p] [−|m|+1,0] [−p,p]
decoder block with anticipation k.
Let x,y ∈ X with ϕ(x) = ϕ(y) = w and ϕ(x) =
[−|w|+1,0] [−|w|+1,0] [1,n+k]
ϕ(y) . Then, x = y = m. Since m is half-
[1,n+k] [−p−|m|+1,−p] [−p−|m|+1,−p]
synchronizing, there are points x′,y′ ∈X as,
z i≤−p,
x′i =(cid:26) xii+p i≥−p+1, yi′ =(cid:26) yzi+p ii≤≥−−pp,+1.
i
x′ and y′ are left transitive and ϕ(x′) = ϕ(y′) . Since ϕ is
(−∞,n+k] (−∞,n+k]
right-closing a.e., x′ =y′ . So x =y .
[−p+1,n] [−p+1,n] [1,n] [1,n]
Now suppose w is a decoder block with anticipation k and y ∈D(Y). So w
appears in y infinitely often to the left and to the right. Since w is a decoder
block, ϕ is 1-1 a.e.. If y = w for some i ∈ Z, a coordinate j > i
[i−|w|+1,i]
of the preimages of y is determined by y . Therefore, ϕ is k-step
[i−|w|+1,j+k]
right-closing a.e.. (cid:3)
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Faculty of MathematicalSciences, Alzahra University
E-mail address: [email protected]