Thomsen in [10] considers a synchronized component of a general subshift and investigates the approximation of entropy from inside of this synchronized component by some certain shift of finite type’s. In fact, many results in [10] are based on this result. To be more specified, suppose W(X) is the set of admissible words of X and Wn(X) the set of admissible words of length n or so called n-words. Thomsen proves that limk→∞ h(Xk) = hsyn(X) where where m is an arbitrary synchronizing word in W(X) and hsyn(X) is half synchronizing entropy also Xk’s are shift of finite type approaching X from inside. In section (3), we extend a version of this result to half synchronized systems.

Now suppose Per(X) be the set of periodic points of X and set R(X) = Per(X). Also suppose S(X) denote the set of synchronizing words for R(X).

Synchronized systems are well known, however, there are not much results for half synchro-nized systems. In this note, we will consider them and in particular, a natural extension of the concept of hsyn will be given for half synchronized systems denoted by hhsyn and we will show that if h(X) = hhsyn(X) then X is almost sofic.

suppose A be a set of non-empty finite symbols called alphabet. The full A-shift denoted by A , is the collection of all bi-infinite sequences of symbols in A. Equip A with discrete topology and A with product topology. A word over A is a finite sequence of symbols from A. It is convenient to include the sequence of no symbols, called the empty word which is denoted by ε. If n ≥ 1, then u denotes the concatenation of n copies of u, and put u = ε. The shift map σ on the full shift A maps a point x to the point y = σ(x) whose i-th coordinate is yi = xi+1. By our topology, σ is a homeomorphism. Suppose F be the collection of all forbidden words over A. The complement of F is the set of admissible words or just words in X. For a full shift A , define XF to be the subset of sequences in A not containing any word from F. A subshift is a subset X of a full shift A such that X = XF for some collection F of forbidden words.

Suppose Wn(X) denote the set of all admissible n-words. The language of X is the collection W(X) = ∪nWn(X). A subshift X is irreducible if for every ordered pair of words u, v ∈ W(X) there is a word w ∈ W(X) so that uwv ∈ W(X). A subshift X is called a shift of finite type if there is a finite set F of forbidden words such that X = XF .

Suppose G be a graph with edge set E = E(G) and the set of vertices V = V(G). The edge shift XG is the subshift over the alphabet A = E defined by

XG = {ξ = (ξi)i∈Z ∈ E : t(ξi) = i(ξi+1)}.

Each edge e initiates at a vertex denoted by i(e) and terminates at a vertex t(e).

A labeled graph is a pair G = (G, L), where G is a graph with edge set E, and the labeling L : E(G) → A assigns to each edge e of G a label L(e) from the finite alphabet A. For a path π = π0 . . . πk, L(π) = L(π0) . . . L(πk) is the label of π. Suppose L∞(ξ) be the sequence of bi-infinite labels of a bi-infinite path ξ in G and set

XG := {L∞(ξ) : ξ ∈ XG} = L∞(XG).

We say G is a presentation or cover for X = XG. In particular, X is sofic if, for a finite graph G, have X = XG [2, Proposition 3.2.10]. Every shift of finite type is sofic [2, Theorem 3.1.5], but the converse is not true [2, Page 67]. A labeled graph G = (G, L) is right-resolving if for each vertex I of G the edges starting at I carry different labels.

Suppose X be a subshift and w ∈ W(X). The follower set F(w) of w is defined by F(w) = {v ∈ W(X) : wv ∈ W(X)}. Suppose x ∈ X. Then, x+ = (xi)i∈Z+ (resp. x− = (xi)i≤0) is called right (resp. left) infinite X-ray. For a left infinite X-ray, say x−, its follower set is w+(x−) = {x+ ∈ X : x−x+ ∈ X}. Consider the collection of all follower sets w+(x−) as the set of vertices of a graph. There is an edge from I1 to I2 labeled a if and only if there is an X-ray x− such that x−a is an X-ray and I1 = w+(x−), I2 = w+(x−a). This labeled graph is called the Krieger graph for X.

A word m ∈ W(X) is synchronizing if whenever um and mv are in W(X), we have umv ∈ W(X). An irreducible subshift X is synchronized system if it has a synchronizing word, or equivalently, if and only if it admit a countable generating graph G such that L∞(XG) is residual in X [3, Theorem 1.1]. A word m ∈ W(X) is (left) half synchronizing if there is a left transitive point x ∈ X such that x[−|m|+1, 0] = m and w+(x(−∞, 0]) = w+(m), that we denote by

((x, m)). (2.1)

If X is a (left) half synchronized system with (left) half synchronizing m, the irreducible com-ponent of the Krieger graph containing the vertex w+(m) is denoted by X and is called the right Fischer cover of X. If for some m ∈ W(X) there is a unique vertex I such that m ∈ F−(m), then m is called a magic word for the Fischer cover.

Suppose X be a subshift. The entropy of X is defined by

h(X) = lim log |Wn(X)|.

A subshift X is almost sofic if there are sofic shifts Xn ⊆ X such that

lim h(Xn) = h(X).

In this section, we will extend half synchronizing entropy to half synchronized components of a subshift with some reservations. First, Suppose us comment that one may verify that without loss of generality S(X) could be taken to be the set of synchronizing words of maximal irreducible components of R(X). For suppose C be such a component and suppose s be synchronizing in C. Then, there is a word u in C that does not appear in any other component. Now a prolon-gation of s including u is synchronizing in S(X). Suppose X be a subshift and call m a half synchronizing word for X, if m is half synchronizing for an irreducible component of X. (Note that if m were defined for whole system, as in synchronizing case, then X had to have just one component.) Before stating our main proposition, suppose X be a half synchronized system and hence irreducible by definition. Set H(X) to be the set of half synchronizing words of X and fix m ∈ H(X). Suppose ((x, m)) be as in (2.1). Suppose’s assume that set M is equal to set

{a ∈ Wn(X) : ∃ ((x, m)) s.t. mam = x[−|mam|+1, 0]}.

Set h(m, X) := lim sup log |M| .

Next proposition shows that h(m, X) is independent of m.

Theorem 3.1. If m is an arbitrary half synchronizing word for X with X its right Fischer cover, then h(m, X) = h(X ). (3.1)

Proof. Pick a left transitive x ∈ X such that x[−|m|+1, 0] = m and w+(x(−∞, 0]) = w+(m) and set θ := w+(x−) , δ := w+(. . . x−|m|−1x−|m|). Suppose π be a finite path in X such that i(π) = θ, t(π) = δ, L(π) = b. Suppose τ is an arbitrary cycle from θ to θ in X with L(τ) = a. Set z− := x−abm. Then, z is left transitive and w+(z−) = w+(m). Thus the number of cycles of length |τ| from θ to θ is at most |Sn| where

Sn = {u ∈ Wn(X) : ∃ ((x, m)) ∋ mum = x[−|mum|+1, 0]}

and n = |τ| + |b|. This means h(X ) ≤ h(m, X).

Conversely, suppose ϵ > 0 and pick {nk : k ∈ N} such that nk < nk+1 and

h(m, X) − ϵ < lim log |Sn | ≤ h(m, X) (3.2) k nk

where Sn = {a , . . . , a } For each k ∈ N and

k 1 jk

1 ≤ i ≤ jk.

(ik)

Choose ((x , m)) such that

(ik) k

x = mai m.

[−|ma m|+1, 0]

k (ik)

Set θi := w+(x− ). We will have

(ik) k

w+(m) = w+(x− ) = w+(θi )[3, P age146].

k k k k k k

Thus w+(θ ) = w+(θ ) = · · · = w+(θj ) and so θ = θ = · · · = θj = w+(m). Since

1 2 k 1 2 k (ik) k (ik) k + k

w+(x− ai m) = w+(x− ), therefore there are cycles Ci in X labeled ai m and initiating at θ = w+(m). Thus jk is at most as large as the number of cycles of length nk + |m| based at θ = w+(m). Set

1 1 k k

H1 = C1 ∪ . . . ∪ Cj , . . . , Hk = Hk−1 ∪ C1 ∪ . . . ∪ Cj .

Then, limk h(Hk) ≥ limk log |Sn | and so by (3.2), there is k ∈ N such that

nk k

h(m, X) − ϵ < h(Hk). (3.3)

But Hk is a subgraph of X . Thus h(Hk) ≤ h(X ) and therefore by (3.3), h(m, X) − ϵ < h(X ). This means that h(m, X) ≤ h(X ).

Define the half synchronized entropy hhsyn(X) to be

hhsyn(X) := h(m, X), (3.4)

where m is an arbitrary half synchronizing word for X. By the above proposition, (3.4) is well defined. Moreover, if X is synchronized, then hhsyn(X) = hsyn(X).

Theorem 3.2. Suppose X = R(X). Then, for any synchronizing word such as m, there is a unique “magic" vertex in X which is the terminal of any path labeled m. Now if x ∈ R(X) have infinitely many synchronizing words is past and future, then there must be a bi-infinite path labeled x, say πx in X passing through m and the magic vertex.

By the above remark, an equivalent statement is

sup{Ix} (3.5)

where Ix = {inf{(j − i) ≥ 0 : x[i,j] ∈ S(X), πx ∈ X }.

Recall that S(X) is the synchronized word of X.

In the next section, we use a definition similar to (3.5) to extend the results to half-synchronizing case.

In this part we intend to extend synchronizing entropy to half synchronized systems, therefore suppose X be half synchronized system. One may use the same routine as in the synchronized case by applying some adjusments. For instance, for say synchronizing entropy, can be replaced by supi∈Z ix where

ix = {inf{(j − i) ≥ 0 : w+(x(−∞,j]) = w+(x[i,j])}} (3.6)

where again πx ∈ X is a path labeled x. Note that

w+(x(−∞,j]) = w+(x[i,j]), πx ∈ X (3.7)

means that x[i,j] is a half synchronizing word.

However, due to irreducibility and therefore the existence of Fischer cover for X, we will choose another proof which we think it gives a better picture.

Fix m ∈ H(X). Then there is a unique vertex I in X such that w+(m) = w+(I) := {L(π) : i(π) = I}. For every n ∈ N, define Cn to be the set of all cycles C in X starting at w+(m) such that |C| = n and m ⊆ L(C).

Set [ H1 := C, . . . , Hn := Hn−1 ∪ ∪C∈C C

C∈C1

and Zm := XH . Observe that Zm = X. Also all C ∈ Cn meet at w+(m) and therefore XH is an irreducible sofic.

Theorem 3.3. Suppose X be a half sychronized system. Then, there is a sequence A1 ⊆ A2 ⊆ · · · of irreducible shift of finite type’s in X such that limn h(An) is equal to

sup{h(A) : A ⊆ Zm is an irreducible shift of finite type}. (3.8)

Also lim h(An) = h(X ) = hhsyn(Zm).

Proof. Set t0 := sup{h(A) : A ⊆ Zm is an irreducible shift of finite type} and suppose A ⊆ Zm be an irreducible shift of finite type. Then, h(A) = hsyn(A) ≤ h(X ) and this implies t0 ≤ h(X ).

To prove h(X ) ≤ t0, fix l ∈ N. There is {lk : k ∈ N} such that lk < lk+1 and

+ 1 1 + h(X ) − < lim log |Cl | ≤ h(X ) (3.9)

0 k 0

l k lk

where Cl is as in Cn. Then, limk h(XH ) ≥ limk log |Cl |. To save the notation, we may

k lk lk k

assume that limk h(XH ) ≥ limk log |Ck| and therefore by (3.9), there is kj ∈ N such that

h(X ) − < h(XH ). (3.10)

0 kj XH ⊆ Zm is an irreducible sofic and by [9, Theorem 3.2], there is a sequence A1 ⊆ A2 ⊆ · · ·

kj

of irreducible shift of finite type’s in XH such that

kj

lim h(Ai) = h (XH ) = hsyn(XH )

kj 0 kj

and moreover as a sofic,

hsyn(XH ) = h(XH )[10, Lemma3.1].

kj kj

Thus limi h(Ai) = h(XH ) and therefore by (3.10), there is il ∈ N such that h(X ) − <

kj 0 l + 1 +

h(Ai ). Thus , h(X )− < t0 and therefore h(X ) ≤ t0 as required and as a result liml h(Ai ) =

l 0 l 0 l

t0. By setting An := Ai , we will have (3.8).

Theorem 3.4. If X is half synchronized and hhsyn(X) = h(X), then X is almost sofic.

Also note that if a general subshift X has a half synchronized component say Y which is not synchronized, then Y ⊆ ∂ X for n ∈ N∪ {0}. In other words, if Z is a subsystem of X such that Z ⊈ ∩n∂ X, then Z is a half synchronized component of X if and only if Z is a synchronized component of X.

A word m is weak synchronizing, when there is a a left ray x− such that if x−m and mw are ad-missible, then x−mw is also admissible. The respective subshifts are called weak synchronized. We, using a rather different approach, to show that this result extends to weak synchronized sys-tems. In exploring the properties of weak synchronizing words and their corresponding subshifts, we uncover a rich structure that highlights the interplay between word formation and admissibil-ity. To illustrate this concept, consider any weakly synchronized system defined by the presence of such a left ray. When we examine the extension, we observe that the admissibility conditions associate the behavior of with that of and. This relationship evokes thoughts of closure prop-erties inherent in subshift dynamics, where concatenation maintains structural constraints that dictate admissible sequences. Consequently, this setup allows us to formulate general principles governing the extension of words in weakly synchronized systems. Moreover, the methodology employed in this study diverges from traditional approaches, positing that the pathways towards establishing weak synchronization can reveal new avenues for understanding dynamic proper-ties of symbolic systems. By approaching the concept of weak synchronization from this fresh perspective, we aim to lay the groundwork for future exploration into associated phenomena, po-tentially unraveling complexities in more elaborate systems. In conclusion, our findings suggest that weak synchronized systems harbor fascinating structural characteristics that intricately link words and admissibility. As we delve deeper into these connections, we anticipate the emergence of novel insights that will enrich the discourse surrounding symbolic dynamics and contribute to the broader field of automata theory.

The authors are grateful to the referees for their careful reading, constructive criticisms, com-ments and suggestions, which have helped us to improve this work significantly.