On the Markov-Dyck shifts of vertex type

For a given finite directed graph $G$, there are two types of Markov-Dyck shifts, the Markov-Dyck shift $D_G^V$ of vertex type and the Markov-Dyck shift $D_G^E$ of edge type. It is shown that, if $G$ does not have multi-edges, the former is a finite-to-one factor of the latter, and they have the same topological entropy. An expression for the zeta function of a Markov-Dyck shift of vertex type is given. It is different from that of the Markov-Dyck shift of edge type.


Introduction
Let Σ be a finite alphabet, and let σ be the left shift on Σ Z defined by σ((x n ) n∈Z ) = (x n+1 ) n∈Z , (x n ) n∈Z ∈ Σ Z . For a closed subset Λ ⊂ Σ Z satisfying σ(Λ) = Λ, the topological dynamical system (Λ, σ) is called a subshift. Denote by B n (Λ) the set of all admissible words appearing in Λ with length n, and by P n (Λ) the set of all n-periodic points of (Λ, σ), respectively. Then the topological entropy h top (Λ) and the zeta function ζ Λ (z) for (Λ, σ) is defined by They are crucial topological conjugacy invariants of (Λ, σ). For an introduction to their theory, which belongs to symbolic dynamics, we refer to [10] and [15]. W. Krieger in [11] has introduced the Dyck shifts from automata theory and language theory in computer science. They are non-sofic subshifts defined by Dyck languages. In [7,11,12,14,17], a class of non-sofic subshifts called Markov-Dyck shifts have been studied (cf. [8]). The subshifts are generalization of Dyck shifts by using finite directed graphs. They have recently come to be studied by computer scientists (cf. [1,2]). For a given finite directed graph G = (V, E), there are two types of Markov-Dyck shifts, the Markov-Dyck shift D V G of vertex type and the Markov-Dyck shift D E G of edge type. Both of them are not sofic subshifts if G is irreducible and not permutive. In the papers [7,11,12,14], the Markov-Dyck shifts mean the Markov-Dyck shifts of edge type. In [14], formulae of topological entropy and zeta functions for Markov-Dyck shifts of edge type have been presented.
In the first part of the paper, we will study relationship between the two types of Markov-Dyck shifts for finite directed graphs, the Markov-Dyck shift D V G of vertex type and the Markov-Dyck shift D E G of edge type. We will show that, if G does not have multiedges, there exists a finite-to-one factor code from D E G to D V G (Proposition 2.9). The factor code can never yield a topological conjugacy unless the transition matrix of the graph is permutation. They have the same topological entropy (Theorem 2.10).
In the second part of the paper, we will present a formula of the zeta function of a Markov-Dyck shift of vertex type (Theorem 3.9). The formula is regarded as a generalization of the formula for Markov-Dyck shifts of edge type [14,Theorem 2.3]. In the final section, the zeta function of the Fibonacci-Dyck shift of vertex type will be presented. It is different from that of the Fibonacci-Dyck shift of edge type. Hence the Fibonacci-Dyck shift of vertex type is not topologically conjugate to the Fibonacci-Dyck shift of edge type.

Markov-Dyck shifts
Throughout this paper N is a fixed positive integer larger than 1. For a finite set S, we denote by |S| its cardinality. We consider the Dyck shift D N with alphabet Σ = Σ − ∪ Σ + where Σ − = {α 1 , . . . , α N }, Σ + = {β 1 , . . . , β N }. The symbols α i , β i correspond to the brackets ( i , ) i respectively, and have the product relations of monoid as follows: for i, j = 1, . . . , N (cf. [12,13]). For a word ω = ω 1 · · · ω n of Σ, we denote byω its reduced form. Namelyω is a word of Σ ∪ {0, 1} obtained after applying the relations (2.1) in ω. Then a word ω of Σ is said to be forbidden in D N if and only ifω = 0. Denote by F N the set of forbidden words. The Dyck shift D N is defined in [11] by a subshift over Σ whose forbidden words are F N , namely ..,N be an N × N matrix with entries in {0, 1}. Throughout this paper, A is assumed to be essential which means that it has no zero rows or columns. Consider the Cuntz-Krieger algebra O A for the matrix A that is the universal C * -algebra generated by N partial isometries t 1 , . . . , t N subject to the following relations: We denote by Σ * the set of all words γ 1 · · · γ n of elements of Σ. Define the set Let G = (V, E) be a finite directed graph with vertex set V and edge set E. We denote by s(e) the initial vertex of e ∈ E and by t(e) the final vertex, respectively. We assume that the cardinalities of V and of E are both finite and write V = {v 1 , . . . , v N 0 } and E = {e 1 , . . . , e N 1 }. We also assume that each vertex of G has at least one in-coming edge and at least one out-going edge. The edge matrix In [14], we have defined the Markov-Dyck shift D G for the graph G as the Markov-Dyck shift D A G for the matrix A G , and presented formulae of the zeta function ζ D G (z) and the topological entropy h(D G ). A finite matrix M with entries in {0, 1} does not necessarily arise from a finite graph as M = A G . The lemma below is easy to prove. For the sake of completeness, we provide its proof.
the ith row vector and the jth column vector for i, j = 1, . . . , N respectively. Then the following three conditions are equivalent: where · | · means the inner product of vectors.
Proof.   For a finite directed G = (V, E), we have another transition matrix A G , which is an The matrix A G is called the vertex matrix for the graph G. It has its entries in {0, 1}. shift of vertex type for G, and written D V G . It is obvious that any finite matrix M with entries in {0, 1} can arise from a finite graph G such that M = A G . By Lemma 2.2, one sees that the class of Markov-Dyck shifts of edge type is a subclass of Markov-Dyck shifts of vertex type. As is well-known that for a finite directed graph G the topological Markov shift X A G defined by the edge matrix A G is topologically conjugate to the topological Markov shift X A G defined by the vertex matrix A G . The Markov-Dyck shifts however do not have this property. Let G 1 be the following graph ( Figure 1). The vertex matrix A G 1 and the edge matrix A G 1 are written as have 4 fixed points as subshifts. The former D V G 1 has 4 periodic points with least period 2. The latter D E G 1 has 6 periodic points with least period 2. Hence D V G 1 is not topologically conjugate to D E G 1 . A Dyck n-path is a continuous broken directed line on the upper half plane consisting of vectors (1, 1) called rise and (1, −1) called fall. It starts at the origin with rise and ends at (2n, 0) with fall (see [5,6], etc.). Let γ = (γ 1 , . . . , γ 2n ) be a Dyck n-path. Hence each γ i is a rise or a fall. If γ i is a rise, there exists the smallest k = 1, 2, . . . , 2n − i satisfying the following two conditions: , which starts at the terminal vertex of γ i and ends at the source vertex of γ i+k .
We call the edge γ i+k the partner of γ i .
Let G = (V, E) be a finite directed graph. Denote by G * = (V * , E * ) the transposed graph of G. The vertex set V * is V and the edge set E * consists of the edges reversing its direction of the edges of G. For an edge e ∈ E, we denote by e * the edge of G * obtained by reversing the direction of e, so that t(e * ) = s(e), s(e * ) = t(e) for e ∈ E. Recall that the edge set E of G is denoted by {e 1 , . . . , e N 1 } and the edge set E * of G * is written as A G-Dyck n-path of edge type for n = 1, 2, . . . is a Dyck n-path (x 1 , . . . , x 2n ) labeled elements of Σ E G satisfying the following rules: (1E) a rise is labeled e * i for some i = 1, . . . , N 1 , (2E) a fall is labeled e i for some i = 1, . . . , N 1 , (3E) the partner of a rise labeled e * i is labeled e i , (4E) a rise labeled e * i follows a rise labeled e * j if and only if t(e * j ) = s(e * i ), (5E) a rise labeled e * i follows a fall labeled e j if and only if t(e j ) = s(e * i ), (6E) a fall labeled e i follows a fall labeled e j if and only if t(e j ) = s(e i ), (7E) a fall labeled e i follows a rise labeled e * j if and only if e j = e i . Similarly, for a vertex v ∈ V , we denote by v * the corresponding vertex of G * obtained by the transposed graph G * = (V * , E * ). The vertex matrix A G * for G * satisfy the relations A G-Dyck n-path of vertex type for n = 1, 2, . . . is a Dyck n-path (x 1 , . . . , x 2n ) labeled elements of Σ V G satisfying the following rules: We note the following lemma (ii) Let t 1 , . . . , t N 0 be the partial isometries satisfying the relations (2.3) for the vertex We remark that a finite path of vertices of a labeled broken directed line of the G-Dyck path of edge type is not necessarily an admissible word of the Dyck shift D V G of vertex type. Consider the following correspondences in G-Dyck paths: a fall e ∈ E −→ the source s(e) ∈ V of e, a rise e * ∈ E * −→ the terminal t(e * ) ∈ V * of e * . (2.9) The rules (1E), . . . , (7E) and (1V ), . . . , (7V ) ensure us the following lemma. (i) Any sequence of vertices of a G-Dyck n-path of edge type yields a labeled sequence by Σ V G of a G-Dyck n-path of vertex type by the correspondence (2.9).
(ii) Any labeled sequence by Σ V G of a G-Dyck n-path of vertex type is realized as a sequence of vertices of a G-Dyck n-path of edge type by the correspondence (2.9).
By the above lemma, it is reasonable to define a 1-block map Φ : In the above situation, we call the vertex v k (= v * k ) a valley. Hence the factor map ϕ : D E G −→ D V G erases the valleys. We will show that the factor map ϕ is finite-to-one, so that the equality of the topological entropy h top (D E G ) = h top (D V G ) holds. We provide the height functions on D E G . These functions on the Dyck shift D N have been first introduced by W. Krieger in [11]. For x = (x n ) n∈Z ∈ D E G , we set the height function is not a relative minimum in x, we have two cases.
Case 1: . We have two cases.
There exists k ∈ Z with m < k < i such that x k−1 , x k ∈ E, and H m (x) = H k (x). We take a vertex v j ∈ V such that Φ(x k ) = v j . We then have t(x m−1 ) = t(x k−1 ) = v j .
Case 2: i < m. There exists l ∈ Z with i < l < m such that x l−1 , x l ∈ E * , and H m (x) = H l (x). We take a vertex v j ∈ V such that Φ(x l−1 ) = v j . We then have t(x m−1 ) = t(x l−1 ) = v j .
(iii) Suppose that two vertices t(x n−1 ) and t(x m−1 ) are both minimum in x, so that H n (x) = H m (x). Assume that n < m. The word (x n , x n+1 , . . . , x m−1 ) is a G-Dyck path of edge type so that the vertices s(x n ) and t(x m−1 ) are the same. This implies that t(x n−1 ) = t(x m−1 ).
(ii) if x has a minimum vertex, then Therefore ϕ : D E G −→ D V G is a finite-to-one factor code.
Proof. (i) Suppose that x = (x n ) n∈Z does not have a miniumum vertex. By (ii) of the above lemma, the sequence ϕ(x) determines the sequence t(x n ), n ∈ Z of vertices. Each symbol x n is an edge of E or of E * , and an edge is determined by the vertices t(x n ), t(x n−1 )(= s(x n )), so that the code ϕ is injective at x.
(ii) Suppose that x has a minimum vertex at t(x m−1 ) for some m ∈ Z. Then the vertex t(x m−1 ) is a valley and x m−1 ∈ E, x m ∈ E * . By (iii) of the above lemma, other minimum vertices are the same as the vertex t(x m−1 ). Hence we have

Theorem 2.10. Suppose that G does not have multi-edges. We then have
Proof. Since there exists a factor code ϕ : between admissible words. It is not necessarily one-to-one at minimal points of words. We then have Concerning embedding of the Markov-Dyck shifts, we have the following proposition. Proof. Let t i , i = 1, . . . , N 0 be partial isometries satisfying the relations (2.3) for the vertex matrix A G . For an edge e n ∈ E with s(e n ) = v i , t(e n ) = v j , define a partial isometry S n = t i t j t * j . It is easy to see that the family S 1 , . . . , S N 1 satisfies the relations (2.3) for the edge matrix A G , This implies that the correspondence Ψ : induces an embedding of D E G into the 3rd power shift (D V G ) [3] of D V G .

The zeta functions of Mrkov-Dyck shifts of vertex type
In what follows, we fix an arbitrary N × N matrix A = [A(i, j)] N i,j=1 with entries in {0, 1}. We will study the Markov-Dyck shift D A and present a formula of the zeta function ζ D A (z). In [14], a formula of the zeta function of the Markov-Dyck shifts of edge type has been presented. The Markov-Dyck shifts of edge type form a subclass of the class of Markov-Dyck shifts. In this section, we will study general Markov-Dyck shift D A and present a formula of its zeta function ζ D A (z). For the N × N matrix A, let v 1 , . . . , v N be N -vertices. Define a directed edge from v i to v j if A(i, j) = 1. We then have a finite directed graph written G = (V, E) such that its vertex matrix A G coincides with the original matrix A.
We put We then see the following lemma. and a map r : Then the quadruplet C + A = (C + A , I,Ã, r) is a circular Markov code in the sense of Keller [9]. We then associate the following shift-invariant subset Ω C + The zeta function ζ(Ω C + A , z) for a shift-invariant set Ω C + A is similarly defined to (1.2) by using a sequence of cardinalities of periodic points of Ω C + A . Following Keller [9], define a for m = 2n + k, and a matrix-valued generating function F (C + A , z) by Denote by I N 2 the identity matrix of size N 2 . By using [9, Theorem 1], we have We then have for (i, j), (p, q) ∈ I Proof. Let U = [U ((i, j), (p, q))] (i,j),(p,q)∈I and V = [V ((i, j), (p, q))] (i,j),(p,q)∈I be I × I matrices defined by , z)) by adding the minus of the (i, N )th column to the (i, j)th column for all j = 1, 2, . . . , N − 1 and i = 1, 2, . . . , N , and the matrix U (I N 2 − F (C + A , z))V is obtained from (I N 2 − F (C + A , z))V by adding the (i, j)th rows to the (i, N )th row for all j = 1, 2, . . . , N − 1 and i = 1, 2, . . . , N . Hence we see Each (p, q)th column for q < N of the matrix U (I N 2 − F (C + A , z))V has 1 on diagonal and zero elsewhere. Since by expanding the matrix U (I N 2 −F (C + A , z))V along the (p, q)th columns for p = 1, 2, . . . , N with q < N , we have As det(U ) = det(V ) = 1, we get the desired equality.
Therefore we have .

(3.2)
Proof. Since we have so that the desired equality holds.
We reach the following formula of the zeta function of a Markov-Dyck shift of vertex type.
Theorem 3.9. Let A be an N × N essential matrix with entries in {0, 1}. Then the zeta function ζ D A (z) of the Markov-Dyck shift D A is given by the following formula: and the functions f A i (z 2 ), i = 1, 2, . . . , N satisfiy the relations (3.1).
Proof. For n, k ∈ N, we define the following set C A,− n,k similarly to C A,+ n,k by Similarly to the previous discussion, we have a circular Markov code C − A = (C − A , I,Ã t , r) and the formula (3.2) for ζ(Ω C − A , z). We then have a disjoint union of periodic points For a finite directed graph G = (V, E) the above formula gives us the formula for the zeta function of the Markov-Dyck shift of vertex type.

The zeta functions of Markov-Dyck shifts of edge type
The Markov-Dyck shifts in the paper [14] are the Markov-Dyck shifts of edge type. In [14], a formula of the zeta functions of Markov-Dyck shifts of edge type has been presented. In this section, we present the formula [14, Theorem 2.3] from Theorem 3.9. We need the following lemma.
Let us denote by c G n (v i ) and c G n (e j ) their cardinalities |C A G n (v i )| and |C A G n (e j )| respectively ( [16, pages 8,9]). Then we have c G n (e j )z n so that f E j (z) = f V i (z) when s(e j ) = v i . Hence we have which implies that D V (z 2 )S = SD E (z 2 ). It then follows that Hence the matrices zD V (z 2 )A G and zD E (z 2 )A G are elementary equivalent (see [15, Definition 7.2.1]), so that det( Therefore we have where f G 1 (z 2 ), . . . , f G N 0 (z 2 ) are the functions satisfying Proof. Since f G i (x) = f V i (x), i = 1, . . . , N 0 and respectively. We then have Proof. It is easy to see that the number of the 2-periodic points of D V G 2 is 6, whereas that of D E G 2 is 7.
The Fibonacci-Dyck shift D E G 2 of edge type is a subshift D A G 2 over six symbols which correspond to the edges of the directed graphs G 2 and G * 2 of Figure 2. The Fibonacci-Dyck shift D A G 2 of vertex type is a subshift D A G 2 over four symbols which correspond to the vertices of the directed graphs of G 2 and G * 2 of Figure 2. Let us denote by α 1 , α 2 and β 1 , β 2 the symbols of D A G 2 . They have the following algebraic relations from the relations (2.3) of operators for A = A G 2 = 1 1 1 0 : α 1 β 1 = β 1 α 1 + β 2 α 2 = 1, α 2 β 2 = β 1 α 1 , β 2 α 2 β 2 = β 2 ,