Markov-Dyck shifts, neutral periodic points and topological conjugacy

We study the neutral periodic points of the Markov-Dyck shifts of finite strongly connected directed graphs. Under certain hypothesis on the structure of the graphs we show, that the topological conjugacy of their Markov-Dyck shifts implies the isomorphism of the graphs.


Introduction
Let Σ be a finite alphabet, and let S be the left shift on Σ Z , The closed shift-invariant subsystems of the shifts S are called subshifts. For an introduction to the theory of subshifts see [Ki] and [LM]. A finite word in the symbols of Σ is called admissible for the subshift X ⊂ Σ Z if it appears somewhere in a point of X. A subshift X ⊂ Σ Z is uniquely determined by its language L(X) of admissible words. In this paper we study the topological conjugacy of subshifts that are constructed from finite directed graphs. We denote a finite directed graph G with vertex set V and edge set E by G(V, E). The source vertex of an edge e ∈ E or of a directed path in the graph we denote by s and its target vertex by t (or by s G and t G , in the case that we have to distinguish between graphs.) We consider strongly connected finite directed graphs G = G(V, E). It is assumed that G is not a cycle. We recall the construction of the Markov-Dyck shift MD(G) of G (see [M1]). Let E − = {e − : e ∈ E} be a copy of E. Reverse the directions of the edges in E − to obtain the edge set E + = {e + : e ∈ E} of the reversed graph of G(V, E − ). In this way one has defined a graph G(V, E − ∪ E + ), where s G (e − ) = s G (e), t G (e − ) = t G (e), s G (e + ) = t G (e), t G (e + ) = s G (e), e ∈ E.
The alphabet of MD(G) is E − ∪ E + , and a word (e k ) 1≤k≤K is admissible for MD(G) precisely if 1≤k≤K e k = 0.
The directed graphs with a single vertex and N > 1 loops yield the Dyck inverse monoids (the "polycycliques" of [NP]) D N and the Dyck shifts D N [Kr1].
The notation ω − has the symmetric meaning. Also set The periodic points in A(X) are called the neutral periodic points of X. In Section 2 we clarify the structure of the set of neutral periodic points of a Markov-Dyck shift. This includes a characterization of the neutral periodic points of the shift. For a finite directed graph G(V, E) we denote by F G the set of edges that are the single incoming edges of their target vertices and we denote by R G the set of V ∈ V G that have more than one incoming edge. The set R G is the set of roots of a set of (possibly degenerate) directed rooted trees. 1 We denote the vertex set of the directed tree with root R ∈ R G by V R , and its edge set by F R . One has The condition, that card(R G ) = 1, is equivalent to the condition, that the graph G(V, F G ) is a (possibly degenerate) directed tree. We denote the graph that is obtained by contracting the non-degenerate trees among the trees G(V R , F R ), R ∈ R G , to their roots R by G(R G , E \ F G ). In the graph G the source vertex of an edge e ∈ E \ F G is the root of the tree that has s G (e) as a leave, and its target vertex is t G (e). In [Kr2] a Property (A) of subshifts, an invariant of topological conjugacy, was introduced, and to a subshift X with Property (A) a semigroup S(X) was invariantly associated. In [HK] it was shown that the Markov-Dyck shift MD(G(V, E)) of a graph G(V, E) has Property (A), and that In Section 2 we show that a topological conjugacy of Markov-Dyck shifts of graphs G(V, E) induces an isomorphism of the graphs G(R G , E \ F G ), that also preserves certain data pertaining to the configuration of the neutral periodic points of the 1 The graph with one vertex and no edges is generally considered to be a tree. We refer to this graph as the degenerate directed tree.
Markov-Dyck shift. For Markov-Motzkin shifts (see [KM2,Section 4.1]) analogous results hold. In Section 3 we consider finite directed graphs G(V, E), such that card(R G ) = 1. In this case, following the terminology, that was introduced in [HI], we say that a periodic point p of MD(G) and its orbit have negative multiplier e ∈ E \ F G , if there exists an i ∈ Z and an M ∈ N, such that The mapping that assigns to a multiplier e ∈ E \ F G the set of periodic points of MD(G) with negative multiplier e is an invariant of topological conjugacy [HIK,Proposition 4.2]. We set We denote for a multiplier e ∈ E \ F G , by I In Section 3 we consider three families, F I , F II and F III of finite directed graphs G(V, E), such that card(R G ) = 1. For the graphs in each of these families we introduce canonical models. Each canonical model is specified by a set of parameters, that we call the "data" of the model. We then establish for the graphs G in each of these families, that the topological conjugacy class of the Markov-Dyck shift of G determines the isomorphism class of the graph G. The proof consists in showing, that the invariants ν(MD(G)), and Λ (e) , e ∈ M(MD(G)), together with I (e) k (MD(G)), Ξ (e) 2k , k ∈ N, e ∈ M(MD(G)) contain sufficient information to determine the "data" of the canonical model of the graph G. We also characterize the Markov-Dyck shifts of the graphs in each of these families within the Markov-Dyck shifts.
The family F I contains the finite strongly connected directed graphs G(V, E), such that card(R G ) = 1, and such that all vertices, except the root of the tree G(V, F G ), have out-degree one.
The family F II contains the finite strongly connected directed graphs G(V, E) such that card(R G ) = 1, and such that all leaves of the tree G(V, F G ) are at level one.
The family F III contains the finite strongly connected directed graphs G(V, E), such that the graph G(V, F G ) is a tree, that has the shape of a "V", and that is such that the two leaves of the tree G(V, F G ) have the same out-degree, and all interior vertices of the tree G(V, F G ) have out-degree one.
It had been known, that for finite directed graphs, in which every vertex has at least two incoming edges, topological conjugacy of their Markov-Dyck shifts implies the isomorphism of the graphs (see [HIK,Section 4] and [Kr3,Corollary 3.2]). Actually, for finite directed graphs, in which every vertex has at least two incoming edges, the flow equivalence of their Markov-Dyck shifts implies the isomorphism of the graphs. For Dyck shifts this follows from [M2], and for the general case see [CS] and [Kr4].
Acknowledgment. Thanks go to the organizers of the research program "Classification of operator algebras: complexity, rigidity, and dynamics" at the Mittag-Leffler Institute, January -April 2016, for the opportunity to work on this paper while both authors were attending the research program. Thanks go to Toshihiro Hamachi for discussions that lead to the improvement of Section 3.

Neutral periodic points of Markov-Dyck shifts
We denote the period of a periodic point of a subshift by π. Given a (nondegenerate) rooted tree T (V, F ), we denote for V ∈ V by b(V ) the path from the root to V .
We continue to consider a strongly connected finite directed graph G = G(V, E), such that card(E \ F G ) > 1. An edge e ∈ E \ F G , determines a generator of S − ( G)(S + ( G)), that we denote by e − ( e + ), and we set A root R ∈ R G determines an idempotent of S( G), that we denote by 1 R , and we set e ∈ E, and, more generally, for a path b = (b i ) 1≤i≤I ∈ L(MD(G)) we use the notation The set of neutral periodic points of MD(G) we denote by and such that Theorem 2.1.
Proof. For the proof let there be given a neutral periodic point p of MD(G). With K + ∈ Z + , K − ∈ Z + , and ).
Note that , and therefore by (2.1) It follows that the word p [ m, m+π(p)) has a suffix e − b, that is uniquely determined by the condition, that e − ∈ E − \ F − G and that b is the empty word, . This contradicts the neutrality of p. Under the assumption that one has the symmetric argument. We have shown that To repeat this reasoning, with K + ∈ Z + , K − ∈ Z + , and Choose an m ∈ [0, π(p)), such that m≤j< m+m Then also m+m≤j< m+π(p) and therefore by (2.3) It follows from this and from (2.2), that there is an This contradicts the periodicity of p. Under the assumption that one has the symmetric argument. This confirms that and completes the proof.
The set of neutral periodic points of a subshift X ⊂ Σ Z carries a pre-order relation (X) (see [Kr2]). For neutral periodic points q and r of X one has q (X)r, if there exists a point in A(X), that is left asymptotic to the orbit of q and right asymptotic to the orbit of r. The equivalence relation that is derived from (X) we denote by ≈(X).
We set P The proof of the following lemma is similar to the proof of Theorem 3.2 of [HK].
Proof. The proof comes in two parts. For the first part let R ∈ R G , let U, W ∈ V R , and q, r ∈ P (0) R , and let j, k ∈ Z, be such that U = s G (p [j,j+π(q)) ), W = s G (r [k,k+π(r)) ), and such that Let a point x ∈ MD(G)) be given by One has x ∈ A max{π(q),π(r)} (MD(G)). This follows since the edges in the paths b(U ) and b(W ) are by construction the only incoming edges of their target vertices. We have proved that q ≈(X)r.
For the second part let R, R ′ ∈ R G , R ′ , and let j, j ′ ∈ Z be such that We prove that p and p ′ are (MD(G))-incomparable. Assume that p (MD(G)) p ′ , and let J ∈ N, and x ∈ A J (MD(G))), (2.5) and m, m ′ ∈ Z, m < m ′ , be such that Let g = e K , be an incoming edge of Q. The word is admissible for M D (G). However the word is not. This contradicts (2.5). Under the assumption that K ′ > 0, one has the symmetric argument. We have shown that p (MD(G))) p ′ .
Denoting by Π n (Y ) the number of points of period n of a shift-invariant set Y ⊂ Σ Z , the zeta function of Y is given by For the finite strongly connected directed graph G(V, E), we vertex weigh the graph G(R G , E \ F G ) by assigning to its vertices R ∈ R G the zeta function of P Proof. There is a canonical projection χ of the set of points in MD(G), that are left-asymptotic to a point in P (0) (MD(G)), and also right-asymptotic to point in P (0) (MD(G)) onto the associated semigroup S(MD(G)) = S( G(R G , E \ F G )).

(compare [Kr3, Section 3] and [Kr2, Section 3]). A topological conjugacy induces an isomorphism of the associated semigroups and it acts accordingly on the inverse images under χ of each of the elements in the set
(see [Kr5,Section 2]). By Lemma 2.2 To a vertex V ∈ V we associate the circular code C V that contains the words (c i ) 1≤i≤I ∈ L(MD(G)) such that The generating function of C V we denote by ϕ V .

Families of finite directed graphs
In this section we consider the case of strongly connected finite directed graphs G = G(V, E), such that card(R G ) = 1. These graphs are precisely the graphs, that have a Dyck inverse monoid as the associated semigroup of their Markov-Dyck shifts. Complete invariants for the isomorphism for these directed graphs are known for the case that all source vertices s(e), e ∈ M(MD(G)), have the same out-degree (see [GM]). We denote the root of F G by V 0 , and the out-degree of V 0 by D(V 0 ). We set M ℓ (MD(G)) = {e ∈ M(MD(G)) : Λ (e) = ℓ}, ℓ ∈ N.
By the use of the notation Λ(MD(G))) we indicate that all lengths Λ (e) (MD(G)), e ∈ E \ F G , are equal and that Λ(MD(G))) is their common value.
Proof. The statement holds if G is a single vertex graph. Consider a graph G = G( V, E), such that the graph G( V, F G ) is a non-degenerate tree. Denote by J G (ℓ) the number of leafs of F G that have level ℓ − 1. One has that Corollary 3.2. For finite directed graphs G = G(V, E) such that the associated semigroup of MD(G) is a Dyck inverse monoid, and such that the topological conjugacy of their Markov-Dyck shifts implies the isomorphism of the graphs.
Proof. In (3.I.2) the data (S ℓ ) ℓ∈N are expressed in terms of invariants of topological conjugacy.

3.2.
A family of finite directed graphs II. We set The data (R, (Q M ) M∈N ) ∈ Π determine canonical models G((R, (Q M ) M∈N )) of the graphs in F II . We define G((R, (Q M ) M∈N )) as the directed graph with a vertex V (0) and with vertices and with edges f M,q , 1 ≤ q ≤ Q M , M ∈ N, and e r , 1 ≤ r ≤ R, and e M,q,m , 1 ≤ m ≤ M, 1 ≤ q ≤ Q M , M ∈ N. The source and target mappings are given by and s(e r ) = t(e r ) = V (0), 1 ≤ r ≤ R, and One has Proof. The assumption, that the associated semigroup of MD(G)) is a Dyck inverse monoid, implies, that the graph G(V, F ) is a (possibly degenerate) tree. In case that the tree G(V, F ) is not degenerate, one has from (3.II.2) that all leaves of the subtree are at level 1. Also, for an e ∈ E \ F G , one has that card(Ξ (e) 4 ) = card({e ′ ∈ E \ F G : s(e ′ ) = s(e)}) + D(V 0 ). This follows from the observation, that in this case every orbit of length 4 with negative multiplier e ∈ E \ F G is obtained by inserting into an orbit of length 2 with negative multiplier e ∈ E \ F G either the word f − f + , f ∈ F G , t(f ) = s(e), or a word e − e + , e ∈ M 1 (G), or a word e − e + , e ∈ M 2 (G), s( e) = s(e). It is D(V 0 ) = I  Note the non-empty intersection of F III with F I and F II .      ℓ+4 (MD(G)) =(Λ(MD(G)) + 1 2 ν(MD(G))) 2 + Λ(MD(G)) + 2ν(MD(G)) − 2, e ∈ E \ F G .
Proof. Let e ∈ E \ F G , and let O (e) be the shortest periodic orbit of MD(G) with negative multiplier e.
All periodic orbits of MD(G) of length Λ(MD(G)) + 2 with multiplier e − are obtained by inserting a word of the form g − g + , where the source vertex of the edge g − is transversed by O (e) , into O (e) . The number of these words is ℓ + M .
All periodic orbits of MD(G) of length Λ(MD(G)) + 4 with multiplier e − are obtained by either inserting two words of the form g − g + , where the source vertex of the edge g − is transversed by O (e) , into O (e) , or by inserting a word of the form g − g − g + g + , t(g) = s( g), into O (e) , where the source vertex of the edge g − is transversed by O (e) , into O (e) , and the number of these words is ℓ + 4M − 2. Proof. Necessity follows from Lemma 3.5, Lemma 3.6. and Lemma 3.7. To prove sufficiency, let G = G(V, E) be a graph that satisfies the conditions of the theorem. We denote the root of the tree G(V, F G ) by V 0 . The out-degree of a vertex we denote by D. It follows from (A) that In the case D(V 0 ) = 2, one has by (A) and (C) that the tree G(V, F G ) has two leaves, that have the same out-degree, and equations (3.III.2) follow. The task is to exclude the case D(V 0 ) = 1.
Assume, that D(V 0 ) = 1. Let L be maximal, such that the tree G(V, F G ) has a single vertex V L at level L. One has that L < Λ(MD(G)) − 2, since otherwise by (A), D(V L ) = Λ(MD(G)), which is either by (C) incompatible with L > 0, or it contradicts (B).
By deriving a contradiction to (A)(B)(C)(D) we will exclude each of the following cases (c1 -5): In case (c3) it follows from (3.III.3) for the cycle b = (e k ) 1≤k≤Λ(MD(G)) , that 1<k≤Λ(MD(G)) (D(s(e k )) − 1) = 1, and by (D) the only other vertex besides s(e 1 ), that is traversed by b, that has an out-degree, that exceeds one (and is equal to two), is necessarily s(e 2 ). This means that G is isomorphic to G 1 With G