On the Quantized Dynamics of Factorial Languages

On the Quantized Dynamics of Factorial Languages Abstract We study local piecewise conjugacy of the quantized dynamics arising from factorial languages. We show that it induces a bijection between allowable words of same length and thus it preserves entropy. In the case of sofic factorial languages we prove that local piecewise conjugacy translates to unlabeled graph isomorphism of the follower set graphs. Moreover, it induces an unlabeled graph isomorphism between the Fischer covers of irreducible subshifts. We verify that local piecewise conjugacy does not preserve finite type nor irreducibility; but it preserves soficity. Moreover, it implies identification (up to a permutation) for factorial languages of type 1 if, and only if, the follower set function is one-to-one on the symbol set. 1. Introduction The fruitful interplay between Symbolic Dynamics and Operator Algebras was established in the seminal paper of Cuntz–Krieger [9]. Following their work, Matsumoto introduced an effective way for associating operators to subshifts and forming Cuntz–Krieger-type C*-algebras [31] that were further examined in-depth in a series of papers [32–34]. This theory was revisited with Carlsen [4, 5] and a new view was exploited in more generality in [34]. These important works clarified a strong connection between intrinsic properties of subshifts with related C*-algebras. Matsumoto operators follow from a Fock representation that accommodates more structures. Shalit–Solel [38] provided such a context for homogeneous ideals in general and established a rigidity programme for the related (non-selfadjoint) tensor algebras. The origins of this framework are traced back to the seminal work of Arveson [1]. Since then, a number of rigidity results have appeared in the literature for tensor algebras of graphs or dynamical systems as in the work of Katsoulis–Kribs [26] and Solel [39], Davidson–Katsoulis [11, 12] that supersedes the work of previous authors [3, 19, 36, 37], Davidson–Roydor [13], Davidson–Ramsey–Shalit [14, 15], Dor-On [16], Katsoulis–Ramsey [28] and the work of the second author with Davidson [10] and Katsoulis [23]. In this endeavour, Davidson–Katsoulis [12] developed the notion of piecewise conjugacy for classical systems as the essential level of equivalence obtained from tensor algebras. Piecewise conjugacy allows for comparisons of the systems locally and thus is more tractable than (global) conjugacy. Recently, it found significant applications to Number Theory and reconstruction of graphs as exhibited in the work of Cornelissen–Marcolli [7, 8] and Cornelissen [6]. Along this line of research, the second author with Shalit examined tensor algebras of factorial languages in [25]. A factorial language Λ* on d symbols is a subset of the free semigroup F+d such that if μ∈Λ* then every subword of μ is also in Λ*. To fix notation, the operators Tμ in discussion act on ℓ2(Λ*) and are defined by   Tμeν≔eμνifμν∈Λ*,0otherwise,for μ∈Λ*. As an intermediate step we use the C*-algebra of checkers  A≔C*(Tμ*Tμ∣μ∈Λ*). The ∗-endomorphisms on A given by αi(a)≔Ti*aTi play an important role in the analysis and the system (A,α1,…,αd) was coined in [25] as the quantized dynamics of Λ*. Two norm-closed subalgebras of B(ℓ2(Λ*)) can be related to the same language Λ*: The A-tensor algebra AΛ*≔alg¯{I,Tμ∣μ∈Λ*} in the sense of Shalit–Solel [38]. The T+-tensor algebra TΛ*+≔alg¯{a,Tμ∣a∈A,μ∈Λ*} in the sense of Muhly–Solel [35].Matsumoto’s C*-algebra is the quotient of C*(T)≔C*(Tμ∣μ∈Λ*) by the compacts K in ℓ2(Λ*). Arveson’s Programme on the C*-envelope (Arveson’s Programme was initiated in [2] and established in [20]. See also its formulation in [24].) provides a solid pathway for researching possible Cuntz–Krieger-type C*-algebras. For example, the natural analogues related to C*-correspondences are exactly the C*-envelopes of the tensor algebras as proven by Katsoulis–Kribs [27]. One of the main results in [25] states that the C*-algebra that fits Arveson’s Programme for both AΛ* and TΛ*+ is the quotient of C*(T) by the generalized compacts; rather than quotienting by all compacts as is done in Matsumoto’s work. In fact the quantized dynamics trigger a dichotomy: the C*-envelope of both AΛ* and TΛ*+ is either the quotient by all compacts or it coincides with C*(T), depending on whether the quantized dynamics is injective or not. This is in full analogy to what holds for graph C*-algebras where sinks or vertices emitting infinite edges are excluded from the Cuntz–Krieger relations. Apart from being a starting point for Cuntz–Krieger-type C*-algebras via the C*-envelope machinery, both AΛ* and TΛ*+ are rigid for factorial languages. It is shown in [25] that they encode the factorial language, yet in two essentially different ways: The A-tensor algebras provide a complete invariant for the factorial languages up to a permutation of the symbols. The T+-tensor algebras provide a complete invariant for local piecewise conjugacy (l.p.c.) of the quantized dynamics.However, it was left open how l.p.c. reflects the initial data: How is l.p.c. interpreted in terms of factorial languages? What properties are (thus) preserved under l.p.c.? What is the impact on sofic factorial languages?In the current paper, we answer these questions that add on the impact of the rigidity results of [25]. Before we move to the description of our results, we stress that languages of subshifts form special examples of factorial languages and several constructions apply to this broader context. Thus, terminology related to subshifts is extended accordingly to cover general factorial languages, when possible. Unlike to Carlsen [4], Matsumoto [31] or Krieger [29], our study is based on the allowable words rather than the points of an induced subshift. In fact the dynamical system of the backward shift is not explicitly used for the Fock space quantization and thus no connections between l.p.c. and topological conjugacy arise. The results and examples herein show that they are incomparable. On one hand, l.p.c. requires for the languages to have the same number of symbols (Definition 4.1) and so it is not implied by topological conjugacy. On the other hand, in Example 5.10, we construct a subshift of finite type that is l.p.c. to the even shift, and so l.p.c. does not imply topological conjugacy. We begin by giving an updated picture of the quantized dynamics (Section 3). We then show that l.p.c. implies a bijection between allowable words of the same length, and thus it respects entropy (Proposition 4.3 and Corollary 4.5). The flexibility of l.p.c. can be seen in the form of this bijection; but has its limitations (see Remark 4.2). Most notably, l.p.c. does not preserve finite type as shown in Example 5.10, nor irreducibility as shown in Example 5.21. Consequently, l.p.c. does not preserve the zeta function (Remark 5.11). Nevertheless, l.p.c. respects soficity (that is the C*-algebra of checkers is finite dimensional) where the theory is rich. There is a well-known construction of a labeled finite graph, that is the follower set graph, that gives a presentation of a sofic factorial language. When the language is of finite type, then this construction can be described by a (terminating) algorithm (Follower Set Graph Algorithm). Labeled graph isomorphism is equivalent then with the factorial languages being the same up to a permutation of symbols (Proposition 5.6). On the other hand, it is the unlabeled graph isomorphism that coincides with l.p.c. (Theorem 5.8). Combining these results with [25], we thus prove the following diagram for two sofic factorial languages Λ* and M*:   The follower set graph construction is rather useful in Theoretical Computer Science as the starting point for computing minimal presentations. Such presentations are unique (up to isomorphism) for irreducible two-sided subshifts and are better known as Fischer covers [17, 18]. We show that l.p.c. induces an unlabeled graph isomorphism between the Fischer covers of irreducible two-sided subshifts (Corollary 5.19). It is quite interesting to notice though that we achieve these results without inducing a bijection between intrinsically synchronizing words. A weaker bijection between the collections of follower sets of such words is induced. This is depicted in Remark 5.20 where we show the limitations of our arguments. The same obstructions do not allow to apply our arguments and prove (or disprove) that l.p.c. respects the mixing property. We further investigate cases where the vertical directions in the diagram above can or cannot be equivalences based on the type and the number of symbols. As commented in [25], these arrows cannot be reversed in general and we extend this remark for two-sided subshifts in Example 5.12. The key in these counterexamples is that the follower set function is not one-to-one. Apparently, this is the only obstruction for type 1 factorial languages. In Theorem 5.13, we show that unlabeled isomorphism for type 1 factorial languages produces a labeled isomorphism if and only if the follower set function is one-to-one on the symbol set. Consequently then, isomorphism of the A-tensor algebras is equivalent to isomorphism of the T+-tensor algebras. This condition is satisfied by edge shifts with invertible adjacency matrix. Injectivity of the follower set function is not required for type 1 factorial languages on two symbols. In Remark 5.16, we describe how an unlabeled graph isomorphism implies a labeled graph isomorphism of the follower set graphs in these cases. These results depend on the low complexity of the system. However, this does not hold when passing to type 2 factorial languages, even when the number of symbols is small (Examples 5.17 and 5.18). To facilitate comparisons, we developed a program that takes as an input a set of forbidden words on two symbols and gives the follower set graph as an output. (We chose to develop this program for the right version of factorial languages, rather than the left we work with here, as it is accustomed in Theoretical Computer Science to concatenate on the right. Nevertheless, the left version follows easily by reversing the words in the input and the output.) The code and the .exe file can be downloaded from the official webpage of the second author, currently at http://www.mas.ncl.ac.uk/~nek29. We remark that our principal objective here was to construct a program that computes easily follower set graphs as a check for our examples and counterexamples. Hence, we were not (extremely) concerned about complexity or the required memory, but only about the fact that it terminates when the type is finite. 2. Preliminaries 2.1. Languages and subshifts Let us fix the terminology and notation we will be using throughout the paper. To this, end F+d denotes the free semigroup on the symbol set Σ≔{1,…,d} with multiplication given by concatenation. For μ=μk…μ1 in F+d, we write ∣μ∣≔k for the length of μ. The empty word ∅ is by default in F+d, it has zero length and plays the role of the unit. For μ,ν∈F+d, we say that ν is a subword of μ if there are w,z∈F+d such that μ=wνz. By default, the empty word is a subword of every μ∈Λ*. A (factorial) language is a subset Λ* of F+d that satisfies the following property:   ifμ∈Λ*theneverysubwordofμisinΛ*. We will simply use the term ‘language’, since we are going to encounter just factorial ones. Without loss of generality, we will always assume that all letters of the symbol set are in Λ*. Otherwise we view Λ* to be defined on less symbols, that is on Σ out the symbols that are not in Λ*. Examples of languages arise from subshifts and below we give a brief description. Several elements from the theory of subshifts apply to the broader context of languages and terminology will be extended to cover languages in general. Apart from two-sided, we also consider one-sided subshifts. Several results that hold for the two-sided version hold also for the one-sided with almost the same proof. We will mainly discuss left subshifts, but similar comments hold for the right subshifts. In order to make sense of the one-sided subshifts and avoid technicalities, we make the following convention. We will write the sequences x=(xn)∈ΣZ+ from right to left, that is   …xn…x1x0.=xand likewise for elements in ΣZ. This is to comply with operator composition which comes by multiplying on the left. We endow ΣZ+ with the product topology and we fix σ:ΣZ+→ΣZ+ be the backward shift with σ((xi))k=xk+1. With our convention, the map σ shifts to the right. The pair (Λ,σ) is called a left subshift if Λ is a closed subset of ΣZ+ with σ(Λ)⊆Λ. Similarly, the pair (Λ,σ) is called a two-sided subshift if Λ is a closed subset of ΣZ with σ(Λ)=Λ. We write x[m+n−1,m] for the block xm+n−1…xm in x∈Λ. A word μ=μ∣μ∣…μ1 is said to occur in some (one-sided or two-sided) sequence x if there is an m such that   x∣μ∣−1+m=μ∣μ∣,…,xm=μ1. If a word occurs in some point of Λ, then it is called allowable. The language of a subshift Λ is defined by   Λ*≔{w∈F+d∣woccursinsomex∈Λ}. Since Λ is σ-invariant, we have that for every allowable word μ there exists an x∈Λ such that x[∣μ∣,0]=μ. We write Bn(Λ*) for the allowable words of length n in Λ*. By following the same arguments as in the two-sided subshifts, one can show that if Λ defines a left subshift, then Λ* is a language such that   foreveryμ∈Λ*thereisa∅≠ν∈Λ*suchthatνμ∈Λ*. Conversely, every language with this property defines uniquely a left subshift (see [38, Proposition 12.3] and [25]). Subshifts can be described also in terms of forbidden words. Let F be a set of words on the symbol set Σ={1,…,d}, and let   ΛF≔{(xn)∈ΣZ+∣noμ∈Foccursin(xn)}. It is known that all two-sided subshifts arise in this way. Likewise this also holds for one-sided subshifts. By setting   Fk≔{μ∈F∣μdoesnotoccurin(xn)∈Λ,∣μ∣≤k},we see that Fk⊆Fk+1 and F=⋃kFk. Then we have that Λ=∩kΛFk, where the intersection is considered inside the full shift space on Σ. The elements in F are called the forbidden words of the subshift. We will call a forbidden word minimal, if all of its proper subwords are allowable. Every set F of forbidden words admits a unique basis F⊆F in the sense that for every μ∈F, there are (unique) ν,w∈F+d and a μ′∈F such that μ=νμ′w and μ′ is minimal. We say that Λ is a subshift of finite type (SFT) if the longest word in the basis of F has finite length. We say that an SFT is of type k if the longest word in the basis of F has length k+1. Hence, if Λ is of type k, then any forbidden word of length strictly greater than k+1 cannot be minimal. The notions of forbidden words, minimality, basis and type pass naturally to any language. For example, given a set F, we can define a language by   ΛF*=F+d⧹{wμν∣μ∈F,w,ν∈F+d}. Not every language is a language of a subshift, but it can be embedded in one by augmenting the symbol set. Suppose that Λ* is defined through a set of forbidden words F in Σ. We introduce a new distinguished symbol ζ and take the symbol set Σ˜={1,…,d,ζ}. Then the augmented subshift (Λ˜,σ)of Λ* is the subshift defined by F in Σ˜Z. Since ζ is not contained in any word in F, it follows that the language of Λ˜ is   Λ˜*={ζnkμk…μ2ζn2μ1ζn1∣n1,…,nk∈Z+,μ1,…,μk∈Λ*}and thus contains Λ*. Recall that a two-sided subshift (Λ,σ) is called sofic if the number of classes in Λ* with respect to the equivalence relation   μ∼ν⇔{w∈Λ*∣wμ∈Λ*}={w∈Λ*∣wν∈Λ*}is finite. Equivalently, if (Λ,σ) is a factor of an SFT [17, 40]. The reader is addressed for example to [30, Theorem 3.2.10] for a modern treatment of sofic subshifts. For languages that do not come from subshifts, we will use the definition of soficity in terms of the equivalence classes. It is shown in [25] that a language Λ* is sofic (resp. of finite type) if and only if its augmented subshift (Λ˜,σ) is sofic (resp. of finite type). This follows by observing that ∅∼ζμ for every μ∈Λ˜*. Every two-sided subshift becomes a compact metric space. Taking the one-sided subshifts to be closed yields the same result in our case. Therefore, every sequence in a subshift has a converging subsequence. This often appears in [30] as the Cantor’s diagonal argument, mainly because metric spaces come later in the presentation of [30]. We preserve this terminology to keep connections with Symbolic Dynamics. However, it is interesting that this argument works to build the one-sided subshift from a set of predetermined forbidden words. The key is that the one-sided full shift is compact and metrizable with the topology given by   ρ(x,y)=2ifx0≠y0,2−kifx≠yandkismaximalsothatx[k,0]=y[k,0],0ifx=y. 2.2. Fock representation We will require some basic theory from Hilbert spaces to show how the quantized dynamics arise from a language. The reader who is not familiar with operator theory may read this subsection in combination with Section 3, where explicit identifications in terms of topological spaces are provided. Operator algebras associated to subshifts were introduced by Matsumoto [31]. Let Λ* be a language on d symbols. Let H=ℓ2(Λ*) and fix the operators Ti such that Tieμ=eiμ if iμ∈Λ* or zero otherwise. We fix   C*(T)≔C*(I,Ti∣i=1,…,d). It is convenient to write TμTν=Tμν even when Tμν=0, that is Tμν=0 if and only if μν∉Λ*. We will also write T∅=I. Likewise we write eμν=0 in ℓ2(Λ*) when μν∉Λ*. The operators Tμ satisfy a list of properties: Tμ*Tμ is an orthogonal projection on span¯{eν∣μν∈Λ*}. TνTν* is an orthogonal projection on span¯{eνμ∣μ∈Λ*}. If ∣μ∣=∣ν∣, then Tμ*Tν=0 if and only if μ≠ν. Tμ*Tμ commutes with Tν*Tν, and with TνTν*. Tμ*Tμ·Ti=Ti·Tμi*Tμi for all i=1,…,d. ∑i=1dTiTi*+P∅=I, where P∅ is the projection on Ce∅. The rank one operator eν↦eμ equals TμP∅Tν*. The ideal K(ℓ2(Λ*)) of compact operators is in C*(T). In [25], the second author with Shalit examines several operator algebras related to the operators Ti. Among them, there are two classes of non-selfadjoint operator algebras: The tensor algebra AΛ* in the sense of Shalit–Solel [38] is defined as the norm-closed subalgebra of B(ℓ2(Λ*)) generated by I and the Ti for i=1,…,d. The tensor algebra TΛ*+ in the sense of Muhly–Solel [35] is defined as the norm-closed subalgebra of B(ℓ2(Λ*)) generated by I, the Ti for i=1,…,d, and the Tμ*Tμ for μ∈Λ*.The relations above imply that   AΛ*=span¯{Tμ∣μ∈Λ*}andTΛ*+=span¯{Tμa∣μ∈Λ*,a∈A}for the unital C*-subalgebra A≔C*(Tμ*Tμ∣μ∈Λ*) of C*(T). 2.3. Q-Projections We will be using the projections generated by the Ti*Ti. To this end we introduce the following enumeration. Write all numbers from 0 to 2d−1 by using 2 as a base, but in reverse order. Hence we write [m]2≡[m]=[m1m2…md] so that 2=[0100…0]. Let the (not necessarily one-to-one) assignment   [m1…md]↦Q[m1…md]≔∏mi=1Ti*Ti·∏mi=0(I−Ti*Ti),where I∈B(ℓ2(Λ*)). For example, we write   Q0=Q[0…0]=∏i=1d(I−Ti*Ti)andQ2d−1=Q[1…1]=∏i=1dTi*Ti. The Q[m] are the minimal projections in the C*-subalgebra C*(I,Ti*Ti∣i=1,…,d) of C*(T). Consequently we obtain ∑[m]=02d−1Q[m]=I. The second author with Shalit have shown in [25] the stronger equality (notice that we sum for [m]≥1 here)   Tμ*Tμ=Tμ*Tμ·∑[m]=12d−1Q[m]=∑[m]=12d−1Q[m]·Tμ*Tμfor all ∅≠μ∈Λ*. It follows that the unit of C*(Ti*Ti∣i=1,…,d) coincides with I∈B(ℓ2(Λ*)) if and only if Λ* induces a left subshift [25, Lemma 4.4]. 3. The quantized dynamics on the allowable words Let Λ* be a language on d symbols. We fix once and for all the unital C*-subalgebra   A≔C*(Tμ*Tμ∣μ∈Λ*)of C*(T). Then A=∪lAl¯ is a unital commutative AF algebra for   Al≔C*(Tμ*Tμ∣μ∈Bl(Λ*)). The C*-algebra A can be characterized by using Λ*. For l≥0, let ∼l be the equivalence relation on Λ* given by the rule   μ∼lν⇔{w∈Bl(Λ*)∣wμ∈Λ*}={w∈Bl(Λ*)∣wν∈Λ*}. Let the discrete space Ωl=Λ*/∼l and write [μ]l for the points in Ωl. Every μ∈Λ* splits Bl(Λ*) into the set of the wi∈Bl(Λ*) for which wiμ∈Λ* and its complement. There is a finite number of such splittings since Bl(Λ*) is finite. They completely identify single points in Ωl, and hence Ωl is a (discrete) finite space. Furthermore, the mapping   ϑ:Ωl+1→Ωl:[μ]l+1↦[μ]lis a well defined (continuous) and onto map. We can then form the projective limit Ω by the directed sequence   for which we obtain the following identification. In [25], the second author with Shalit have shown that   Al≃C(Ωl)andA≃C(Ω). We write ΩΛ* for Ω when we want to highlight the language Λ* to which Ω is related. From now on, we will tacitly identify Al with C(Ωl) and A with C(Ω). In the sequel we will use the same notation Q[m] for the subspaces of Ω that correspond to the projections Q[m] of C*(T). To make this precise, if [m]=[m1…md] is the binary expansion of a number from 0 to 2d−1, then the subspace corresponding to Q[m] consists of the points [μ]∈Ω for which: iμ∈Λ* for all i with mi=1; and iμ∉Λ* for all i with mi=0.We record here the following proposition from [25] for future reference. Proposition 3.1 [25] Let Λ*be a language on d symbols. Then Ωis finite if and only if Λ*is a sofic language. If, in particular, Λ*is of finite type k, then Al=Akfor all l≥k+1. Therefore, if Λ* is sofic, then there exists a stabilizing step k for which Al=Ak for all l≥k+1. When Λ* is in particular of finite type, then we get the following proposition for the possible equivalent classes. Proposition 3.2 Let Λ*be a language on dsymbols. If Λ*is of finite type k, then for all μ∈Λ*with ∣μ∣≥kwe have that  [μ1…μ∣μ∣]k=[μ1…μk]k. Proof Fix an allowable word μ=μ1…μk…μ∣μ∣. By the properties of the language, if wμ∈Λ* then wμ1…μk∈Λ* as well. Conversely, let w∈Λ* such that wμ1…μk∈Λ*. To reach contradiction suppose that wμ∉Λ*. Therefore, there is an n1 and an n2 such that   ν≔wn1…wkμ1…μkμk+1…μn2∉Λ*. Choose n1 and n2 so that ν is a minimal forbidden word. Since μ∈Λ* (hence μ1…μn2∈Λ*) we have that wn1…wk≠∅; hence ∣w∣+1−n1≥1. Similarly, since wμ1…μk∈Λ* (hence wn1…wkμ1…μk∈Λ*) we have that μk+1…μn2≠∅; hence n2≥k+1. Therefore, ν is a forbidden word of length at least   (∣w∣+1−n1)+n2≥k+2.Since Λ* is of type k, the word ν cannot be minimal, which gives the required contradiction.□ We define the maps αi:A→A such that αi(a)=Ti*aTi. It is clear that every αi is a positive map and takes values in A since   αi(Tμ*Tμ)=Tμi*Tμi∈Afor all μ∈Λ*. In fact every αi is an ∗-endomorphism of A, since Tμ*Tμ commutes with TiTi* and TiTi*Ti=Ti. The αi induce the required covariant relation   aTi=Tiαi(a)foralla∈A,i=1,…,dused in the study of TΛ*+. We use the identification of A with C(Ω) to get a translation of each αi as a continuous map φi partially defined on Ω. Let Ai be the direct summand Ti*TiA of A, with unit Ti*Ti. Then Ai is the direct limit of Ali=Al∩Ai, and the corresponding projective limit Ωi is determined by the spaces   Ωli≔{[μ]l∈Ωl∣iμ∈Λ*}and the map   ϑ:Ωl+1i→Ωli:[μ]l+1→[μ]l. Hence αi:A→Ai is a unit preserving map from A=C(Ω) into Ai≔Ti*TiA=C(Ωi), and, therefore, induces a continuous map φi:Ωi→Ω. Proposition 3.3 Let Λ*be a language on dsymbols. With the above notation we have that αi∣Al:Al→Aliis induced by φi:Ωli→Ωlsuch that φi([μ]l+1)=[iμ]l. Proof We have to show that αi(f)=fφi for all f∈Al=C(Ωl). It suffices to do so for f=Tμ*Tμ with μ∈Bl(Λ*). To this end, we have that   αi(Tμ*Tμ)=Tμi*Tμi=χAfor the set A={[w]l+1∣μiw∈Λ*}. On the other hand, we have that Tμ*Tμ=χB for the set B={[w]l∣μw∈Λ*}. Hence we compute   χB(φi([w]l+1))=χB([iw]l)=1ifμiw∈Λ*,0otherwise,which shows that χBφi=χA=αi(χB).□ The universal property of the projective limit implies that this information is enough to describe φi. Indeed, we have that   Ω={(ωn)n≥0∣ωn=[μ]n,μ∈Λ*}.Therefore, for ω∈Ωi, that is ωn=[μ]n such that iμ∈Λ*, we get   φi(ω)=([iμ]n). In the particular case, when Λ* is sofic, let k be the step so that Al=Ak for all l≥k+1. Then we have that Ωl=Ωk for l≥k+1, and hence   Ω≃{[μ]k∣μ∈Λ*}.The projective limit description gives that the induced φi is then given by [μ]k↦[iμ]k, for μ∈Λ* with iμ∈Λ*. Definition 3.4 Let Λ* be a language on d symbols. We call the (A,α)≡(A,α1,…,αd), or alternatively the (Ω,φ)≡(Ω,φ1,…,φd), the quantized dynamics of Λ*. Remark 3.5 The intersection of the subspaces corresponding to the Tμ*Tμ is always non-empty since ∏μ∈Λ*Tμ*Tμe∅=e∅. In fact the intersection consists of the single point ([∅]n)∈Ω. It is evident that ([∅]n) is in the domain of all φμ with μ∈Λ*. 4. Local piecewise conjugacy Let Λ* and M* be languages. Fix their associated quantized dynamics (ΩΛ*,φ) and (ΩM*,ψ), and let Q and P be the corresponding systems of projections. Recall that an ∗-isomorphism γ:C(ΩM*)→C(ΩΛ*) induces a homeomorphism γs:ΩΛ*→ΩM* on the spectra. We have the following notion of conjugacy for the quantized dynamics. For convenience, we use the notation   supp[m]≔{i∈{1,…,d}∣mi=1},where [m]=[m1…md] is the binary expansion of m∈{0,1,…,2d−1}. Definition 4.1 We say that the systems (ΩΛ*,φ) and (ΩM*,ψ) are Q- P-locally piecewise conjugate if there exists a homeomorphism γs:ΩΛ*→ΩM*; and for every x∈Q[m] there is a neighbourhood x∈U⊆Q[m] and an [n]∈{0,1,…,2d−1} such that ∣supp[n]∣=∣supp[m]∣, γs(U)⊆P[n], and   γsφi∣U=ψπ(i)γs∣U,for a bijection π:supp[m]→supp[n]. Equivalently, every Q[m]⊆ΩΛ* has an open cover {Uπ}π indexed by the one-to-one correspondences π:supp[m]→{1,…,d} such that γs(Uπ)⊆P[n] for all [n] with supp[n]=π(supp[m]), and γsφi∣Uπ=ψπ(i)γs∣Uπ. We do not exclude the case where Uπ=∅ for some π. As an immediate consequence, if an ω∈Q[m] is in the intersection of Uπ with Uπ′, then   ψπ(i)γs(ω)=γsφi(ω)=ψπ′(i)γs(ω),for all i∈supp[m]. Remark 4.2 We can use local piecewise conjugacy to define inductively maps fn:Bn(Λ*)→Bn(M*). For every step, we start at the point ω≔([∅]n) which is in the intersection of all subspaces corresponding to Tμ*Tμ for μ∈Λ*. By definition, there exists a neighborhood Uπ⊆Q[1…1] containing ω such that   γsφi(ω)=ψπ(i)γs(ω)foralli=1,…,d. We write π∅,0 for the bijection π and set f1=π∅,0. Consequently, we obtain   γsφμ1(ω)=ψπ∅,0(μ1)γs(ω). For f2, let a word μ=μ2μ1∈B2(Λ*). Since μ2μ1∈Λ*, then ω is in the domain of φμ2φμ1=φμ1μ2. We apply the same argument for ω1=φμ1(ω) and find a bijection πμ,1 coming from the neighborhood Uπμ,1 of ω1 such that   γsφμ2(φμ1(ω))=ψπμ,1(μ2)γs(φμ1(ω))=ψπμ,1(μ2)ψπ∅,0(μ1)γs(ω). In particular, we get that γs(ω) is in the domain of   ψπμ,1(μ2)ψπ∅,0(μ1)=ψπμ,1(μ2)π∅,0(μ1),and hence the word πμ,1(μ2)π∅,0(μ1) is in M*. This procedure gives a well-defined map f2:B2(Λ*)→B(M*). The same argument applies for a word μ1 of length n (instead of just a letter) and a letter μ2, and gives a map fn:Bn(Λ*)→Bn(M*) that is defined by   fn(μn…μ1)≔πμ,n−1(μn)…π∅,0(μ1). The notation πμ,k≡πμk denotes that this bijection comes from the neighborhood Uπμk of the point φμk…μ1(ω). Notice that each fn depends on the point ω≔([∅]n). However, it also depends on the orbit of ω. For a word μ=μn…μ1∈Λ*, we have to keep track where we are at under fn−1, since the nth bijection depends on where φμn−1⋯φμ1(ω) sits. It is clear that   fn(μn…μ1)=πμ,n−1(μn)fn−1(μn−1…μ1),but in general fn(μn…μ1) may be different than f1(μn)fn−1(μn−1…μ1). We present such a case in Example 5.10. Local piecewise conjugacy respects certain properties of languages. This is very pleasing as several invariants for the usual topological conjugacy of subshifts depend on these data. Proposition 4.3 If Λ*and M*are locally piecewise conjugate languages, then the maps fndefined in Remark4.2are bijections. Consequently, ∣Bn(Λ*)∣=∣Bn(M*)∣for all n∈N. Proof First we show that the fn are one-to-one. To this end suppose that   πμ,n−1(μn)…π∅,0(μ1)=πν,n−1(νn)…π∅,0(ν1).Then we get that πμ,i−1(μi)=πν,i−1(νi) for all i=1,…,n. In particular we have that π∅,0(μ1)=π∅,0(ν1) and therefore μ1=ν1. Consequently we get that πμ,1=πν,1, hence μ2=ν2. Inductively we have that μi=νi for all i=1,…n. Therefore we get ∣Bn(Λ*)∣≤∣Bn(M*)∣. By symmetry we obtain equality which implies that the fn are bijections.□ Remark 4.4 Since the functions fn are bijections, then γs(ω) is in the domain of all ψν for ν∈M*, for ω=([∅]n). Hence γs fixes the points ([∅]n) of ΩΛ* and ΩM*. The entropy of a language Λ* is given by h(Λ*)≔limnn−1log2∣Bn(Λ*)∣. The following corollary is immediate. Corollary 4.5 Local piecewise conjugate languages share the same entropy. Local piecewise conjugacy respects the class of sofic languages. However, as we will see in Example 5.10, it does not preserve the class of SFT’s. Proposition 4.6 Let Λ*and M*be locally piecewise conjugate languages. If Λ*is sofic, then so is M*. Proof Immediate by Proposition 3.1.□ 5. Applications to the follower set graph The quantized dynamics can be described by a finite graph in the case of the sofic languages. Fix a sofic language Λ* on d symbols. For every μ∈Λ*, let   F(μ)≔{w∈Λ*∣wμ∈Λ*}be the follower set of μ. We write FΛ*(μ) when we want to highlight the language to which we refer. (We will rarely use this notation, as the language will be clear from the context.) Recall that the elements in Ω are of the form ([μ]n) for μ∈Λ*. Since ([μ]n)=([ν]n) if and only if F(μ)=F(ν) we have a bijection between Ω and the set {F(μ)∣μ∈Λ*}. By definition, soficity is equivalent to Ω being finite. In this case, Proposition 3.1 provides the existence of a stabilizing step k such that Ωl=Ωk for all l≥k+1; that is every F(μ) can be identified with [μ]k. The follower set graph GΛ*=(GΛ*,L) of a sofic language on d symbols is an edge-labeled graph, where L is a coloring map from the edge set GΛ*(1) onto {1,…,d}, defined as follows: The vertices of GΛ* are given by the follower sets. We draw an edge labeled i from F(μ) to F(iμ) if and only if F(iμ)≠∅.Therefore, we have an edge (labeled i) between ω=[μ]k and ω′=[ν]k if [iμ]k=[ν]k, that is, if φi(ω)=ω′. It follows that the labeled graph gives a representation of the quantized dynamics. Notice that this procedure gives also the follower set graph in the case of a two-sided subshift [30, Section 3.2] (when everything is written in the opposite direction). Notice that we do not use soficity for the construction of the follower set graph, and one may be tempted to produce a follower set graph for any language. However, this is not the right thing to do. As we saw in Section 3, the spectrum of A is totally disconnected in general and configuring the dynamics with a discrete structure would not comply with the topology. By Proposition 3.1, it is exactly when Λ* is sofic that the spectrum is discrete and we are able to do so without losing information. Example 5.1 Let Λ* be of type 1. Then Ω≃Ω1={[i1]1,…,[ir]1} for some symbols i1,…,ir∈{1,…,d}. For the follower set graph, we have an edge labeled j from [i]1=F(i) to [ji]1=[j]1=F(j) whenever ji∈Λ*. Notice here the use of Proposition 3.2. The case of type 1 languages requires less information to store, as the label of an edge coincides with the one-lettered word that labels the vertex where the edge terminates. Remark 5.2 There is a strong connection between the follower set graph of a sofic language Λ* and that coming from its augmented subshift (Λ˜,σ). Suppose that Λ* is on d symbols and G is its follower set graph. Recall that F(∅)=F(ζμ) for every μ∈Λ˜* for the added distinguished symbol ζ. Therefore, the follower set graph of Λ˜* contains G, shares the same vertex set and contains in addition edges from every vertex to F(∅) labeled by ζ. When the language is of finite type then we can find a finite set of forbidden words that describes it. By using this, we can determine the follower set graph in a finite number of steps. Indeed the dual of Proposition 3.2 suggests that the determination of every F(μ) can be achieved in finite steps, even though F(μ) may be infinite in principle. Proposition 5.3 Let Λ*be a language of finite type k. For every ∅≠μ∈Λ*and wn…w1∈Λ*with n≥k, we have that  wμ∈Λ*ifandonlyifwk…w1μ∈Λ*. Proof The proof is the same to that of Proposition 3.2 once the words are reversed.□ Therefore, in order to distinguish between the F(μ), it suffices to distinguish between the finite sets   {w∈Bk(Λ*)∣wμ∈Λ*}. Consequently, we obtain the following algorithm for constructing the follower set graph of a language of type k. Follower Set Graph Algorithm. Let Λ* be a language of finite type and let F be a finite set of forbidden words that defines Λ*. The algorithm takes F as an input and has output the follower set graph. We have two cases: Case 1. If F=∅, then Λ*=F+d and its follower set graph is the Hawaiian ring on d edges (lines 1–9 in the pseudocode). Case 2. IfF is non-empty, we write  k≔max{∣μ∣∣μ∈F}−1. Then Λ* is a language of type k (lines 10–16 in the pseudocode). We then generate the set Bk(Λ*) of allowable words of length k (lines 17–24 in the pseudocode). We then proceed to forming the follower set graph in two steps: Step 1: Determining the vertex set {F(μ)∣μ∈Λ*} (lines 25–44 in the pseudocode). It suffices to compute the F(μ) for ∣μ∣≤k (Proposition 3.2). Nevertheless, we will also compute the F(μ) for ∣μ∣=k+1. This will be helpful for writing the edges. To this end, we construct a table indexed by Bk(Λ*) whose entries show whether an allowable word can follow another one. If two rows are the same, then the follower sets of the corresponding words determine the same vertex on the graph. It is convenient to multi-label a vertex by using all the F(μ), for ∣μ∣≤k+1, that coincide. – Form a table indexed by Bk(Λ*) (followerTable). We fix an enumeration {μ1,…,μN} of these words, including the void word which we set to be μ1. For the (μi,μj) entry of the table write μjμi. If μjμi∈Λ*, then write TRUE for the entry; otherwise write FALSE. In this way, we create a second table (truthTable). By Proposition 5.3, these finite entries are sufficient for identifying the follower sets. – Compute the F(μ) for μ of length k+1 (lines 45–56 in the pseudocode). If μ is forbidden, then do nothing. If μ=μ1μk…μk+1 is allowable, then add the label F(μ) to the vertex that has F(μ1…μk) among its labels. Step 2: Determining the edges of the follower set graph (lines 57–69 in the pseudocode). We read the rows of the truthTable from top to bottom. If we pass to a row that is the same with one already read, we move to the next (that is we identify the words with the same follower sets). – Start from μl=μ1=∅ and repeat for l=2,…,N. Let μl be a word from {μ1,…,μN}. Start with i=1 and repeat for i=2,…,d. Write an edge labeled i from the vertex with label F(μl) to the vertex with label F(iμl) if there is a vertex that contains the label F(iμl). Otherwise do nothing and go to i+1. After we finish for i=d (the last step), we repeat for μl+1. Remark 5.4 In the following pages, we give the pseudocode for the algorithm. Notice that in the output, each node is a set of words, and each word associated with a node generates the same follower set. We remind that we were not concerned about the technical features of this algorithm, but more about that it does terminate. Algorithm Generate FollowerSetGraph(F, symbolSet)   Input: F = the list of forbidden words generating the language.    symbolSet={1,…,d}.    Output : Graph=(Nodes,Edges) where:    – each node in Nodes is the set of words of length less than or equal to k+1 which have the same follower set; and    – each edge in Edges is of the form (source, destination, label).    //Deal with the case that Fis the empty set.   1  if F is empty then   2 3 4 5 6 7 8  initializenodeasalistcontainingsymbolSetandtheemptywordinitializenodesasalistcontainingonlynodeinitializeedgesasemptylistforeachletter∈symbolSetdo∣insert(node,node,letter)intoedgesendreturn(nodes,edges)   9  end    //IfFis non-empty find value ofkfromF.  10  initialize maxLength as 1  11  for each word∈Fdo  121314  iflengthofword>maxLengththen∣maxLength⟵lengthofwordend  15  end  16  initialize k as maxLength−1    //Generate mu, the set of allowed words of length at most k.  17  initialize mu as a list containing only the empty word  18  foreach n∈{1,2,…,k}do  1920212223  foreachword∈setofallpossiblewordsoversymbolSetoflengthndoifwordcontainsnoelementofFasasubwordthen∣insertwordintomuendend  24  end    //Form tables. Indexed by base word and concatenated word.  25  for each word a∈mudo  2627282930313233  foreachwordb∈mudofollowerTable[a][b]⟵concatenatebwithaiffollowerTable[a][b]containsnoelementofFasasubwordthen∣truthTable[a][b]⟵Trueelse∣truthTable[a][b]⟵Falseendend  34  end    //Each node of the graph is a set of words sharing the same follower set.  35  initialize Nodes as empty list;  36  for each row∈unique rows of truthTabledo  37383940414243  initializenodeasanemptylist;foreachworda∈mudoiftruthTable[a]=rowthen∣insertaintonode;endendinsertnodeintoNodes;  44  end    //Add words of length k+1to the appropriate node.  45  foreachnode∈Nodesdo  46474849505152535455  foreachword∈nodedoiflengthofword=kthenforeachletter∈symbolSetdonewWord⟵concatenatewordwithletter;ifnewWordcontainsnoelementofFasasubwordthen∣insertnewWordintonode;endendendend  56  end    //Create edges.  57  initialize Edges as empty list;  58  for each node∈Nodesdo  59606162636465666768  foreachletter∈symbolSetdonewWord⟵concatenateletterwithanywordinnodeoflengthlessthank+1;//𝙵𝚒𝚗𝚍 𝚠𝚑𝚒𝚌𝚑 𝚗𝚘𝚍𝚎newWord𝚋𝚎𝚕𝚘𝚗𝚐𝚜 𝚝𝚘,𝚒𝚏 𝚊𝚗𝚢foreachnode′∈NodesdoifnewWord∈node′thenedge⟵(node,node′,letter);insertedgeintoEdges;break;endendend  69  end      70  return (Nodes, Edges);    Input: F = the list of forbidden words generating the language.    symbolSet={1,…,d}.    Output : Graph=(Nodes,Edges) where:    – each node in Nodes is the set of words of length less than or equal to k+1 which have the same follower set; and    – each edge in Edges is of the form (source, destination, label).    //Deal with the case that Fis the empty set.   1  if F is empty then   2 3 4 5 6 7 8  initializenodeasalistcontainingsymbolSetandtheemptywordinitializenodesasalistcontainingonlynodeinitializeedgesasemptylistforeachletter∈symbolSetdo∣insert(node,node,letter)intoedgesendreturn(nodes,edges)   9  end    //IfFis non-empty find value ofkfromF.  10  initialize maxLength as 1  11  for each word∈Fdo  121314  iflengthofword>maxLengththen∣maxLength⟵lengthofwordend  15  end  16  initialize k as maxLength−1    //Generate mu, the set of allowed words of length at most k.  17  initialize mu as a list containing only the empty word  18  foreach n∈{1,2,…,k}do  1920212223  foreachword∈setofallpossiblewordsoversymbolSetoflengthndoifwordcontainsnoelementofFasasubwordthen∣insertwordintomuendend  24  end    //Form tables. Indexed by base word and concatenated word.  25  for each word a∈mudo  2627282930313233  foreachwordb∈mudofollowerTable[a][b]⟵concatenatebwithaiffollowerTable[a][b]containsnoelementofFasasubwordthen∣truthTable[a][b]⟵Trueelse∣truthTable[a][b]⟵Falseendend  34  end    //Each node of the graph is a set of words sharing the same follower set.  35  initialize Nodes as empty list;  36  for each row∈unique rows of truthTabledo  37383940414243  initializenodeasanemptylist;foreachworda∈mudoiftruthTable[a]=rowthen∣insertaintonode;endendinsertnodeintoNodes;  44  end    //Add words of length k+1to the appropriate node.  45  foreachnode∈Nodesdo  46474849505152535455  foreachword∈nodedoiflengthofword=kthenforeachletter∈symbolSetdonewWord⟵concatenatewordwithletter;ifnewWordcontainsnoelementofFasasubwordthen∣insertnewWordintonode;endendendend  56  end    //Create edges.  57  initialize Edges as empty list;  58  for each node∈Nodesdo  59606162636465666768  foreachletter∈symbolSetdonewWord⟵concatenateletterwithanywordinnodeoflengthlessthank+1;//𝙵𝚒𝚗𝚍 𝚠𝚑𝚒𝚌𝚑 𝚗𝚘𝚍𝚎newWord𝚋𝚎𝚕𝚘𝚗𝚐𝚜 𝚝𝚘,𝚒𝚏 𝚊𝚗𝚢foreachnode′∈NodesdoifnewWord∈node′thenedge⟵(node,node′,letter);insertedgeintoEdges;break;endendend  69  end      70  return (Nodes, Edges);  Let us illustrate with an example how the follower set graph is constructed from the Follower Set Graph Algorithm. Example 5.5 Let the symbol set be {0,1}. Let Λ* be the language defined by the words F={101,110}. We form the followerTable of the Follower Set Graph algorithm with all the words of length at most two. As we add words on the left, it is convenient to put the labels at the right of the followerTable:     For simplicity, we can just indicate the forbidden words by a black square in the truthTable of the Follower Set Graph Algorithm:     This gives the following five vertices:   F(∅),F(0)=F(00)=F(000)=F(001),F(1)=F(11)=F(111),F(10)=F(100),F(01)=F(010)=F(011),where we used Proposition 3.2 to provide the classes for the allowable words of length 3. Notice here that F(101) and F(110) are meaningless as 101 and 110 are forbidden words. Then the follower set graph is   By definition, the follower set graph is (left-)resolving, that is different edges with the same source carry different labels. Recall that a graph is called follower-separated if distinct vertices have distinct follower sets (set of paths starting on the vertex). This is also a property that the follower set graph has. Indeed, if two vertices, say F(μ) and F(ν) have the same follower sets on the graph then every path w that starts from μ can also start at ν. This shows that wμ∈Λ* if and only if wν∈Λ* for all w∈Λ*, hence F(μ)=F(ν). Resolving graphs go by the name of Shannon graphs in the literature. Another type of a Shannon graph for subshifts is produced through the Krieger cover [29]. The Krieger cover is constructed by taking the follower sets on one-way infinite words. This is the important difference with the follower set graph here, as we consider the vertices to be the follower sets on the finite words. For a nice exposition (among others) on the Krieger cover, the reader can see also [21]. Given two languages, we can consider their labeled graphs or the ambient unlabeled graphs. Graph isomorphism within each class is translated to a different level of equivalence. 5.1. Labeled graph isomorphism Let G1=(G1,L1) and G2=(G2,L2) be two edge-labeled graphs such that the labels L1,L2 take values on the same symbol set {1,…,d}. The labeled graphs G1 and G2 are called labeled graph isomorphic if there is a graph isomorphism (∂γ,γ):G1→G2 and a bijection π on the symbol sets such that   L2(γ(e))=π(L1(e))foralledgese.(We use the notation ∂γ:G1(0)→G2(0) and γ:G1(1)→G2(1) for the graph isomorphism.) Proposition 5.6 Let Λ*and M*be sofic languages. Then their follower set graphs are isomorphic if and only if there is a bijection on their symbol sets that preserves the allowable words. Proof Suppose there is an edge-labeled graph isomorphism. By construction, there is at least one vertex, that is the vertex FΛ*(∅) (resp. FM*(∅)), that emits edges of all labels. Hence max{s−1(v)∣vvertex} coincides with the number of the symbols, thus d=d′. The graph isomorphism identifies FΛ*(∅) with some FM*(ν). Then the edge e=(FΛ*(∅),FΛ*(i)) labeled i corresponds to the edge   γ(e)=(FM*(ν),FM*(π(i)ν))labeled π(i). Suppose ji∈Λ*. Then the edge f=(FΛ*(i),FΛ*(ji)) labeled j corresponds to the edge   γ(f)=(FM*(π(i)ν),FM*(π(j)π(i)ν))labeled π(j). Therefore, we get that π(j)π(i)∈M*. Hence π respects the allowable words of length 2. Inductively, we get that permutation of the edges along the vertices extends to a match on the allowable words of any length. Notice here that π−1 is the associated labeling for the inverse graph isomorphism. Conversely, the existence of a bijection π on the symbol sets implies that d=d′. Furthermore, π extends to the allowable words such that π(μν)=π(μ)π(ν). We define the edge-labeled graph homomorphism by   ∂γ(FΛ*(μ))≔FM*(π(μ)). The map ∂γ is well defined. Indeed if FΛ*(μ)=FΛ*(ν), then we get that   {w∈Λ*∣wμ∈Λ*}={w∈Λ*∣wν∈Λ*}. Applying π and working backwards we have that FM*(π(μ))=FM*(π(ν)). It follows that ∂γ is also one-to-one, hence a bijection due to the properties of π. If there is an edge e=(FΛ*(μ),FΛ*(iμ)) labeled i, then we have that iμ∈Λ* and thus π(iμ)=π(i)π(μ)∈M*. Therefore, there is an edge labeled π(i) connecting FM*(π(μ)) and FM*(π(iμ)); we write γ(e) for that edge. Since the edges on the same source have different labels we have that γ(e) is unique. In this way, we extend ∂γ to a map γ on the edges. Then (∂γ,γ) gives the required edge-labeled graph isomorphism.□ Remark 5.7 Proposition 5.6 is straightforward when Λ* and M* are languages of subshifts. In this case, (XΛ*)*=Λ* and the bijection gives a 1-block code between the subshifts. However, here we provide the same result even when (XΛ*)*⊂Λ* and with no reference to possible subshifts that the languages induce. 5.2. Unlabeled graph isomorphisms By an unlabeled graph isomorphism between two edge-labeled graphs we will mean a simple isomorphism of the ambient graphs with the labels omitted. Theorem 5.8 Let Λ*and M*be sofic languages. Then their unlabeled follower set graphs are isomorphic if and only if their quantized dynamics are locally piecewise conjugate. Proof By assumption, the spaces ΩΛ* and ΩM* are (discrete) finite spaces. First suppose that the unlabeled follower set graphs are isomorphic. Then we obtain an induced bijection γs on the vertices, and thus a homeomorphism γs:ΩΛ*→ΩM*. Let a point ω∈Q[m]. Then the number of the edges emitted by ω coincides with the number of edges emitted by γs(ω). Due to the follower set graph construction, this implies that γs(ω)∈P[n], with ∣supp[n]∣ coinciding with the number of emitted edges from γs(ω), thus with ∣supp[m]∣. Now the graph isomorphism implies a bijection, say π, between edges. For convenience, let supp[m]={i1,…,ir} and supp[n]={j1,…,jr}, such that π(il)=jl for all l=1,…r. Then the terminal vertex of il is mapped to the terminal of π(il). Consequently, we obtain   γsϕil(ω)=ψπ(il)γs(ω)foralll=1,…,r.Taking U={ω} gives the required local piecewise conjugacy. Conversely, suppose that (ΩΛ*,φ) and (ΩM*,ψ) are locally piecewise conjugate. Then the ambient spaces are homeomorphic, that is there is a bijection on the vertex sets. Moreover, for every point ω∈Q[m], we have γs(ω)∈P[n] with ∣supp[m]∣=∣supp[n]∣, such that   γsφi(ω)=ψπ(i)γs(ω)foralli∈supp[m]. Recall that φi is defined on ω if and only if i∈supp[m]. Hence local piecewise conjugacy gives that the number of edges that ω emits is ∣supp[m]∣, and thus it equals the number of edges that γs(ω) emits, which is ∣supp[n]∣. The above equation then shows that for the edge labeled i there is a unique edge labeled π(i), such that the end point of the i-edge is mapped to the endpoint of the π(i)-edge. As edges are preserved under γs, we get the required graph isomorphism.□ Remark 5.9. There is a conceptual difference between labeled and unlabeled graph isomorphism. In the first case, the isomorphism on the edges is given by the bijection on the labels, and is the same on all vertices. However, in the second case, the bijection changes each time we pass to another vertex. Removing the labels from a graph representation G=(G,L) of a subshift Λ does not preserve type. In the case of sofic subshifts, this procedure amounts to producing finite covers and hints that unlabeled graph isomorphisms of the follower set graphs should not (Notice the subtle point here: If Λ is a sofic subshift, then removing all labels from its follower set graph G=(G,L) produces an edge-shift XG, and Λ is a factor of XG. However, it is not ensured that G is the follower set graph of XG.) preserve SFT’s. The following example clarifies this point. Example 5.10 Let Λ* be the language of the even shift on {0,1}, that is the forbidden words are of the form 102n+11 for all n≥0. Then the vertices of the follower set graph are of the form   F(μ)=F(∅)ifμcontainsno1’s,F(1)ifμbeginswith02k1forsomek≥0,F(01)ifμbeginswith02k+11forsomek≥0,for Λ*. Therefore, the follower set graph takes up the form   For more details, see [30, Example 3.2.7, Figure 3.2.2]. On the other hand, let M* on {0,1} be defined by the forbidden word 001. Then the truthTable of the Follower Set Graph algorithm is given by:     (with FALSE indicated by a black box). Therefore, we have the vertices   F(∅)=F(0)=F(00)=F(000),F(1)=F(10)=F(11)=F(100)=F(101)=F(110)=F(111),F(01)=F(010)=F(011).Then the follower set graph for M* is   It is immediate that the unlabeled graphs for Λ* and M* are isomorphic. However, Λ* is not a language of finite type. Notice also how the functions fn:Bn(Λ*)→Bn(M*) of Remark 4.2 depend on n in this example. For example, we have that   f4(0010)=1010≠0010=f1(0)f3(010).Recall that the image of w under fn is taken through the unlabeled graph isomorphism, when w corresponds to the path w beginning at ω=F(∅). Remark 5.11 Recall that a point x in a two-sided subshift Λ has period n if there exists an n∈N such that σn(x)=x. The number of points with period n is denoted by pn(Λ). The zeta function of Λ is given by   ζΛ(t)≔exp(∑n=1∞pn(Λ)ntn). The zeta function is not preserved by local piecewise conjugacy. In Example 5.10, we show that the even shift is locally piecewise conjugate to a subshift of finite type. By [30, Theorem 6.4.6], the zeta function of any subshift of finite type is the reciprocal of a polynomial, whereas [30, Example 6.4.5] implies that the zeta function of the even shift is ζ(t)=(1+t)(1−t−t2)−1. We would like to thank Ian Putnam for this remark. 5.3. Graph isomorphism for type 1 languages Let us examine further the case of type 1 languages. We begin with an example of two languages with no isomorphic labeled follower set graphs. Example 5.12 Consider the language Λ* on 5 symbols determined by the forbidden words   {11,21,31,41,12,22,32,42,13,33,24,44}.Then the resulting follower set graph is   for Λ*. Consider also the language M* on 5 symbols determined by the forbidden words   {11,21,31,41,12,22,32,42,13,33,14,44}.The only difference with the forbidden words of Λ* is to consider 14 in place of 24. Similarly we get the follower set graph   for M*. The unlabeled graphs are isomorphic. The only difference in the labeled graphs is the lower right arrow which carries different labels. As the follower set graphs are irreducible, it can be seen that the languages Λ* and M* are also the languages of two-sided irreducible subshifts. In fact these are the augmented versions of [25, Example 9.8]. A key feature in the example above is that there are two symbols that have the same follower sets. It appears that this is the only obstruction. Theorem 5.13 Let Λ*and M*be languages of type 1. Suppose that Λ*is on dsymbols and that FΛ*(i)≠FΛ*(j)for i≠j, with i,j=1,…d. The following are equivalent: The follower set graphs are isomorphic. The unlabeled follower set graphs are isomorphic. Proof Of course item (i) implies item (ii). For the converse, recall that unlabeled graph isomorphism imposes that M* is on d symbols as well. Furthermore, the vertex sets must have the same size. We have two cases. If FΛ*(i)≠FΛ*(∅) for all i, then the same must hold for the graph of M*, as d+1 is the maximum size of the vertex sets. In this case, both FΛ*(∅) and FM*(∅) are the unique sources for the graph, hence related by the graph isomorphism. If there is an i such that FΛ*(i)=FΛ*(∅), then the number of the FM*(j) is d, and thus there exists a unique j such that FM*(j)=FM*(∅). In any case, we get that the graph isomorphism induces a bijection between {FΛ*(i)∣i=1,…,d} and {FM*(i)∣i=1,…,d}. Without loss of generality, we may relabel for M* so that this bijection sends FΛ*(i) to FM*(i). As we remarked in Example 5.1, the labels on the edges are pre-determined by their range. Notice that by hypothesis every vertex receives at most one edge. Therefore, the unlabeled graph isomorphism respects the label of the edges, and the proof is complete.□ Remark 5.14 For type 1 subshifts, there is a strong connection between the followerTable of the Follower Set Graph algorithm and the representation of the subshift as an edge shift. Let us recall how this follows from [30, Theorem 2.3.2]. To allow comparisons, we denote the graph of the edge shift by Ge(Λ). Let Λ be a two-sided subshift of type 1. The vertices of Ge(Λ) are the symbols of Λ. We write an edge between the vertex i and j if (and only if) ji∈Λ*, and label the resulting edge by j. It is evident that this graph is given by a 0–1 adjacency matrix. If A is the followerTable of the Follower Set Graph algorithm where we replace the allowable words by 1 and the forbidden words by 0, then the adjacency matrix of Ge(Λ) is taken by deleting the row and the column that corresponds to ∅ in A (which all have entries equal to 1). Working under the condition that F is one-to-one on the symbol set (of Theorem 5.13) we distinguish two cases: Case 1. If F(∅)=F(i) for some symbol i, then Ge(Λ) coincides with the graph G of the follower set graph G=(G,L) of Λ. Case 2. If F(∅)≠F(i) for all symbols i, then Ge(Λ) coincides with the subgraph G of the follower set graph G=(G,L) of Λ, once we erase the vertex F(∅) and the emitting edges. Remark 5.15 Theorem 5.13 applies to edge shifts with invertible adjacency matrices. Indeed let Ae be the adjacency matrix of Ge(Λ). If there are i≠j with F(i)=F(j), then we have that two rows of the matrix from the Follower Set Graph algorithm coincide. Thus the same holds for Ae and thus detAe=0. Remark 5.16 Theorem 5.13 holds in the particular case of languages of type 1 on two symbols {0,1} without the assumption on the follower set function. Indeed the cases where F(0)=F(1) produce the following graphs:   for the sets of forbidden words ∅ and {00,10,01,11}, and   for the sets of forbidden words {00,01} and {11,10}, respectively. It follows that also in these cases, the follower set graphs are unique up to a permutation of symbols. However, one direction of Theorem 5.13 does not hold in general even for languages of finite type. We highlight this in the following two examples. Example 5.17 Let the language Λ* on two symbols {0,1} be defined by the forbidden words   000,010,001,101,011.Then the corresponding truthTable of the Follower Set Graph algorithm is:     and gives the vertices   v0=F(∅),v1=F(1),v2=F(0),v3=F(01),v4=F(00)=F(10)=F(11)=F(100)=F(110)=F(111).Then the follower set graph of Λ* is   On the other hand, let the language M* on two symbols {0,1} be defined by the forbidden words   000,010,001,100,011.Then the corresponding truthTable of the Follower Set Graph algorithm is:     and gives the vertices   w0=F(∅),w1=F(0),w2=F(1),w3=F(00),w4=F(10)=F(01)=F(11)=F(101)=F(111)=F(110).Then the follower set graph of M* is   It is clear that there is only one unlabeled graph isomorphism; the one sending vi to wi. If it lifted to a labeled graph isomorphism, then the 0 label would match to the 0 label, as it appears from v4 and w4. However, this does not comply with the labels on v0 and w0. The two-sided subshifts coming from the languages Λ* and M* are formed on a single point. For creating a more interesting counterexample in the category of two-sided subshifts, we may use the augmentations defined in {0,1,ζ}. In this case, we obtain   as the follower set graph of Λ˜* and   as the follower set graph of M˜*. Again there is not a labeled graph isomorphism between those. Example 5.18 Let the language Λ* on two symbols {0,1} be defined by the forbidden words   000,100,010,101,011,111.Then the corresponding truthTable of the Follower Set Graph algorithm is:     and gives the vertices   v0=F(∅),v1=F(0),v2=F(1),v3=F(10),v4=F(01)v5=F(00)=F(11)=F(110)=F(001).Then the follower set graph of Λ* is   On the other hand, let the language M* on two symbols {0,1} be defined by the forbidden words   000,110,010,101,001,111.Then the corresponding truthTable of the Follower Set Graph algorithm is:     and gives the vertices   w0=F(∅),w1=F(0),w2=F(1),w3=F(00),w4=F(11)w5=F(10)=F(01)=F(100)=F(011).Then the follower set graph of M* is   We see that that there are two unlabeled graph isomorphisms. The first one sends vi to wi and the second one is the composition with the reflection along the vertical line that passes through w0 and w5. Both of them do not lift to a labeled graph isomorphism as the path 11 connecting v1 with v5 consists of two edges with the same label whereas its image 10 has two edges of different labels. The languages Λ* and M* do not arise from subshifts, but once more we can use their augmentations to produce a counterexample in this class. We thus have   for Λ˜*, and   for M˜*. Again the follower set graphs are not labeled graph isomorphic. 5.4. Irreducible two-sided sofic subshifts Let us examine further the case of irreducible two-sided sofic subshifts. Recall that a subshift is called irreducible if it has a presentation through a labeled graph G=(G,L) so that G is irreducible. Among all presentations, Fischer [17, 18] has shown that there exists a minimal resolving one, that is unique up to label graph isomorphism; see also [30, Theorem 3.3.18]. Minimality is taken with respect to the number of vertices of the ambient graph. This presentation is also known as the Fischer cover of the subshift. Fischer covers are follower-separated; for example see [30, Corollary 3.319]. Uniqueness of the Fischer cover fails for reducible sofic subshifts; Jonoska [22] provides such a counterexample for subshifts of finite type. The Fischer cover can be induced by the Krieger cover or by the follower set graph. A way to obtain it from the follower set graph is as follows. Recall that a word μ is called intrinsically synchronizing if:   wheneverνμ∈Λ*andμν′∈Λ*thenνμν′∈Λ*. Then the minimal resolving presentation is the labeled subgraph of the follower set graph formed by using just the follower sets of the intrinsically synchronizing words (It is worth mentioning that a subshift is of finite type if and only if all sufficiently long words are intrinsically synchronizing [30, Exercise 3.3.5]. This property emphasizes the value of Jonoska’s [22] counterexample.) (see [30, Exercise 3.3.4]). We will see how the graph isomorphism we get from Theorem 5.8 induces an unlabeled graph isomorphism of the Fischer covers. Moreover, that it respects the vertices labeled by follower sets of intrinsically synchronizing words. For the latter, we will require some terminology and results from [30]. Recall that we follow the left version of their notation. Given a labeled graph G=(G,L) we say that a path μ is synchronizing for G if:   allpathslabeledμterminateatthesamevertex. Recall that we read paths from right to left. Suppose in addition that G is resolving and follower separated. In this case, every path w can be extended on the left to a synchronizing path μ=uw for G by [30, Proposition 3.3.16]. Under the same assumption, if μ is synchronizing for G, then every path uμ is synchronizing for G by [30, Lemma 3.3.15]. The connection between synchronizing paths and intrinsically synchronizing words is given in [30, Exercise 3.3.3]. That is, if G is the minimal resolving presentation of a two-sided irreducible sofic shift Λ then a path w is synchronizing for G if and only if the word w is intrinsically synchronizing for Λ. Now we have set the context for proving the next corollary. Corollary 5.19. Let Λand Mbe two-sided irreducible sofic subshifts. If Λand Mare locally piecewise conjugate, then there is an unlabeled graph isomorphism between their Fischer covers. Furthermore, the unlabeled graph isomorphism induces a bijection between the collections  {FΛ(μ)∣μisanintrinsicallysynchronizingwordforΛ}and  {FM(ν)∣νisanintrinsicallysynchronizingwordforM}. Proof Let GΛ=(GΛ,L1) and GM=(GM,L2) be the follower set graphs of Λ and M, respectively. Let HΛ be the labeled graph that remains from GΛ by using only the intrinsically synchronizing words, i.e. the Fischer cover of Λ. The graph isomorphism of Theorem 5.8 then gives an isomorphism of the ambient graph HΛ of HΛ onto a subgraph HM of GM. Let HM be the labeled subgraph induced by HM inside GM. Notice that both HΛ and HM are resolving as subgraphs of the resolving follower set graphs and thus   h(X(HΛ))=h(X(HΛ))andh(X(HM))=h(X(HM))by [30, Proposition 4.13]. First, we claim that HM gives a presentation of M. If X(HM)≠M then X(HM) is a proper subshift of M and, therefore, [30, Corollary 4.4.9] yields   h(X(HM))<h(M). Since graph isomorphism of HΛ with HM respects entropy, we get that   h(Λ)=h(X(HΛ))=h(X(HΛ))=h(X(HM))=h(X(HM))<h(M). However, this contradicts h(M)=h(Λ) of Proposition 4.3. Hence HM is a presentation of M. Notice here that the number of vertices of HM coincides with the minimal number of vertices required to describe Λ, that is   ∣(HM)(0)∣=∣(HM)(0)∣=∣(HΛ)(0)∣=∣(HΛ)(0)∣. Secondly, we claim that HM is minimal for M with respect to the number of vertices. Otherwise we could find a subgraph HM′ of GM on less vertices than that of HM. Then the unlabeled graph isomorphism would carry over, as above, to a presentation HΛ′ of Λ, giving   ∣(HΛ′)(0)∣=∣(HM′)(0)∣<∣(HM)(0)∣=∣(HΛ)(0)∣. However, this contradicts minimality of HΛ for Λ, and thus HM is a minimal presentation of M. So far we have proved that HM is a minimal resolving presentation of M. Initially, HM is isomorphic to the subgraph of GM obtained by using the vertices labeled by follower sets of intrinsically synchronizing words. We will show now that these two sub-graphs of GM are actually equal. Due to minimality, it suffices to show that the vertices of HM correspond to follower sets of intrinsically synchronizing words. We will use that HM is follower-separated and irreducible as a minimal presentation of M [30, Corollary 3.3.19]. Let F(w) be a vertex of HM. We will show that   ∃anintrinsicallysynchronizingwordν∈M*suchthatF(w)=F(ν). Consider a labeled path u1 in HM starting at a vertex J and extend it to a synchronizing path u2u1 in HM. That is, all paths labeled u2u1 end at the same vertex, say J′ in HM. By irreducibility of HM, there is a path u3 connecting J′ with F(w) so that the path u3u2u1 is allowable in HM. Moreover, this is an extension of a synchronizing path and thus it is a synchronizing path in HM, that is all paths in HM representing u3u2u1 end at F(w). Minimality of HM implies that u3u2u1 is an intrinsically synchronizing word for M. As HM represents M, the follower set of u3u2u1 in M coincides with the paths in HM starting at the vertex F(w). However, the collection of these paths is exactly the set F(w) due to the follower set graph construction. Hence we conclude that F(w)=F(u3u2u1) and the proof is complete.□ Remark 5.20 Referring to the proof of Corollary 5.19, we do not claim a direct bijection between intrinsically synchronizing words. It is unclear whether the function of Proposition 4.3 respects this property. The main obstacle is that the word   fn−1(f∣ν∣(ν)μf∣ν′∣(ν′))forn=∣ν∣+∣μ∣+∣ν′∣gives a word ν″μ′ν′ instead of νμ′ν′ (as indicated in Example 5.10). A similar obstacle does not allow checking directly whether the mixing property is also preserved under local piecewise conjugacy. Example 5.21 We require both Λ and M be irreducible in Corollary 5.19. This is because local piecewise conjugacy does not preserve irreducibility. For an example recall the even shift and the subshift of finite type constructed in Example 5.10. The even shift Λ is irreducible and the subgraph   gives its Fischer cover. However, the subshift M on {001} is not irreducible. Any subgraph on less vertices produces a proper subshift of M. In particular, the unlabeled graph isomorphism from the Fischer cover of Λ to M produces the irreducible subgraph   which does not represent 0∞∈M. Example 5.22 The converse of Corollary 5.19 does not hold. That is, irreducible subshifts that are not locally piecewise conjugate may have Fischer covers that admit an unlabeled graph isomorphism. For a counterexample, let M be defined by the forbidden word {00}. Then the truthTable of M is     and thus its follower set graph is given by   where v0=F(∅)=F(1) and v1=F(0). Consequently, M is irreducible (having one irreducible presentation) and its Fischer cover coincides with its follower set graph. Comparing with the even shift Λ, we see that the Fischer covers of Λ and M admit an unlabeled graph isomorphism, but this does not hold for their follower set graphs. Acknowledgements This project started as a continuation of with Orr Shalit. Following his suggestion, it was decided for the paper to go with just two authors. We thank Orr for the numerous comments and corrections on earlier drafts of the paper. We also thank Vissarion Fisikopoulos for his valuable feedback and corrections on earlier versions of the algorithm and the program that computes the follower set graphs. A friend in need is a friend indeed. This paper is part of the MMath project of the first author. The authors would like to thank the anonymous referee for her/his remarks and comments that helped improve the current paper. References 1 W. Arveson, Operator algebras and measure preserving automorphisms, Acta Math.  118 ( 1967), 95– 109. Google Scholar CrossRef Search ADS   2 W. B. Arveson, Subalgebras of C*-algebras, Acta Math.  123 ( 1969), 141– 224. Google Scholar CrossRef Search ADS   3 W. Arveson and K. Josephson, Operator algebras and measure preserving automorphisms II, J. Funct. Anal.  4 ( 1969), 100– 134. Google Scholar CrossRef Search ADS   4 T. M. Carlsen, C*-algebras associated with subshifts, Internat. J. Math.  19 ( 2008), 47– 70. Google Scholar CrossRef Search ADS   5 T. M. Carlsen and K. 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On the Quantized Dynamics of Factorial Languages

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Abstract

Abstract We study local piecewise conjugacy of the quantized dynamics arising from factorial languages. We show that it induces a bijection between allowable words of same length and thus it preserves entropy. In the case of sofic factorial languages we prove that local piecewise conjugacy translates to unlabeled graph isomorphism of the follower set graphs. Moreover, it induces an unlabeled graph isomorphism between the Fischer covers of irreducible subshifts. We verify that local piecewise conjugacy does not preserve finite type nor irreducibility; but it preserves soficity. Moreover, it implies identification (up to a permutation) for factorial languages of type 1 if, and only if, the follower set function is one-to-one on the symbol set. 1. Introduction The fruitful interplay between Symbolic Dynamics and Operator Algebras was established in the seminal paper of Cuntz–Krieger [9]. Following their work, Matsumoto introduced an effective way for associating operators to subshifts and forming Cuntz–Krieger-type C*-algebras [31] that were further examined in-depth in a series of papers [32–34]. This theory was revisited with Carlsen [4, 5] and a new view was exploited in more generality in [34]. These important works clarified a strong connection between intrinsic properties of subshifts with related C*-algebras. Matsumoto operators follow from a Fock representation that accommodates more structures. Shalit–Solel [38] provided such a context for homogeneous ideals in general and established a rigidity programme for the related (non-selfadjoint) tensor algebras. The origins of this framework are traced back to the seminal work of Arveson [1]. Since then, a number of rigidity results have appeared in the literature for tensor algebras of graphs or dynamical systems as in the work of Katsoulis–Kribs [26] and Solel [39], Davidson–Katsoulis [11, 12] that supersedes the work of previous authors [3, 19, 36, 37], Davidson–Roydor [13], Davidson–Ramsey–Shalit [14, 15], Dor-On [16], Katsoulis–Ramsey [28] and the work of the second author with Davidson [10] and Katsoulis [23]. In this endeavour, Davidson–Katsoulis [12] developed the notion of piecewise conjugacy for classical systems as the essential level of equivalence obtained from tensor algebras. Piecewise conjugacy allows for comparisons of the systems locally and thus is more tractable than (global) conjugacy. Recently, it found significant applications to Number Theory and reconstruction of graphs as exhibited in the work of Cornelissen–Marcolli [7, 8] and Cornelissen [6]. Along this line of research, the second author with Shalit examined tensor algebras of factorial languages in [25]. A factorial language Λ* on d symbols is a subset of the free semigroup F+d such that if μ∈Λ* then every subword of μ is also in Λ*. To fix notation, the operators Tμ in discussion act on ℓ2(Λ*) and are defined by   Tμeν≔eμνifμν∈Λ*,0otherwise,for μ∈Λ*. As an intermediate step we use the C*-algebra of checkers  A≔C*(Tμ*Tμ∣μ∈Λ*). The ∗-endomorphisms on A given by αi(a)≔Ti*aTi play an important role in the analysis and the system (A,α1,…,αd) was coined in [25] as the quantized dynamics of Λ*. Two norm-closed subalgebras of B(ℓ2(Λ*)) can be related to the same language Λ*: The A-tensor algebra AΛ*≔alg¯{I,Tμ∣μ∈Λ*} in the sense of Shalit–Solel [38]. The T+-tensor algebra TΛ*+≔alg¯{a,Tμ∣a∈A,μ∈Λ*} in the sense of Muhly–Solel [35].Matsumoto’s C*-algebra is the quotient of C*(T)≔C*(Tμ∣μ∈Λ*) by the compacts K in ℓ2(Λ*). Arveson’s Programme on the C*-envelope (Arveson’s Programme was initiated in [2] and established in [20]. See also its formulation in [24].) provides a solid pathway for researching possible Cuntz–Krieger-type C*-algebras. For example, the natural analogues related to C*-correspondences are exactly the C*-envelopes of the tensor algebras as proven by Katsoulis–Kribs [27]. One of the main results in [25] states that the C*-algebra that fits Arveson’s Programme for both AΛ* and TΛ*+ is the quotient of C*(T) by the generalized compacts; rather than quotienting by all compacts as is done in Matsumoto’s work. In fact the quantized dynamics trigger a dichotomy: the C*-envelope of both AΛ* and TΛ*+ is either the quotient by all compacts or it coincides with C*(T), depending on whether the quantized dynamics is injective or not. This is in full analogy to what holds for graph C*-algebras where sinks or vertices emitting infinite edges are excluded from the Cuntz–Krieger relations. Apart from being a starting point for Cuntz–Krieger-type C*-algebras via the C*-envelope machinery, both AΛ* and TΛ*+ are rigid for factorial languages. It is shown in [25] that they encode the factorial language, yet in two essentially different ways: The A-tensor algebras provide a complete invariant for the factorial languages up to a permutation of the symbols. The T+-tensor algebras provide a complete invariant for local piecewise conjugacy (l.p.c.) of the quantized dynamics.However, it was left open how l.p.c. reflects the initial data: How is l.p.c. interpreted in terms of factorial languages? What properties are (thus) preserved under l.p.c.? What is the impact on sofic factorial languages?In the current paper, we answer these questions that add on the impact of the rigidity results of [25]. Before we move to the description of our results, we stress that languages of subshifts form special examples of factorial languages and several constructions apply to this broader context. Thus, terminology related to subshifts is extended accordingly to cover general factorial languages, when possible. Unlike to Carlsen [4], Matsumoto [31] or Krieger [29], our study is based on the allowable words rather than the points of an induced subshift. In fact the dynamical system of the backward shift is not explicitly used for the Fock space quantization and thus no connections between l.p.c. and topological conjugacy arise. The results and examples herein show that they are incomparable. On one hand, l.p.c. requires for the languages to have the same number of symbols (Definition 4.1) and so it is not implied by topological conjugacy. On the other hand, in Example 5.10, we construct a subshift of finite type that is l.p.c. to the even shift, and so l.p.c. does not imply topological conjugacy. We begin by giving an updated picture of the quantized dynamics (Section 3). We then show that l.p.c. implies a bijection between allowable words of the same length, and thus it respects entropy (Proposition 4.3 and Corollary 4.5). The flexibility of l.p.c. can be seen in the form of this bijection; but has its limitations (see Remark 4.2). Most notably, l.p.c. does not preserve finite type as shown in Example 5.10, nor irreducibility as shown in Example 5.21. Consequently, l.p.c. does not preserve the zeta function (Remark 5.11). Nevertheless, l.p.c. respects soficity (that is the C*-algebra of checkers is finite dimensional) where the theory is rich. There is a well-known construction of a labeled finite graph, that is the follower set graph, that gives a presentation of a sofic factorial language. When the language is of finite type, then this construction can be described by a (terminating) algorithm (Follower Set Graph Algorithm). Labeled graph isomorphism is equivalent then with the factorial languages being the same up to a permutation of symbols (Proposition 5.6). On the other hand, it is the unlabeled graph isomorphism that coincides with l.p.c. (Theorem 5.8). Combining these results with [25], we thus prove the following diagram for two sofic factorial languages Λ* and M*:   The follower set graph construction is rather useful in Theoretical Computer Science as the starting point for computing minimal presentations. Such presentations are unique (up to isomorphism) for irreducible two-sided subshifts and are better known as Fischer covers [17, 18]. We show that l.p.c. induces an unlabeled graph isomorphism between the Fischer covers of irreducible two-sided subshifts (Corollary 5.19). It is quite interesting to notice though that we achieve these results without inducing a bijection between intrinsically synchronizing words. A weaker bijection between the collections of follower sets of such words is induced. This is depicted in Remark 5.20 where we show the limitations of our arguments. The same obstructions do not allow to apply our arguments and prove (or disprove) that l.p.c. respects the mixing property. We further investigate cases where the vertical directions in the diagram above can or cannot be equivalences based on the type and the number of symbols. As commented in [25], these arrows cannot be reversed in general and we extend this remark for two-sided subshifts in Example 5.12. The key in these counterexamples is that the follower set function is not one-to-one. Apparently, this is the only obstruction for type 1 factorial languages. In Theorem 5.13, we show that unlabeled isomorphism for type 1 factorial languages produces a labeled isomorphism if and only if the follower set function is one-to-one on the symbol set. Consequently then, isomorphism of the A-tensor algebras is equivalent to isomorphism of the T+-tensor algebras. This condition is satisfied by edge shifts with invertible adjacency matrix. Injectivity of the follower set function is not required for type 1 factorial languages on two symbols. In Remark 5.16, we describe how an unlabeled graph isomorphism implies a labeled graph isomorphism of the follower set graphs in these cases. These results depend on the low complexity of the system. However, this does not hold when passing to type 2 factorial languages, even when the number of symbols is small (Examples 5.17 and 5.18). To facilitate comparisons, we developed a program that takes as an input a set of forbidden words on two symbols and gives the follower set graph as an output. (We chose to develop this program for the right version of factorial languages, rather than the left we work with here, as it is accustomed in Theoretical Computer Science to concatenate on the right. Nevertheless, the left version follows easily by reversing the words in the input and the output.) The code and the .exe file can be downloaded from the official webpage of the second author, currently at http://www.mas.ncl.ac.uk/~nek29. We remark that our principal objective here was to construct a program that computes easily follower set graphs as a check for our examples and counterexamples. Hence, we were not (extremely) concerned about complexity or the required memory, but only about the fact that it terminates when the type is finite. 2. Preliminaries 2.1. Languages and subshifts Let us fix the terminology and notation we will be using throughout the paper. To this, end F+d denotes the free semigroup on the symbol set Σ≔{1,…,d} with multiplication given by concatenation. For μ=μk…μ1 in F+d, we write ∣μ∣≔k for the length of μ. The empty word ∅ is by default in F+d, it has zero length and plays the role of the unit. For μ,ν∈F+d, we say that ν is a subword of μ if there are w,z∈F+d such that μ=wνz. By default, the empty word is a subword of every μ∈Λ*. A (factorial) language is a subset Λ* of F+d that satisfies the following property:   ifμ∈Λ*theneverysubwordofμisinΛ*. We will simply use the term ‘language’, since we are going to encounter just factorial ones. Without loss of generality, we will always assume that all letters of the symbol set are in Λ*. Otherwise we view Λ* to be defined on less symbols, that is on Σ out the symbols that are not in Λ*. Examples of languages arise from subshifts and below we give a brief description. Several elements from the theory of subshifts apply to the broader context of languages and terminology will be extended to cover languages in general. Apart from two-sided, we also consider one-sided subshifts. Several results that hold for the two-sided version hold also for the one-sided with almost the same proof. We will mainly discuss left subshifts, but similar comments hold for the right subshifts. In order to make sense of the one-sided subshifts and avoid technicalities, we make the following convention. We will write the sequences x=(xn)∈ΣZ+ from right to left, that is   …xn…x1x0.=xand likewise for elements in ΣZ. This is to comply with operator composition which comes by multiplying on the left. We endow ΣZ+ with the product topology and we fix σ:ΣZ+→ΣZ+ be the backward shift with σ((xi))k=xk+1. With our convention, the map σ shifts to the right. The pair (Λ,σ) is called a left subshift if Λ is a closed subset of ΣZ+ with σ(Λ)⊆Λ. Similarly, the pair (Λ,σ) is called a two-sided subshift if Λ is a closed subset of ΣZ with σ(Λ)=Λ. We write x[m+n−1,m] for the block xm+n−1…xm in x∈Λ. A word μ=μ∣μ∣…μ1 is said to occur in some (one-sided or two-sided) sequence x if there is an m such that   x∣μ∣−1+m=μ∣μ∣,…,xm=μ1. If a word occurs in some point of Λ, then it is called allowable. The language of a subshift Λ is defined by   Λ*≔{w∈F+d∣woccursinsomex∈Λ}. Since Λ is σ-invariant, we have that for every allowable word μ there exists an x∈Λ such that x[∣μ∣,0]=μ. We write Bn(Λ*) for the allowable words of length n in Λ*. By following the same arguments as in the two-sided subshifts, one can show that if Λ defines a left subshift, then Λ* is a language such that   foreveryμ∈Λ*thereisa∅≠ν∈Λ*suchthatνμ∈Λ*. Conversely, every language with this property defines uniquely a left subshift (see [38, Proposition 12.3] and [25]). Subshifts can be described also in terms of forbidden words. Let F be a set of words on the symbol set Σ={1,…,d}, and let   ΛF≔{(xn)∈ΣZ+∣noμ∈Foccursin(xn)}. It is known that all two-sided subshifts arise in this way. Likewise this also holds for one-sided subshifts. By setting   Fk≔{μ∈F∣μdoesnotoccurin(xn)∈Λ,∣μ∣≤k},we see that Fk⊆Fk+1 and F=⋃kFk. Then we have that Λ=∩kΛFk, where the intersection is considered inside the full shift space on Σ. The elements in F are called the forbidden words of the subshift. We will call a forbidden word minimal, if all of its proper subwords are allowable. Every set F of forbidden words admits a unique basis F⊆F in the sense that for every μ∈F, there are (unique) ν,w∈F+d and a μ′∈F such that μ=νμ′w and μ′ is minimal. We say that Λ is a subshift of finite type (SFT) if the longest word in the basis of F has finite length. We say that an SFT is of type k if the longest word in the basis of F has length k+1. Hence, if Λ is of type k, then any forbidden word of length strictly greater than k+1 cannot be minimal. The notions of forbidden words, minimality, basis and type pass naturally to any language. For example, given a set F, we can define a language by   ΛF*=F+d⧹{wμν∣μ∈F,w,ν∈F+d}. Not every language is a language of a subshift, but it can be embedded in one by augmenting the symbol set. Suppose that Λ* is defined through a set of forbidden words F in Σ. We introduce a new distinguished symbol ζ and take the symbol set Σ˜={1,…,d,ζ}. Then the augmented subshift (Λ˜,σ)of Λ* is the subshift defined by F in Σ˜Z. Since ζ is not contained in any word in F, it follows that the language of Λ˜ is   Λ˜*={ζnkμk…μ2ζn2μ1ζn1∣n1,…,nk∈Z+,μ1,…,μk∈Λ*}and thus contains Λ*. Recall that a two-sided subshift (Λ,σ) is called sofic if the number of classes in Λ* with respect to the equivalence relation   μ∼ν⇔{w∈Λ*∣wμ∈Λ*}={w∈Λ*∣wν∈Λ*}is finite. Equivalently, if (Λ,σ) is a factor of an SFT [17, 40]. The reader is addressed for example to [30, Theorem 3.2.10] for a modern treatment of sofic subshifts. For languages that do not come from subshifts, we will use the definition of soficity in terms of the equivalence classes. It is shown in [25] that a language Λ* is sofic (resp. of finite type) if and only if its augmented subshift (Λ˜,σ) is sofic (resp. of finite type). This follows by observing that ∅∼ζμ for every μ∈Λ˜*. Every two-sided subshift becomes a compact metric space. Taking the one-sided subshifts to be closed yields the same result in our case. Therefore, every sequence in a subshift has a converging subsequence. This often appears in [30] as the Cantor’s diagonal argument, mainly because metric spaces come later in the presentation of [30]. We preserve this terminology to keep connections with Symbolic Dynamics. However, it is interesting that this argument works to build the one-sided subshift from a set of predetermined forbidden words. The key is that the one-sided full shift is compact and metrizable with the topology given by   ρ(x,y)=2ifx0≠y0,2−kifx≠yandkismaximalsothatx[k,0]=y[k,0],0ifx=y. 2.2. Fock representation We will require some basic theory from Hilbert spaces to show how the quantized dynamics arise from a language. The reader who is not familiar with operator theory may read this subsection in combination with Section 3, where explicit identifications in terms of topological spaces are provided. Operator algebras associated to subshifts were introduced by Matsumoto [31]. Let Λ* be a language on d symbols. Let H=ℓ2(Λ*) and fix the operators Ti such that Tieμ=eiμ if iμ∈Λ* or zero otherwise. We fix   C*(T)≔C*(I,Ti∣i=1,…,d). It is convenient to write TμTν=Tμν even when Tμν=0, that is Tμν=0 if and only if μν∉Λ*. We will also write T∅=I. Likewise we write eμν=0 in ℓ2(Λ*) when μν∉Λ*. The operators Tμ satisfy a list of properties: Tμ*Tμ is an orthogonal projection on span¯{eν∣μν∈Λ*}. TνTν* is an orthogonal projection on span¯{eνμ∣μ∈Λ*}. If ∣μ∣=∣ν∣, then Tμ*Tν=0 if and only if μ≠ν. Tμ*Tμ commutes with Tν*Tν, and with TνTν*. Tμ*Tμ·Ti=Ti·Tμi*Tμi for all i=1,…,d. ∑i=1dTiTi*+P∅=I, where P∅ is the projection on Ce∅. The rank one operator eν↦eμ equals TμP∅Tν*. The ideal K(ℓ2(Λ*)) of compact operators is in C*(T). In [25], the second author with Shalit examines several operator algebras related to the operators Ti. Among them, there are two classes of non-selfadjoint operator algebras: The tensor algebra AΛ* in the sense of Shalit–Solel [38] is defined as the norm-closed subalgebra of B(ℓ2(Λ*)) generated by I and the Ti for i=1,…,d. The tensor algebra TΛ*+ in the sense of Muhly–Solel [35] is defined as the norm-closed subalgebra of B(ℓ2(Λ*)) generated by I, the Ti for i=1,…,d, and the Tμ*Tμ for μ∈Λ*.The relations above imply that   AΛ*=span¯{Tμ∣μ∈Λ*}andTΛ*+=span¯{Tμa∣μ∈Λ*,a∈A}for the unital C*-subalgebra A≔C*(Tμ*Tμ∣μ∈Λ*) of C*(T). 2.3. Q-Projections We will be using the projections generated by the Ti*Ti. To this end we introduce the following enumeration. Write all numbers from 0 to 2d−1 by using 2 as a base, but in reverse order. Hence we write [m]2≡[m]=[m1m2…md] so that 2=[0100…0]. Let the (not necessarily one-to-one) assignment   [m1…md]↦Q[m1…md]≔∏mi=1Ti*Ti·∏mi=0(I−Ti*Ti),where I∈B(ℓ2(Λ*)). For example, we write   Q0=Q[0…0]=∏i=1d(I−Ti*Ti)andQ2d−1=Q[1…1]=∏i=1dTi*Ti. The Q[m] are the minimal projections in the C*-subalgebra C*(I,Ti*Ti∣i=1,…,d) of C*(T). Consequently we obtain ∑[m]=02d−1Q[m]=I. The second author with Shalit have shown in [25] the stronger equality (notice that we sum for [m]≥1 here)   Tμ*Tμ=Tμ*Tμ·∑[m]=12d−1Q[m]=∑[m]=12d−1Q[m]·Tμ*Tμfor all ∅≠μ∈Λ*. It follows that the unit of C*(Ti*Ti∣i=1,…,d) coincides with I∈B(ℓ2(Λ*)) if and only if Λ* induces a left subshift [25, Lemma 4.4]. 3. The quantized dynamics on the allowable words Let Λ* be a language on d symbols. We fix once and for all the unital C*-subalgebra   A≔C*(Tμ*Tμ∣μ∈Λ*)of C*(T). Then A=∪lAl¯ is a unital commutative AF algebra for   Al≔C*(Tμ*Tμ∣μ∈Bl(Λ*)). The C*-algebra A can be characterized by using Λ*. For l≥0, let ∼l be the equivalence relation on Λ* given by the rule   μ∼lν⇔{w∈Bl(Λ*)∣wμ∈Λ*}={w∈Bl(Λ*)∣wν∈Λ*}. Let the discrete space Ωl=Λ*/∼l and write [μ]l for the points in Ωl. Every μ∈Λ* splits Bl(Λ*) into the set of the wi∈Bl(Λ*) for which wiμ∈Λ* and its complement. There is a finite number of such splittings since Bl(Λ*) is finite. They completely identify single points in Ωl, and hence Ωl is a (discrete) finite space. Furthermore, the mapping   ϑ:Ωl+1→Ωl:[μ]l+1↦[μ]lis a well defined (continuous) and onto map. We can then form the projective limit Ω by the directed sequence   for which we obtain the following identification. In [25], the second author with Shalit have shown that   Al≃C(Ωl)andA≃C(Ω). We write ΩΛ* for Ω when we want to highlight the language Λ* to which Ω is related. From now on, we will tacitly identify Al with C(Ωl) and A with C(Ω). In the sequel we will use the same notation Q[m] for the subspaces of Ω that correspond to the projections Q[m] of C*(T). To make this precise, if [m]=[m1…md] is the binary expansion of a number from 0 to 2d−1, then the subspace corresponding to Q[m] consists of the points [μ]∈Ω for which: iμ∈Λ* for all i with mi=1; and iμ∉Λ* for all i with mi=0.We record here the following proposition from [25] for future reference. Proposition 3.1 [25] Let Λ*be a language on d symbols. Then Ωis finite if and only if Λ*is a sofic language. If, in particular, Λ*is of finite type k, then Al=Akfor all l≥k+1. Therefore, if Λ* is sofic, then there exists a stabilizing step k for which Al=Ak for all l≥k+1. When Λ* is in particular of finite type, then we get the following proposition for the possible equivalent classes. Proposition 3.2 Let Λ*be a language on dsymbols. If Λ*is of finite type k, then for all μ∈Λ*with ∣μ∣≥kwe have that  [μ1…μ∣μ∣]k=[μ1…μk]k. Proof Fix an allowable word μ=μ1…μk…μ∣μ∣. By the properties of the language, if wμ∈Λ* then wμ1…μk∈Λ* as well. Conversely, let w∈Λ* such that wμ1…μk∈Λ*. To reach contradiction suppose that wμ∉Λ*. Therefore, there is an n1 and an n2 such that   ν≔wn1…wkμ1…μkμk+1…μn2∉Λ*. Choose n1 and n2 so that ν is a minimal forbidden word. Since μ∈Λ* (hence μ1…μn2∈Λ*) we have that wn1…wk≠∅; hence ∣w∣+1−n1≥1. Similarly, since wμ1…μk∈Λ* (hence wn1…wkμ1…μk∈Λ*) we have that μk+1…μn2≠∅; hence n2≥k+1. Therefore, ν is a forbidden word of length at least   (∣w∣+1−n1)+n2≥k+2.Since Λ* is of type k, the word ν cannot be minimal, which gives the required contradiction.□ We define the maps αi:A→A such that αi(a)=Ti*aTi. It is clear that every αi is a positive map and takes values in A since   αi(Tμ*Tμ)=Tμi*Tμi∈Afor all μ∈Λ*. In fact every αi is an ∗-endomorphism of A, since Tμ*Tμ commutes with TiTi* and TiTi*Ti=Ti. The αi induce the required covariant relation   aTi=Tiαi(a)foralla∈A,i=1,…,dused in the study of TΛ*+. We use the identification of A with C(Ω) to get a translation of each αi as a continuous map φi partially defined on Ω. Let Ai be the direct summand Ti*TiA of A, with unit Ti*Ti. Then Ai is the direct limit of Ali=Al∩Ai, and the corresponding projective limit Ωi is determined by the spaces   Ωli≔{[μ]l∈Ωl∣iμ∈Λ*}and the map   ϑ:Ωl+1i→Ωli:[μ]l+1→[μ]l. Hence αi:A→Ai is a unit preserving map from A=C(Ω) into Ai≔Ti*TiA=C(Ωi), and, therefore, induces a continuous map φi:Ωi→Ω. Proposition 3.3 Let Λ*be a language on dsymbols. With the above notation we have that αi∣Al:Al→Aliis induced by φi:Ωli→Ωlsuch that φi([μ]l+1)=[iμ]l. Proof We have to show that αi(f)=fφi for all f∈Al=C(Ωl). It suffices to do so for f=Tμ*Tμ with μ∈Bl(Λ*). To this end, we have that   αi(Tμ*Tμ)=Tμi*Tμi=χAfor the set A={[w]l+1∣μiw∈Λ*}. On the other hand, we have that Tμ*Tμ=χB for the set B={[w]l∣μw∈Λ*}. Hence we compute   χB(φi([w]l+1))=χB([iw]l)=1ifμiw∈Λ*,0otherwise,which shows that χBφi=χA=αi(χB).□ The universal property of the projective limit implies that this information is enough to describe φi. Indeed, we have that   Ω={(ωn)n≥0∣ωn=[μ]n,μ∈Λ*}.Therefore, for ω∈Ωi, that is ωn=[μ]n such that iμ∈Λ*, we get   φi(ω)=([iμ]n). In the particular case, when Λ* is sofic, let k be the step so that Al=Ak for all l≥k+1. Then we have that Ωl=Ωk for l≥k+1, and hence   Ω≃{[μ]k∣μ∈Λ*}.The projective limit description gives that the induced φi is then given by [μ]k↦[iμ]k, for μ∈Λ* with iμ∈Λ*. Definition 3.4 Let Λ* be a language on d symbols. We call the (A,α)≡(A,α1,…,αd), or alternatively the (Ω,φ)≡(Ω,φ1,…,φd), the quantized dynamics of Λ*. Remark 3.5 The intersection of the subspaces corresponding to the Tμ*Tμ is always non-empty since ∏μ∈Λ*Tμ*Tμe∅=e∅. In fact the intersection consists of the single point ([∅]n)∈Ω. It is evident that ([∅]n) is in the domain of all φμ with μ∈Λ*. 4. Local piecewise conjugacy Let Λ* and M* be languages. Fix their associated quantized dynamics (ΩΛ*,φ) and (ΩM*,ψ), and let Q and P be the corresponding systems of projections. Recall that an ∗-isomorphism γ:C(ΩM*)→C(ΩΛ*) induces a homeomorphism γs:ΩΛ*→ΩM* on the spectra. We have the following notion of conjugacy for the quantized dynamics. For convenience, we use the notation   supp[m]≔{i∈{1,…,d}∣mi=1},where [m]=[m1…md] is the binary expansion of m∈{0,1,…,2d−1}. Definition 4.1 We say that the systems (ΩΛ*,φ) and (ΩM*,ψ) are Q- P-locally piecewise conjugate if there exists a homeomorphism γs:ΩΛ*→ΩM*; and for every x∈Q[m] there is a neighbourhood x∈U⊆Q[m] and an [n]∈{0,1,…,2d−1} such that ∣supp[n]∣=∣supp[m]∣, γs(U)⊆P[n], and   γsφi∣U=ψπ(i)γs∣U,for a bijection π:supp[m]→supp[n]. Equivalently, every Q[m]⊆ΩΛ* has an open cover {Uπ}π indexed by the one-to-one correspondences π:supp[m]→{1,…,d} such that γs(Uπ)⊆P[n] for all [n] with supp[n]=π(supp[m]), and γsφi∣Uπ=ψπ(i)γs∣Uπ. We do not exclude the case where Uπ=∅ for some π. As an immediate consequence, if an ω∈Q[m] is in the intersection of Uπ with Uπ′, then   ψπ(i)γs(ω)=γsφi(ω)=ψπ′(i)γs(ω),for all i∈supp[m]. Remark 4.2 We can use local piecewise conjugacy to define inductively maps fn:Bn(Λ*)→Bn(M*). For every step, we start at the point ω≔([∅]n) which is in the intersection of all subspaces corresponding to Tμ*Tμ for μ∈Λ*. By definition, there exists a neighborhood Uπ⊆Q[1…1] containing ω such that   γsφi(ω)=ψπ(i)γs(ω)foralli=1,…,d. We write π∅,0 for the bijection π and set f1=π∅,0. Consequently, we obtain   γsφμ1(ω)=ψπ∅,0(μ1)γs(ω). For f2, let a word μ=μ2μ1∈B2(Λ*). Since μ2μ1∈Λ*, then ω is in the domain of φμ2φμ1=φμ1μ2. We apply the same argument for ω1=φμ1(ω) and find a bijection πμ,1 coming from the neighborhood Uπμ,1 of ω1 such that   γsφμ2(φμ1(ω))=ψπμ,1(μ2)γs(φμ1(ω))=ψπμ,1(μ2)ψπ∅,0(μ1)γs(ω). In particular, we get that γs(ω) is in the domain of   ψπμ,1(μ2)ψπ∅,0(μ1)=ψπμ,1(μ2)π∅,0(μ1),and hence the word πμ,1(μ2)π∅,0(μ1) is in M*. This procedure gives a well-defined map f2:B2(Λ*)→B(M*). The same argument applies for a word μ1 of length n (instead of just a letter) and a letter μ2, and gives a map fn:Bn(Λ*)→Bn(M*) that is defined by   fn(μn…μ1)≔πμ,n−1(μn)…π∅,0(μ1). The notation πμ,k≡πμk denotes that this bijection comes from the neighborhood Uπμk of the point φμk…μ1(ω). Notice that each fn depends on the point ω≔([∅]n). However, it also depends on the orbit of ω. For a word μ=μn…μ1∈Λ*, we have to keep track where we are at under fn−1, since the nth bijection depends on where φμn−1⋯φμ1(ω) sits. It is clear that   fn(μn…μ1)=πμ,n−1(μn)fn−1(μn−1…μ1),but in general fn(μn…μ1) may be different than f1(μn)fn−1(μn−1…μ1). We present such a case in Example 5.10. Local piecewise conjugacy respects certain properties of languages. This is very pleasing as several invariants for the usual topological conjugacy of subshifts depend on these data. Proposition 4.3 If Λ*and M*are locally piecewise conjugate languages, then the maps fndefined in Remark4.2are bijections. Consequently, ∣Bn(Λ*)∣=∣Bn(M*)∣for all n∈N. Proof First we show that the fn are one-to-one. To this end suppose that   πμ,n−1(μn)…π∅,0(μ1)=πν,n−1(νn)…π∅,0(ν1).Then we get that πμ,i−1(μi)=πν,i−1(νi) for all i=1,…,n. In particular we have that π∅,0(μ1)=π∅,0(ν1) and therefore μ1=ν1. Consequently we get that πμ,1=πν,1, hence μ2=ν2. Inductively we have that μi=νi for all i=1,…n. Therefore we get ∣Bn(Λ*)∣≤∣Bn(M*)∣. By symmetry we obtain equality which implies that the fn are bijections.□ Remark 4.4 Since the functions fn are bijections, then γs(ω) is in the domain of all ψν for ν∈M*, for ω=([∅]n). Hence γs fixes the points ([∅]n) of ΩΛ* and ΩM*. The entropy of a language Λ* is given by h(Λ*)≔limnn−1log2∣Bn(Λ*)∣. The following corollary is immediate. Corollary 4.5 Local piecewise conjugate languages share the same entropy. Local piecewise conjugacy respects the class of sofic languages. However, as we will see in Example 5.10, it does not preserve the class of SFT’s. Proposition 4.6 Let Λ*and M*be locally piecewise conjugate languages. If Λ*is sofic, then so is M*. Proof Immediate by Proposition 3.1.□ 5. Applications to the follower set graph The quantized dynamics can be described by a finite graph in the case of the sofic languages. Fix a sofic language Λ* on d symbols. For every μ∈Λ*, let   F(μ)≔{w∈Λ*∣wμ∈Λ*}be the follower set of μ. We write FΛ*(μ) when we want to highlight the language to which we refer. (We will rarely use this notation, as the language will be clear from the context.) Recall that the elements in Ω are of the form ([μ]n) for μ∈Λ*. Since ([μ]n)=([ν]n) if and only if F(μ)=F(ν) we have a bijection between Ω and the set {F(μ)∣μ∈Λ*}. By definition, soficity is equivalent to Ω being finite. In this case, Proposition 3.1 provides the existence of a stabilizing step k such that Ωl=Ωk for all l≥k+1; that is every F(μ) can be identified with [μ]k. The follower set graph GΛ*=(GΛ*,L) of a sofic language on d symbols is an edge-labeled graph, where L is a coloring map from the edge set GΛ*(1) onto {1,…,d}, defined as follows: The vertices of GΛ* are given by the follower sets. We draw an edge labeled i from F(μ) to F(iμ) if and only if F(iμ)≠∅.Therefore, we have an edge (labeled i) between ω=[μ]k and ω′=[ν]k if [iμ]k=[ν]k, that is, if φi(ω)=ω′. It follows that the labeled graph gives a representation of the quantized dynamics. Notice that this procedure gives also the follower set graph in the case of a two-sided subshift [30, Section 3.2] (when everything is written in the opposite direction). Notice that we do not use soficity for the construction of the follower set graph, and one may be tempted to produce a follower set graph for any language. However, this is not the right thing to do. As we saw in Section 3, the spectrum of A is totally disconnected in general and configuring the dynamics with a discrete structure would not comply with the topology. By Proposition 3.1, it is exactly when Λ* is sofic that the spectrum is discrete and we are able to do so without losing information. Example 5.1 Let Λ* be of type 1. Then Ω≃Ω1={[i1]1,…,[ir]1} for some symbols i1,…,ir∈{1,…,d}. For the follower set graph, we have an edge labeled j from [i]1=F(i) to [ji]1=[j]1=F(j) whenever ji∈Λ*. Notice here the use of Proposition 3.2. The case of type 1 languages requires less information to store, as the label of an edge coincides with the one-lettered word that labels the vertex where the edge terminates. Remark 5.2 There is a strong connection between the follower set graph of a sofic language Λ* and that coming from its augmented subshift (Λ˜,σ). Suppose that Λ* is on d symbols and G is its follower set graph. Recall that F(∅)=F(ζμ) for every μ∈Λ˜* for the added distinguished symbol ζ. Therefore, the follower set graph of Λ˜* contains G, shares the same vertex set and contains in addition edges from every vertex to F(∅) labeled by ζ. When the language is of finite type then we can find a finite set of forbidden words that describes it. By using this, we can determine the follower set graph in a finite number of steps. Indeed the dual of Proposition 3.2 suggests that the determination of every F(μ) can be achieved in finite steps, even though F(μ) may be infinite in principle. Proposition 5.3 Let Λ*be a language of finite type k. For every ∅≠μ∈Λ*and wn…w1∈Λ*with n≥k, we have that  wμ∈Λ*ifandonlyifwk…w1μ∈Λ*. Proof The proof is the same to that of Proposition 3.2 once the words are reversed.□ Therefore, in order to distinguish between the F(μ), it suffices to distinguish between the finite sets   {w∈Bk(Λ*)∣wμ∈Λ*}. Consequently, we obtain the following algorithm for constructing the follower set graph of a language of type k. Follower Set Graph Algorithm. Let Λ* be a language of finite type and let F be a finite set of forbidden words that defines Λ*. The algorithm takes F as an input and has output the follower set graph. We have two cases: Case 1. If F=∅, then Λ*=F+d and its follower set graph is the Hawaiian ring on d edges (lines 1–9 in the pseudocode). Case 2. IfF is non-empty, we write  k≔max{∣μ∣∣μ∈F}−1. Then Λ* is a language of type k (lines 10–16 in the pseudocode). We then generate the set Bk(Λ*) of allowable words of length k (lines 17–24 in the pseudocode). We then proceed to forming the follower set graph in two steps: Step 1: Determining the vertex set {F(μ)∣μ∈Λ*} (lines 25–44 in the pseudocode). It suffices to compute the F(μ) for ∣μ∣≤k (Proposition 3.2). Nevertheless, we will also compute the F(μ) for ∣μ∣=k+1. This will be helpful for writing the edges. To this end, we construct a table indexed by Bk(Λ*) whose entries show whether an allowable word can follow another one. If two rows are the same, then the follower sets of the corresponding words determine the same vertex on the graph. It is convenient to multi-label a vertex by using all the F(μ), for ∣μ∣≤k+1, that coincide. – Form a table indexed by Bk(Λ*) (followerTable). We fix an enumeration {μ1,…,μN} of these words, including the void word which we set to be μ1. For the (μi,μj) entry of the table write μjμi. If μjμi∈Λ*, then write TRUE for the entry; otherwise write FALSE. In this way, we create a second table (truthTable). By Proposition 5.3, these finite entries are sufficient for identifying the follower sets. – Compute the F(μ) for μ of length k+1 (lines 45–56 in the pseudocode). If μ is forbidden, then do nothing. If μ=μ1μk…μk+1 is allowable, then add the label F(μ) to the vertex that has F(μ1…μk) among its labels. Step 2: Determining the edges of the follower set graph (lines 57–69 in the pseudocode). We read the rows of the truthTable from top to bottom. If we pass to a row that is the same with one already read, we move to the next (that is we identify the words with the same follower sets). – Start from μl=μ1=∅ and repeat for l=2,…,N. Let μl be a word from {μ1,…,μN}. Start with i=1 and repeat for i=2,…,d. Write an edge labeled i from the vertex with label F(μl) to the vertex with label F(iμl) if there is a vertex that contains the label F(iμl). Otherwise do nothing and go to i+1. After we finish for i=d (the last step), we repeat for μl+1. Remark 5.4 In the following pages, we give the pseudocode for the algorithm. Notice that in the output, each node is a set of words, and each word associated with a node generates the same follower set. We remind that we were not concerned about the technical features of this algorithm, but more about that it does terminate. Algorithm Generate FollowerSetGraph(F, symbolSet)   Input: F = the list of forbidden words generating the language.    symbolSet={1,…,d}.    Output : Graph=(Nodes,Edges) where:    – each node in Nodes is the set of words of length less than or equal to k+1 which have the same follower set; and    – each edge in Edges is of the form (source, destination, label).    //Deal with the case that Fis the empty set.   1  if F is empty then   2 3 4 5 6 7 8  initializenodeasalistcontainingsymbolSetandtheemptywordinitializenodesasalistcontainingonlynodeinitializeedgesasemptylistforeachletter∈symbolSetdo∣insert(node,node,letter)intoedgesendreturn(nodes,edges)   9  end    //IfFis non-empty find value ofkfromF.  10  initialize maxLength as 1  11  for each word∈Fdo  121314  iflengthofword>maxLengththen∣maxLength⟵lengthofwordend  15  end  16  initialize k as maxLength−1    //Generate mu, the set of allowed words of length at most k.  17  initialize mu as a list containing only the empty word  18  foreach n∈{1,2,…,k}do  1920212223  foreachword∈setofallpossiblewordsoversymbolSetoflengthndoifwordcontainsnoelementofFasasubwordthen∣insertwordintomuendend  24  end    //Form tables. Indexed by base word and concatenated word.  25  for each word a∈mudo  2627282930313233  foreachwordb∈mudofollowerTable[a][b]⟵concatenatebwithaiffollowerTable[a][b]containsnoelementofFasasubwordthen∣truthTable[a][b]⟵Trueelse∣truthTable[a][b]⟵Falseendend  34  end    //Each node of the graph is a set of words sharing the same follower set.  35  initialize Nodes as empty list;  36  for each row∈unique rows of truthTabledo  37383940414243  initializenodeasanemptylist;foreachworda∈mudoiftruthTable[a]=rowthen∣insertaintonode;endendinsertnodeintoNodes;  44  end    //Add words of length k+1to the appropriate node.  45  foreachnode∈Nodesdo  46474849505152535455  foreachword∈nodedoiflengthofword=kthenforeachletter∈symbolSetdonewWord⟵concatenatewordwithletter;ifnewWordcontainsnoelementofFasasubwordthen∣insertnewWordintonode;endendendend  56  end    //Create edges.  57  initialize Edges as empty list;  58  for each node∈Nodesdo  59606162636465666768  foreachletter∈symbolSetdonewWord⟵concatenateletterwithanywordinnodeoflengthlessthank+1;//𝙵𝚒𝚗𝚍 𝚠𝚑𝚒𝚌𝚑 𝚗𝚘𝚍𝚎newWord𝚋𝚎𝚕𝚘𝚗𝚐𝚜 𝚝𝚘,𝚒𝚏 𝚊𝚗𝚢foreachnode′∈NodesdoifnewWord∈node′thenedge⟵(node,node′,letter);insertedgeintoEdges;break;endendend  69  end      70  return (Nodes, Edges);    Input: F = the list of forbidden words generating the language.    symbolSet={1,…,d}.    Output : Graph=(Nodes,Edges) where:    – each node in Nodes is the set of words of length less than or equal to k+1 which have the same follower set; and    – each edge in Edges is of the form (source, destination, label).    //Deal with the case that Fis the empty set.   1  if F is empty then   2 3 4 5 6 7 8  initializenodeasalistcontainingsymbolSetandtheemptywordinitializenodesasalistcontainingonlynodeinitializeedgesasemptylistforeachletter∈symbolSetdo∣insert(node,node,letter)intoedgesendreturn(nodes,edges)   9  end    //IfFis non-empty find value ofkfromF.  10  initialize maxLength as 1  11  for each word∈Fdo  121314  iflengthofword>maxLengththen∣maxLength⟵lengthofwordend  15  end  16  initialize k as maxLength−1    //Generate mu, the set of allowed words of length at most k.  17  initialize mu as a list containing only the empty word  18  foreach n∈{1,2,…,k}do  1920212223  foreachword∈setofallpossiblewordsoversymbolSetoflengthndoifwordcontainsnoelementofFasasubwordthen∣insertwordintomuendend  24  end    //Form tables. Indexed by base word and concatenated word.  25  for each word a∈mudo  2627282930313233  foreachwordb∈mudofollowerTable[a][b]⟵concatenatebwithaiffollowerTable[a][b]containsnoelementofFasasubwordthen∣truthTable[a][b]⟵Trueelse∣truthTable[a][b]⟵Falseendend  34  end    //Each node of the graph is a set of words sharing the same follower set.  35  initialize Nodes as empty list;  36  for each row∈unique rows of truthTabledo  37383940414243  initializenodeasanemptylist;foreachworda∈mudoiftruthTable[a]=rowthen∣insertaintonode;endendinsertnodeintoNodes;  44  end    //Add words of length k+1to the appropriate node.  45  foreachnode∈Nodesdo  46474849505152535455  foreachword∈nodedoiflengthofword=kthenforeachletter∈symbolSetdonewWord⟵concatenatewordwithletter;ifnewWordcontainsnoelementofFasasubwordthen∣insertnewWordintonode;endendendend  56  end    //Create edges.  57  initialize Edges as empty list;  58  for each node∈Nodesdo  59606162636465666768  foreachletter∈symbolSetdonewWord⟵concatenateletterwithanywordinnodeoflengthlessthank+1;//𝙵𝚒𝚗𝚍 𝚠𝚑𝚒𝚌𝚑 𝚗𝚘𝚍𝚎newWord𝚋𝚎𝚕𝚘𝚗𝚐𝚜 𝚝𝚘,𝚒𝚏 𝚊𝚗𝚢foreachnode′∈NodesdoifnewWord∈node′thenedge⟵(node,node′,letter);insertedgeintoEdges;break;endendend  69  end      70  return (Nodes, Edges);  Let us illustrate with an example how the follower set graph is constructed from the Follower Set Graph Algorithm. Example 5.5 Let the symbol set be {0,1}. Let Λ* be the language defined by the words F={101,110}. We form the followerTable of the Follower Set Graph algorithm with all the words of length at most two. As we add words on the left, it is convenient to put the labels at the right of the followerTable:     For simplicity, we can just indicate the forbidden words by a black square in the truthTable of the Follower Set Graph Algorithm:     This gives the following five vertices:   F(∅),F(0)=F(00)=F(000)=F(001),F(1)=F(11)=F(111),F(10)=F(100),F(01)=F(010)=F(011),where we used Proposition 3.2 to provide the classes for the allowable words of length 3. Notice here that F(101) and F(110) are meaningless as 101 and 110 are forbidden words. Then the follower set graph is   By definition, the follower set graph is (left-)resolving, that is different edges with the same source carry different labels. Recall that a graph is called follower-separated if distinct vertices have distinct follower sets (set of paths starting on the vertex). This is also a property that the follower set graph has. Indeed, if two vertices, say F(μ) and F(ν) have the same follower sets on the graph then every path w that starts from μ can also start at ν. This shows that wμ∈Λ* if and only if wν∈Λ* for all w∈Λ*, hence F(μ)=F(ν). Resolving graphs go by the name of Shannon graphs in the literature. Another type of a Shannon graph for subshifts is produced through the Krieger cover [29]. The Krieger cover is constructed by taking the follower sets on one-way infinite words. This is the important difference with the follower set graph here, as we consider the vertices to be the follower sets on the finite words. For a nice exposition (among others) on the Krieger cover, the reader can see also [21]. Given two languages, we can consider their labeled graphs or the ambient unlabeled graphs. Graph isomorphism within each class is translated to a different level of equivalence. 5.1. Labeled graph isomorphism Let G1=(G1,L1) and G2=(G2,L2) be two edge-labeled graphs such that the labels L1,L2 take values on the same symbol set {1,…,d}. The labeled graphs G1 and G2 are called labeled graph isomorphic if there is a graph isomorphism (∂γ,γ):G1→G2 and a bijection π on the symbol sets such that   L2(γ(e))=π(L1(e))foralledgese.(We use the notation ∂γ:G1(0)→G2(0) and γ:G1(1)→G2(1) for the graph isomorphism.) Proposition 5.6 Let Λ*and M*be sofic languages. Then their follower set graphs are isomorphic if and only if there is a bijection on their symbol sets that preserves the allowable words. Proof Suppose there is an edge-labeled graph isomorphism. By construction, there is at least one vertex, that is the vertex FΛ*(∅) (resp. FM*(∅)), that emits edges of all labels. Hence max{s−1(v)∣vvertex} coincides with the number of the symbols, thus d=d′. The graph isomorphism identifies FΛ*(∅) with some FM*(ν). Then the edge e=(FΛ*(∅),FΛ*(i)) labeled i corresponds to the edge   γ(e)=(FM*(ν),FM*(π(i)ν))labeled π(i). Suppose ji∈Λ*. Then the edge f=(FΛ*(i),FΛ*(ji)) labeled j corresponds to the edge   γ(f)=(FM*(π(i)ν),FM*(π(j)π(i)ν))labeled π(j). Therefore, we get that π(j)π(i)∈M*. Hence π respects the allowable words of length 2. Inductively, we get that permutation of the edges along the vertices extends to a match on the allowable words of any length. Notice here that π−1 is the associated labeling for the inverse graph isomorphism. Conversely, the existence of a bijection π on the symbol sets implies that d=d′. Furthermore, π extends to the allowable words such that π(μν)=π(μ)π(ν). We define the edge-labeled graph homomorphism by   ∂γ(FΛ*(μ))≔FM*(π(μ)). The map ∂γ is well defined. Indeed if FΛ*(μ)=FΛ*(ν), then we get that   {w∈Λ*∣wμ∈Λ*}={w∈Λ*∣wν∈Λ*}. Applying π and working backwards we have that FM*(π(μ))=FM*(π(ν)). It follows that ∂γ is also one-to-one, hence a bijection due to the properties of π. If there is an edge e=(FΛ*(μ),FΛ*(iμ)) labeled i, then we have that iμ∈Λ* and thus π(iμ)=π(i)π(μ)∈M*. Therefore, there is an edge labeled π(i) connecting FM*(π(μ)) and FM*(π(iμ)); we write γ(e) for that edge. Since the edges on the same source have different labels we have that γ(e) is unique. In this way, we extend ∂γ to a map γ on the edges. Then (∂γ,γ) gives the required edge-labeled graph isomorphism.□ Remark 5.7 Proposition 5.6 is straightforward when Λ* and M* are languages of subshifts. In this case, (XΛ*)*=Λ* and the bijection gives a 1-block code between the subshifts. However, here we provide the same result even when (XΛ*)*⊂Λ* and with no reference to possible subshifts that the languages induce. 5.2. Unlabeled graph isomorphisms By an unlabeled graph isomorphism between two edge-labeled graphs we will mean a simple isomorphism of the ambient graphs with the labels omitted. Theorem 5.8 Let Λ*and M*be sofic languages. Then their unlabeled follower set graphs are isomorphic if and only if their quantized dynamics are locally piecewise conjugate. Proof By assumption, the spaces ΩΛ* and ΩM* are (discrete) finite spaces. First suppose that the unlabeled follower set graphs are isomorphic. Then we obtain an induced bijection γs on the vertices, and thus a homeomorphism γs:ΩΛ*→ΩM*. Let a point ω∈Q[m]. Then the number of the edges emitted by ω coincides with the number of edges emitted by γs(ω). Due to the follower set graph construction, this implies that γs(ω)∈P[n], with ∣supp[n]∣ coinciding with the number of emitted edges from γs(ω), thus with ∣supp[m]∣. Now the graph isomorphism implies a bijection, say π, between edges. For convenience, let supp[m]={i1,…,ir} and supp[n]={j1,…,jr}, such that π(il)=jl for all l=1,…r. Then the terminal vertex of il is mapped to the terminal of π(il). Consequently, we obtain   γsϕil(ω)=ψπ(il)γs(ω)foralll=1,…,r.Taking U={ω} gives the required local piecewise conjugacy. Conversely, suppose that (ΩΛ*,φ) and (ΩM*,ψ) are locally piecewise conjugate. Then the ambient spaces are homeomorphic, that is there is a bijection on the vertex sets. Moreover, for every point ω∈Q[m], we have γs(ω)∈P[n] with ∣supp[m]∣=∣supp[n]∣, such that   γsφi(ω)=ψπ(i)γs(ω)foralli∈supp[m]. Recall that φi is defined on ω if and only if i∈supp[m]. Hence local piecewise conjugacy gives that the number of edges that ω emits is ∣supp[m]∣, and thus it equals the number of edges that γs(ω) emits, which is ∣supp[n]∣. The above equation then shows that for the edge labeled i there is a unique edge labeled π(i), such that the end point of the i-edge is mapped to the endpoint of the π(i)-edge. As edges are preserved under γs, we get the required graph isomorphism.□ Remark 5.9. There is a conceptual difference between labeled and unlabeled graph isomorphism. In the first case, the isomorphism on the edges is given by the bijection on the labels, and is the same on all vertices. However, in the second case, the bijection changes each time we pass to another vertex. Removing the labels from a graph representation G=(G,L) of a subshift Λ does not preserve type. In the case of sofic subshifts, this procedure amounts to producing finite covers and hints that unlabeled graph isomorphisms of the follower set graphs should not (Notice the subtle point here: If Λ is a sofic subshift, then removing all labels from its follower set graph G=(G,L) produces an edge-shift XG, and Λ is a factor of XG. However, it is not ensured that G is the follower set graph of XG.) preserve SFT’s. The following example clarifies this point. Example 5.10 Let Λ* be the language of the even shift on {0,1}, that is the forbidden words are of the form 102n+11 for all n≥0. Then the vertices of the follower set graph are of the form   F(μ)=F(∅)ifμcontainsno1’s,F(1)ifμbeginswith02k1forsomek≥0,F(01)ifμbeginswith02k+11forsomek≥0,for Λ*. Therefore, the follower set graph takes up the form   For more details, see [30, Example 3.2.7, Figure 3.2.2]. On the other hand, let M* on {0,1} be defined by the forbidden word 001. Then the truthTable of the Follower Set Graph algorithm is given by:     (with FALSE indicated by a black box). Therefore, we have the vertices   F(∅)=F(0)=F(00)=F(000),F(1)=F(10)=F(11)=F(100)=F(101)=F(110)=F(111),F(01)=F(010)=F(011).Then the follower set graph for M* is   It is immediate that the unlabeled graphs for Λ* and M* are isomorphic. However, Λ* is not a language of finite type. Notice also how the functions fn:Bn(Λ*)→Bn(M*) of Remark 4.2 depend on n in this example. For example, we have that   f4(0010)=1010≠0010=f1(0)f3(010).Recall that the image of w under fn is taken through the unlabeled graph isomorphism, when w corresponds to the path w beginning at ω=F(∅). Remark 5.11 Recall that a point x in a two-sided subshift Λ has period n if there exists an n∈N such that σn(x)=x. The number of points with period n is denoted by pn(Λ). The zeta function of Λ is given by   ζΛ(t)≔exp(∑n=1∞pn(Λ)ntn). The zeta function is not preserved by local piecewise conjugacy. In Example 5.10, we show that the even shift is locally piecewise conjugate to a subshift of finite type. By [30, Theorem 6.4.6], the zeta function of any subshift of finite type is the reciprocal of a polynomial, whereas [30, Example 6.4.5] implies that the zeta function of the even shift is ζ(t)=(1+t)(1−t−t2)−1. We would like to thank Ian Putnam for this remark. 5.3. Graph isomorphism for type 1 languages Let us examine further the case of type 1 languages. We begin with an example of two languages with no isomorphic labeled follower set graphs. Example 5.12 Consider the language Λ* on 5 symbols determined by the forbidden words   {11,21,31,41,12,22,32,42,13,33,24,44}.Then the resulting follower set graph is   for Λ*. Consider also the language M* on 5 symbols determined by the forbidden words   {11,21,31,41,12,22,32,42,13,33,14,44}.The only difference with the forbidden words of Λ* is to consider 14 in place of 24. Similarly we get the follower set graph   for M*. The unlabeled graphs are isomorphic. The only difference in the labeled graphs is the lower right arrow which carries different labels. As the follower set graphs are irreducible, it can be seen that the languages Λ* and M* are also the languages of two-sided irreducible subshifts. In fact these are the augmented versions of [25, Example 9.8]. A key feature in the example above is that there are two symbols that have the same follower sets. It appears that this is the only obstruction. Theorem 5.13 Let Λ*and M*be languages of type 1. Suppose that Λ*is on dsymbols and that FΛ*(i)≠FΛ*(j)for i≠j, with i,j=1,…d. The following are equivalent: The follower set graphs are isomorphic. The unlabeled follower set graphs are isomorphic. Proof Of course item (i) implies item (ii). For the converse, recall that unlabeled graph isomorphism imposes that M* is on d symbols as well. Furthermore, the vertex sets must have the same size. We have two cases. If FΛ*(i)≠FΛ*(∅) for all i, then the same must hold for the graph of M*, as d+1 is the maximum size of the vertex sets. In this case, both FΛ*(∅) and FM*(∅) are the unique sources for the graph, hence related by the graph isomorphism. If there is an i such that FΛ*(i)=FΛ*(∅), then the number of the FM*(j) is d, and thus there exists a unique j such that FM*(j)=FM*(∅). In any case, we get that the graph isomorphism induces a bijection between {FΛ*(i)∣i=1,…,d} and {FM*(i)∣i=1,…,d}. Without loss of generality, we may relabel for M* so that this bijection sends FΛ*(i) to FM*(i). As we remarked in Example 5.1, the labels on the edges are pre-determined by their range. Notice that by hypothesis every vertex receives at most one edge. Therefore, the unlabeled graph isomorphism respects the label of the edges, and the proof is complete.□ Remark 5.14 For type 1 subshifts, there is a strong connection between the followerTable of the Follower Set Graph algorithm and the representation of the subshift as an edge shift. Let us recall how this follows from [30, Theorem 2.3.2]. To allow comparisons, we denote the graph of the edge shift by Ge(Λ). Let Λ be a two-sided subshift of type 1. The vertices of Ge(Λ) are the symbols of Λ. We write an edge between the vertex i and j if (and only if) ji∈Λ*, and label the resulting edge by j. It is evident that this graph is given by a 0–1 adjacency matrix. If A is the followerTable of the Follower Set Graph algorithm where we replace the allowable words by 1 and the forbidden words by 0, then the adjacency matrix of Ge(Λ) is taken by deleting the row and the column that corresponds to ∅ in A (which all have entries equal to 1). Working under the condition that F is one-to-one on the symbol set (of Theorem 5.13) we distinguish two cases: Case 1. If F(∅)=F(i) for some symbol i, then Ge(Λ) coincides with the graph G of the follower set graph G=(G,L) of Λ. Case 2. If F(∅)≠F(i) for all symbols i, then Ge(Λ) coincides with the subgraph G of the follower set graph G=(G,L) of Λ, once we erase the vertex F(∅) and the emitting edges. Remark 5.15 Theorem 5.13 applies to edge shifts with invertible adjacency matrices. Indeed let Ae be the adjacency matrix of Ge(Λ). If there are i≠j with F(i)=F(j), then we have that two rows of the matrix from the Follower Set Graph algorithm coincide. Thus the same holds for Ae and thus detAe=0. Remark 5.16 Theorem 5.13 holds in the particular case of languages of type 1 on two symbols {0,1} without the assumption on the follower set function. Indeed the cases where F(0)=F(1) produce the following graphs:   for the sets of forbidden words ∅ and {00,10,01,11}, and   for the sets of forbidden words {00,01} and {11,10}, respectively. It follows that also in these cases, the follower set graphs are unique up to a permutation of symbols. However, one direction of Theorem 5.13 does not hold in general even for languages of finite type. We highlight this in the following two examples. Example 5.17 Let the language Λ* on two symbols {0,1} be defined by the forbidden words   000,010,001,101,011.Then the corresponding truthTable of the Follower Set Graph algorithm is:     and gives the vertices   v0=F(∅),v1=F(1),v2=F(0),v3=F(01),v4=F(00)=F(10)=F(11)=F(100)=F(110)=F(111).Then the follower set graph of Λ* is   On the other hand, let the language M* on two symbols {0,1} be defined by the forbidden words   000,010,001,100,011.Then the corresponding truthTable of the Follower Set Graph algorithm is:     and gives the vertices   w0=F(∅),w1=F(0),w2=F(1),w3=F(00),w4=F(10)=F(01)=F(11)=F(101)=F(111)=F(110).Then the follower set graph of M* is   It is clear that there is only one unlabeled graph isomorphism; the one sending vi to wi. If it lifted to a labeled graph isomorphism, then the 0 label would match to the 0 label, as it appears from v4 and w4. However, this does not comply with the labels on v0 and w0. The two-sided subshifts coming from the languages Λ* and M* are formed on a single point. For creating a more interesting counterexample in the category of two-sided subshifts, we may use the augmentations defined in {0,1,ζ}. In this case, we obtain   as the follower set graph of Λ˜* and   as the follower set graph of M˜*. Again there is not a labeled graph isomorphism between those. Example 5.18 Let the language Λ* on two symbols {0,1} be defined by the forbidden words   000,100,010,101,011,111.Then the corresponding truthTable of the Follower Set Graph algorithm is:     and gives the vertices   v0=F(∅),v1=F(0),v2=F(1),v3=F(10),v4=F(01)v5=F(00)=F(11)=F(110)=F(001).Then the follower set graph of Λ* is   On the other hand, let the language M* on two symbols {0,1} be defined by the forbidden words   000,110,010,101,001,111.Then the corresponding truthTable of the Follower Set Graph algorithm is:     and gives the vertices   w0=F(∅),w1=F(0),w2=F(1),w3=F(00),w4=F(11)w5=F(10)=F(01)=F(100)=F(011).Then the follower set graph of M* is   We see that that there are two unlabeled graph isomorphisms. The first one sends vi to wi and the second one is the composition with the reflection along the vertical line that passes through w0 and w5. Both of them do not lift to a labeled graph isomorphism as the path 11 connecting v1 with v5 consists of two edges with the same label whereas its image 10 has two edges of different labels. The languages Λ* and M* do not arise from subshifts, but once more we can use their augmentations to produce a counterexample in this class. We thus have   for Λ˜*, and   for M˜*. Again the follower set graphs are not labeled graph isomorphic. 5.4. Irreducible two-sided sofic subshifts Let us examine further the case of irreducible two-sided sofic subshifts. Recall that a subshift is called irreducible if it has a presentation through a labeled graph G=(G,L) so that G is irreducible. Among all presentations, Fischer [17, 18] has shown that there exists a minimal resolving one, that is unique up to label graph isomorphism; see also [30, Theorem 3.3.18]. Minimality is taken with respect to the number of vertices of the ambient graph. This presentation is also known as the Fischer cover of the subshift. Fischer covers are follower-separated; for example see [30, Corollary 3.319]. Uniqueness of the Fischer cover fails for reducible sofic subshifts; Jonoska [22] provides such a counterexample for subshifts of finite type. The Fischer cover can be induced by the Krieger cover or by the follower set graph. A way to obtain it from the follower set graph is as follows. Recall that a word μ is called intrinsically synchronizing if:   wheneverνμ∈Λ*andμν′∈Λ*thenνμν′∈Λ*. Then the minimal resolving presentation is the labeled subgraph of the follower set graph formed by using just the follower sets of the intrinsically synchronizing words (It is worth mentioning that a subshift is of finite type if and only if all sufficiently long words are intrinsically synchronizing [30, Exercise 3.3.5]. This property emphasizes the value of Jonoska’s [22] counterexample.) (see [30, Exercise 3.3.4]). We will see how the graph isomorphism we get from Theorem 5.8 induces an unlabeled graph isomorphism of the Fischer covers. Moreover, that it respects the vertices labeled by follower sets of intrinsically synchronizing words. For the latter, we will require some terminology and results from [30]. Recall that we follow the left version of their notation. Given a labeled graph G=(G,L) we say that a path μ is synchronizing for G if:   allpathslabeledμterminateatthesamevertex. Recall that we read paths from right to left. Suppose in addition that G is resolving and follower separated. In this case, every path w can be extended on the left to a synchronizing path μ=uw for G by [30, Proposition 3.3.16]. Under the same assumption, if μ is synchronizing for G, then every path uμ is synchronizing for G by [30, Lemma 3.3.15]. The connection between synchronizing paths and intrinsically synchronizing words is given in [30, Exercise 3.3.3]. That is, if G is the minimal resolving presentation of a two-sided irreducible sofic shift Λ then a path w is synchronizing for G if and only if the word w is intrinsically synchronizing for Λ. Now we have set the context for proving the next corollary. Corollary 5.19. Let Λand Mbe two-sided irreducible sofic subshifts. If Λand Mare locally piecewise conjugate, then there is an unlabeled graph isomorphism between their Fischer covers. Furthermore, the unlabeled graph isomorphism induces a bijection between the collections  {FΛ(μ)∣μisanintrinsicallysynchronizingwordforΛ}and  {FM(ν)∣νisanintrinsicallysynchronizingwordforM}. Proof Let GΛ=(GΛ,L1) and GM=(GM,L2) be the follower set graphs of Λ and M, respectively. Let HΛ be the labeled graph that remains from GΛ by using only the intrinsically synchronizing words, i.e. the Fischer cover of Λ. The graph isomorphism of Theorem 5.8 then gives an isomorphism of the ambient graph HΛ of HΛ onto a subgraph HM of GM. Let HM be the labeled subgraph induced by HM inside GM. Notice that both HΛ and HM are resolving as subgraphs of the resolving follower set graphs and thus   h(X(HΛ))=h(X(HΛ))andh(X(HM))=h(X(HM))by [30, Proposition 4.13]. First, we claim that HM gives a presentation of M. If X(HM)≠M then X(HM) is a proper subshift of M and, therefore, [30, Corollary 4.4.9] yields   h(X(HM))<h(M). Since graph isomorphism of HΛ with HM respects entropy, we get that   h(Λ)=h(X(HΛ))=h(X(HΛ))=h(X(HM))=h(X(HM))<h(M). However, this contradicts h(M)=h(Λ) of Proposition 4.3. Hence HM is a presentation of M. Notice here that the number of vertices of HM coincides with the minimal number of vertices required to describe Λ, that is   ∣(HM)(0)∣=∣(HM)(0)∣=∣(HΛ)(0)∣=∣(HΛ)(0)∣. Secondly, we claim that HM is minimal for M with respect to the number of vertices. Otherwise we could find a subgraph HM′ of GM on less vertices than that of HM. Then the unlabeled graph isomorphism would carry over, as above, to a presentation HΛ′ of Λ, giving   ∣(HΛ′)(0)∣=∣(HM′)(0)∣<∣(HM)(0)∣=∣(HΛ)(0)∣. However, this contradicts minimality of HΛ for Λ, and thus HM is a minimal presentation of M. So far we have proved that HM is a minimal resolving presentation of M. Initially, HM is isomorphic to the subgraph of GM obtained by using the vertices labeled by follower sets of intrinsically synchronizing words. We will show now that these two sub-graphs of GM are actually equal. Due to minimality, it suffices to show that the vertices of HM correspond to follower sets of intrinsically synchronizing words. We will use that HM is follower-separated and irreducible as a minimal presentation of M [30, Corollary 3.3.19]. Let F(w) be a vertex of HM. We will show that   ∃anintrinsicallysynchronizingwordν∈M*suchthatF(w)=F(ν). Consider a labeled path u1 in HM starting at a vertex J and extend it to a synchronizing path u2u1 in HM. That is, all paths labeled u2u1 end at the same vertex, say J′ in HM. By irreducibility of HM, there is a path u3 connecting J′ with F(w) so that the path u3u2u1 is allowable in HM. Moreover, this is an extension of a synchronizing path and thus it is a synchronizing path in HM, that is all paths in HM representing u3u2u1 end at F(w). Minimality of HM implies that u3u2u1 is an intrinsically synchronizing word for M. As HM represents M, the follower set of u3u2u1 in M coincides with the paths in HM starting at the vertex F(w). However, the collection of these paths is exactly the set F(w) due to the follower set graph construction. Hence we conclude that F(w)=F(u3u2u1) and the proof is complete.□ Remark 5.20 Referring to the proof of Corollary 5.19, we do not claim a direct bijection between intrinsically synchronizing words. It is unclear whether the function of Proposition 4.3 respects this property. The main obstacle is that the word   fn−1(f∣ν∣(ν)μf∣ν′∣(ν′))forn=∣ν∣+∣μ∣+∣ν′∣gives a word ν″μ′ν′ instead of νμ′ν′ (as indicated in Example 5.10). A similar obstacle does not allow checking directly whether the mixing property is also preserved under local piecewise conjugacy. Example 5.21 We require both Λ and M be irreducible in Corollary 5.19. This is because local piecewise conjugacy does not preserve irreducibility. For an example recall the even shift and the subshift of finite type constructed in Example 5.10. The even shift Λ is irreducible and the subgraph   gives its Fischer cover. However, the subshift M on {001} is not irreducible. Any subgraph on less vertices produces a proper subshift of M. In particular, the unlabeled graph isomorphism from the Fischer cover of Λ to M produces the irreducible subgraph   which does not represent 0∞∈M. Example 5.22 The converse of Corollary 5.19 does not hold. That is, irreducible subshifts that are not locally piecewise conjugate may have Fischer covers that admit an unlabeled graph isomorphism. For a counterexample, let M be defined by the forbidden word {00}. Then the truthTable of M is     and thus its follower set graph is given by   where v0=F(∅)=F(1) and v1=F(0). Consequently, M is irreducible (having one irreducible presentation) and its Fischer cover coincides with its follower set graph. Comparing with the even shift Λ, we see that the Fischer covers of Λ and M admit an unlabeled graph isomorphism, but this does not hold for their follower set graphs. Acknowledgements This project started as a continuation of with Orr Shalit. Following his suggestion, it was decided for the paper to go with just two authors. We thank Orr for the numerous comments and corrections on earlier drafts of the paper. 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