Kengo Matsumoto has introduced lambda-graph systems and strong shift equivalence of lambda-graph systems [Doc.Math.4 (1999), 285-340]. We associate to a subshift of a subshift a lambda-graph system. The strong shift equivalence class of the associated lambda-graph system is an invariant of subsystem equivalence. Wolfgang Krieger and Kengo Matsumoto have introduced the lambda$-entropy of a lambda-graph system and have shown its invariance under strong shift equivalence [Ergod.Th.&Dynam.Sys. 24 (2004) 1155 - 1172]. A separation entropy of a subshift of a subshift is introduced as the lambda-entropy of the associated lambda-graph system.
A certain synchronizing property for subshifts called $λ$-synchronization yields $λ$-graph systems called the $λ$-synchronizing $λ$-graph systems for the subshifts. The $λ$-synchronizing $λ$-graph system is a left Fischer cover analogue for a $λ$-synchronizing subshift. We will study algebraic structure of the $C^*$-algebra associated with the $λ$-synchronizing $λ$-graph system and prove that the stable isomorphism class of the $C^*$-algebra with its Cartan subalgebra is invariant under flow equivalence of $λ$-synchronizing subshifts.
The notions of symbolic matrix system and $λ$-graph system for a subshift are generalizations of symbolic matrix and $λ$-graph (= finite symbolic matrix) for a sofic shift respectively ([Doc. Math. 4(1999), 285-340]). M. Nasu introduced the notion of textile system for a pair of graph homomorphisms to study automorphisms and endomorphisms of topological Markov shifts ([Mem. Amer. Math. Soc. 546,114(1995)]). In this paper, we formulate textile systems on $λ$-graph systems and study automorphisms on subshifts. We will prove that for a forward automorphism $ϕ$ of a subshift $(Λ,σ)$, the automorph…
We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.
Subshifts with property $(A)$ are constructed from a class of directed graphs. As special cases the Markov-Dyck shifts are shown to have property $(A)$. The semigroups, that are associated to $\mathcal R$-graph shifts with Property (A), are determined. Also results on the reconstruction of $\mathcal R$-graphs from their $\mathcal R$-graph shifts are obtained.
A $λ$-graph system is a labeled Bratteli diagram with some additional structure, which presents a subshift and yields a $C^*$-algebra. In this paper, we construct a $λ$-graph system from a pushdown automaton, such that the accepted language by the automaton coincides with the language of admissible words of the presented subshift by the $λ$-graph system. The $λ$-graph systems for pushdown automata accepting the languages of Markov-Dyck shifts and sofic-Dyck shifts are presented.
We consider partitioned graphs, by which we mean finite strongly connected directed graphs with a partitioned edge set $ {\mathcal E} ={\mathcal E}^- \cup{\mathcal E}^+$. With additionally given a relation $\mathcal R$ between the edges in ${\mathcal E}^-$ and the edges in $\mathcal E^+ $, and denoting the vertex set of the graph by ${\frak P}$, we speak of an an ${\mathcal R}$-graph ${\mathcal G}_{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+) $. From ${\mathcal R}$-graphs ${\mathcal G}_{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+) $ we construct semigroups (with zero) ${\mathca…
A $λ$-graph system $\frak L$ is a labeled Bratteli diagram with shift operation. It is a generalized notion of finite labeled graph and presents a subshifts. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and $C^*$-algebras constructed from $λ$-graph systems.
We show that Property $(A)$ of subshifts and the semigroup, that is associated to subshifts with Property (A), are invariants of flow equivalence. We show for certain $\mathcal R$-graphs that their isomorphism is implied by the flow equivalence of their $\mathcal R$-graph shifts.
We develop some aspects of a general theory of presentations of subshifts by labelled directed graphs, in particular by compact graphs. Also considered are synchronization properties of subshifts that lead to presentations by countable graphs.
We present a new sufficient criterion to prove that a non-sofic half-synchronized subshift is direct prime. The criterion is based on conjugacy invariant properties of Fischer graphs of half-synchronized shifts. We use this criterion to show as a new result that all n-Dyck shifts are direct prime, and we also give new proofs of direct primeness of non-sofic beta-shifts and non-sofic S-gap shifts. We also construct a class of non-sofic synchronized direct prime subshifts which additionally admit reversible cellular automata with all directions sensitive.
$λ$-graph systems are labeled Bratteli diagram with shift operations. They present subshifts. Their matrix presentations are called symbolic matrix systems. We define skew products of $λ$-graph systems and study extensions of subshifts by finite groups. We prove that two canonical symbolic matrix systems are $G$-strong shift equivalent if and only if their presented subshifts are $G$-conjugate. $G$-equivalent classes of subshifts are classified by the cohomology classes of their associated skewing functions.
The paper gives a characterisation of the chain relation of a sofic subshift. Every sofic subshift $Σ$ can be described by a labelled graph $G$. Factorising $G$ in a suitable way we obtain the graph $G/_\approx$ that offers insight into some properties of the original subshift. Using $G/_\approx$ we describe first the chain relation in $Σ$, then characterise chain-transitive sofic subshifts, chain-mixing sofic subshifts and finally the attractors of the subshift dynamic system. At the end we present (straightforward) algorithms deciding chain-transitivity and chain-mixing properties of a sofic…
We discuss a method of calculating the zeta function of subshifts which have a presentation by a finite directed graph labeled by elements of the associated inverse semigroup. This class of subshifts is introduced as a class of property A subshifts(T.Hamachi, K.Inoue and W.Krieger/Subsystems of finite type and semigroup invariants of subshifts, preprint), and the Dyck shift and the Motzkin shift are representative subshifts in this class. The exact number of the periodic points and entropies are also given by this method, and these values are used in the embedding condition for an irreducible …
This paper is a continuation of the paper entitled "Subshifts, $λ$-graph bisystems and $C^*$-algebras", arXiv:1904.06464. A $λ$-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying certain compatibility condition on their edge labeling. For any two-sided subshift $Λ$, there exists a $λ$-graph bisystem satisfying a special property called FPCC. We will construct an AF-algebra ${\mathcal{F}}_{\frak L}$ with shift automorphism $ρ_{\frak L}$ from a $λ$-graph bisystem $({\frak L}^-,{\frak L}^+)$, and define a $C^*$-algebra ${\mathcal R}_{\frak L}$ by the crossed product ${\…
We consider minimal, aperiodic symbolic subshifts and show how to characterize the combinatorial property of bounded powers by means of a metric property. For this purpose we construct a family of graphs which all approximate the subshift space, and define a metric on each graph which extends to a metric on the subshift space. The characterization of bounded powers is then given by the Lipschitz equivalence of a suitably defined infimum metric with the corresponding supremum metric. We also introduce zeta-functions and relate their abscissa of convergence to various exponents of complexity of …
We introduce a notion of $λ$-graph bisystem. It consists of a pair $({\frak L}^-, {\frak L}^+)$ of two labeled Bratteli diagrams ${\frak L}^-, {\frak L}^+$ over alphabets $Σ^-, Σ^+$, respectively, and satisfy certain compatibility condition of their labeling on edges. Its matrix presentation is called a symbolic matrix bisystem. We first show that any $λ$-graph bisystem presents subshifts and conversely any subshift is presented by a $λ$-graph bisystem, called the canonical $λ$-graph bisystem for the subshift. We introduce a notion of properly strong shift equivalence on symbolic matrix bisyst…
Given a $k$-graph $Λ$ we construct a Markov space $M_Λ$, and a collection of $k$ pairwise commuting cellular automata on $M_Λ$, providing for a factorization of Markov's shift. Iterating these maps we obtain an action of ${\mathbb N}^k$ on $M_Λ$ which is then used to form a semidirect product groupoid $M_Λ\rtimes {\mathbb N}^k$. This groupoid turns out to be identical to the path groupoid constructed by Kumjian and Pask, and hence its C*-algebra is isomorphic to the higher rank graph C*-algebra of $Λ$.
We show that a synchronized coded system $X$ is intrinsically ergodic of full support if and only if $ h(Y)$, the topological entropy of $Y$, is less than $h(X) $ whenever $Y$ is a proper subsystem of $ X $. We also show that like systems with specification property, SVGL's the non-mixing extension of systems with specification property, are intrinsically ergodic of full support. Moreover, we compute the entropy of the underlying graph of the Fischer cover of a synchronized system.
Higher-rank graphs (or $k$-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger $C^*$-algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the $C^*$-algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their $K$-theory. We find examples whose $C^*$-algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic …
A $λ$-graph bisystem $\mathcal{L}$ consists of two labeled Bratteli diagrams $(\mathcal{L}^-,\mathcal{L}^+)$, that presents a two-sided subshift $Λ_\mathcal{L}$. We will construct a compact totally disconnected metric space with a shift homeomorphism consisting of two-dimensional configurations from a $λ$-graph bisystem. The configuration space has a certain étale AF-equivalence relation written $R_{\mathcal{L}}$ with a natural shift homeomorphism $σ_{\mathcal{L}}$ coming from the shift homeomorphism $σ_{Λ_{\mathcal{L}}}$ on the subshift $Λ_{\mathcal{L}}$. The equivalence relation $R_{\mathcal…
A $λ$-graph system ${\frak L}$ is a generalization of a finite labeled graph and presents a subshift. We will prove that the topological dynamical systems $(X_{{\frak L}_1},σ_{{\frak L}_1})$ and $(X_{{\frak L}_2},σ_{{\frak L}_2})$ for $λ$-graph systems ${\frak L}_1$ and ${\frak L}_2$ are continuously orbit equivalent if and only if there exists an isomorphism between the associated $C^*$-algebras ${\Cal O}_{{\frak L}_1}$ and ${\Cal O}_{{\frak L}_2}$ keeping their commutative $C^*$-subalgebras $C(X_{{\frak L}_1})$ and $C(X_{{\frak L}_2})$. It is also equivalent to the condition that there exist…
This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proven. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterised in terms of combinatorial quantities, based on a recent result of Liu and Qu. Particular simple characterisations are provided for simple Toeplitz subshifts that correspond to the orbital Schrei…
We introduce a class of subshifts under the name of "standard one-counter shifts". The standard one-counter shifts are the Markov coded systems of certain Markov codes that belong to the family of one-counter languages. We study topological conjugacy and flow equivalence of standard one-counter shifts. To subshifts there are associated C*-algebras by their $λ$-graph systems. We describe a class of standard one-counter shifts with the property that the C*-algebra associated to them is simple, while the C*-algebra that is associated to their inverse is not. This gives examples of subshifts that …
Let O_{Lambda} be a higher rank graph C*-algebra of rank r. For every tuple p of non-negative integers there is a canonical completely positive map Phi^p on O_{Lambda} and a subshift T^p on the path space X of the graph. We show that ht(Phi^p)=h(T^p), where ht is Voiculescu's approximation entropy and h the classical topological entropy. For a higher rank Cuntz-Krieger algebra O_M we obtain ht(Phi^p)= log r(M_1^{p_1}M_2^{p_2} ... M_r^{p_r}), r being the spectral radius. This generalises Boca and Goldstein's result for Cuntz-Krieger algebras.
We study the density of periodic configurations for shift spaces defined on (the Cayley graph of) a finitely generated group. We prove that in the case of a full shift on a residually finite group and in that of a group shift space on an abelian group, the periodic configurations are dense. In the one-dimensional case we prove the density for irreducible sofic shifts. In connection with this we study the surjunctivity of cellular automata and local selfmappings. Some related decision problems for shift spaces of finite type are also investigated.
In this paper we associate an abelian category to a finite directed graph and prove the categories arising from two graphs are equivalent if the incidence matrices of the graphs are shift equivalent. The abelian category is the quotient of the category of graded vector space representations of the quiver obtained by making the graded representations that are the sum of their finite dimensional submodules isomorphic to zero. Actually, the main result in this paper is that the abelian categories are equivalent if the incidence matrices are strong shift equivalent. That result is combined with an…
Let $X$ be an irreducible shift of finite type (SFT) of positive entropy, and let $B_n(X)$ be its set of words of length $n$. Define a random subset $ω$ of $B_n(X)$ by independently choosing each word from $B_n(X)$ with some probability $α$. Let $X_ω$ be the (random) SFT built from the set $ω$. For each $0\leq α\leq1$ and $n$ tending to infinity, we compute the limit of the likelihood that $X_ω$ is empty, as well as the limiting distribution of entropy for $X_ω$. For $α$ near 1 and $n$ tending to infinity, we show that the likelihood that $X_ω$ contains a unique irreducible component of positi…
Sofic shifts are symbolic dynamical systems defined by the set of bi-infinite sequences on an edge-labeled directed graph, called a presentation. We study the computational complexity of an array of natural decision problems about presentations of sofic shifts, such as whether a given graph presents a shift of finite type, or an irreducible shift; whether one graph presents a subshift of another; and whether a given presentation is minimal, or has a synchronizing word. Leveraging connections to automata theory, we first observe that these problems are all decidable in polynomial time when the …
In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow Γ$ and vector towers $\overleftarrow v$ on $\overleftarrow Γ$, which allows us to efficiently describe all invariant measures $μ= μ^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$…
We show that for a sofic shift Lambda, Matsumoto's C*-algebra O_Lambda is isomorphic to the Cuntz-Krieger algebra of the left Krieger cover graph of Lambda.
