On subshift presentations
Authors: KRIEGER WOLFGANG 
Published: March 2016

We consider partitioned graphs, by which we mean finite directed graphs with a partitioned edge set${\mathcal{E}}={\mathcal{E}}^{-}\cup {\mathcal{E}}^{+}$. Additionally given a relation${\mathcal{R}}$between the edges in${\mathcal{E}}^{-}$and the edges in${\mathcal{E}}^{+}$, and under the appropriate assumptions on${\mathcal{E}}^{-},{\mathcal{E}}^{+}$and${\mathcal{R}}$, denoting the vertex set of the graph by$\mathfrak{P}$, we speak of an${\mathcal{R}}$-graph${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$. From${\mathcal{R}}$-graphs${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$we construct semigroups (with zero)${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$that we call${\mathcal{R}}$-graph semigroups. We write a list of conditions on a topologically transitive subshift with property$(A)$that together are sufficient for the subshift to have an${\mathcal{R}}$-graph semigroup as its associated semigroup.

Generalizing previous constructions, we describe a method of presenting subshifts by means of suitably structured finite labeled directed graphs$({\mathcal{V}},~\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706}~)$with vertex set${\mathcal{V}}$, edge set$\unicode[STIX]{x1D6F4}$, and a label map that assigns to the edges in$\unicode[STIX]{x1D6F4}$labels in an${\mathcal{R}}$-graph semigroup${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$. We denote the presented subshift by$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$and call$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$an${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$-presentation.

We introduce a property$(B)$of subshifts that describes a relationship between contexts of admissible words of a subshift, and we introduce a property$(c)$of subshifts that in addition describes a relationship between the past and future contexts and the context of admissible words of a subshift. Property$(B)$and the simultaneous occurrence of properties$(B)$and$(c)$are invariants of topological conjugacy.

We consider subshifts in which every admissible word has a future context that is compatible with its entire past context. Such subshifts we call right instantaneous. We introduce a property$RI$of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a right instantaneous presentation. We consider also subshifts in which every admissible word has a future context that is compatible with its entire past context, and also a past context that is compatible with its entire future context. Such subshifts we call bi-instantaneous. We introduce a property$BI$of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a bi-instantaneous presentation.

We define a subshift as strongly bi-instantaneous if it has for every sufficiently long admissible word$a$an admissible word$c$, that is contained in both the future context of$a$and the past context of$a$, and that is such that the word$ca$is a word in the future context of$a$that is compatible with the entire past context of$a$, and the word$ac$is a word in the past context of$a$, that is compatible with the entire future context of$a$. We show that a topologically transitive subshift with property$(A)$, and associated semigroup a graph inverse semigroup${\mathcal{S}}$, has an${\mathcal{S}}$-presentation, if and only if it has properties$(c)$and$BI$, and a strongly bi-instantaneous presentation, if and only if it has properties$(c)$and$BI$, and all of its bi-instantaneous presentations are strongly bi-instantaneous.

We construct a class of subshifts with property$(A)$, to which certain graph inverse semigroups${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$are associated, that do not have${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$-presentations.

We associate to the labeled directed graphs$({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$topological Markov chains and Markov codes, and we derive an expression for the zeta function of$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$in terms of the zeta functions of the topological Markov shifts and the generating functions of the Markov codes.