m at h . D S ] 2 3 Ju n 20 05 The K - group of Substitutional Systems

In another article we associated a dynamical system to a non-properly ordered Bratteli diagram. In this article we describe how to compute the K−group K 0 of the dynamical system in terms of the Bratteli diagram. In the case of properly ordered Bratteli diagrams this description coincides with what is already known, namely the so-called dimension group of the Bratteli diagram. The new ordered group defined here is more relevant for non-properly ordered Bratteli diagrams. We use our main result to describe K 0 of a substitutional system. An important tool in the study of Cantor minimal dynamical systems (X, T) is its K-theory; in particular the K 0 −group K 0 (X, T), which is an ordered group, is an important invariant. After the celebrated Vershik-Herman-Putnam-Skau approach of codifying minimal Cantor dynamical systems by using the so-called ordered Brat-teli diagrams, it became relevant to understand the group K 0 directly through diagrams. This is achieved in [HPS, Thm.5.4 and Cor.6.3] when properly ordered Bratteli diagrams are employed. Recently, we showed how to associate dynamical systems to non-properly ordered Bratteli diagrams. We generalise the above result of [HPS] by a careful modification of the notion of dimension group of an ordered Bratteli diagram. In doing this we have employed the " tripling " construction that was first introduced in [EP]. Since our description of the modification of dimension group in the case of non-properly ordered Bratteli diagrams depends heavily on the key constructions that were first introduced in [EP] we summarize the same for the benefit of the reader following closely the text of the first chapter of [EP]. Some of the basic definitions and concepts in the study of Cantor dynamical systems are also recalled in this section. A topological dynamical system is a pair (X, ϕ) where X is a compact metric space and ϕ is a homeomorphism in X. We say that ϕ is minimal if for any x ∈ X, the ϕ-orbit of x := {ϕ n (x) | n ∈ Z} is dense in X. We say that (X, ϕ) is a Cantor dynamical system if X is a Cantor set, i.e. X is totally disconnected without isolated points. (X, ϕ) is a Cantor minimal dynamical system if, in addition, ϕ is minimal. Some of the basic concepts of the theory are recalled below, mostly from the more detailed sources [DHS] and [HPS].