ar X iv : m at h / 05 06 30 5 v 3 [ m at h . D S ] 1 F eb 2 00 6 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 (see 3.1) 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]. The result which describes the group K 0 in the case of a substitutional system arising from a primitive aperiodic non-proper substitution is described in theorem 3.12. It may be remarked that a method of computing K 0 even in the case of non-proper substitutions is indicated in [DHS, sections 5,6,7]; it relies on showing that the substitutional dynamical system is iso-morphic to another one arising from a proper substitution. The proof of [DHS, proposition 20] and [DHS, proposition 23] relies heavily on 'return words' and 'de-rivative sequences' (loc.cit). But this method seems to us to be quite indirect and not entirely transparent; 'return words' are essentially of an existential nature and hence do not afford a feasible method by which to compute efffectively the dimension groups or even the Bratteli diagrams of the preferred proper substitutional system. In contrast we feel that our description in theorem 3.12 for non-proper substitutions is direct and closer in its approach and simplicity to the above cited Herman-Putnam-Skau description for proper substitutions. It eliminates the handicap of having to first work …