Ordered K-groups associated to substitutional dynamics

The Matsumoto K0-group is an interesting invariant of flow equivalence for symbolic dynamical systems. Because of its origin as the K-theory of a certain C∗-algebra, which is also a flow invariant, this group comes equipped with a flow invariant order structure. We emphasize this order structure and demonstrate how methods from operator algebra and symbolic dynamics combine to allow a computation of it in certain cases, including Sturmian and primitive substitutional shifts. In the latter case we show by example that the ordered group is a strictly finer invariant than the group itself.