[Joint work with Mike Boyle] We are interested in zed-actions on Cantor sets, but have to work in the more general setting of zero-dimensional spaces. Unlike the case of minimal actions, where the ordered K_0 (of the crossed product C*-algebra, or equivalently, the ordered Cech cohomology) is well understood and described, the ordered K_0 group for non-minimal systems is wild---very few are dimension groups, and interestingly, many dimension groups with order unit cannot be realized. The typical interesting non-minimal examples arises are subshifts of finite type (SFT).

There is a large class of small, tractible systems (on atomic spaces: these are finite compactifications of countable discrete spaces) corresponding to directed graphs, for which many test questions can be applied; the (graph-theoretic ordered edge) cohomology of the graph appears, but is not the whole story.

Moreover, given a large space (such as a mixing SFT), there are typically lots of useful maps from the ordered K_0-group of the big space to these tiny ones, which yield a number of properties (mostly negative, such as the group not being a dimension group) of the big space. The order ideal structure is particularly rich, but revealing---for example, the order ideals correspond to chain recurrent subsets, and if nonempty intersections chain transitive closed invariant subsets are not themselves again chain transitive (which happens generically), then the dimension group property fails, and in addition, the set of traces on the order ideal is not going to be a simplex space.