ORBIT EQUIVALENCE, FLOW EQUIVALENCE AND ORDERED COHOMOLOGY

We study self-homeomorphisms of zero dimensional metrizable compact Ha usdorff spaces by means of the ordered first cohomology group, particu larly in the light of the recent work of Giordano, Putnam, and Skau on minimal homeomorphisms. We show that Row equivalence of systems is an alogous to Morita equivalence between algebras, and this is reflected in the ordered cohomology group. We show that the ordered cohomology g roup is a complete invariant for flow equivalence between irreducible shifts of finite type; it follows that orbit equivalence implies flow equivalence for this class of systems. The cohomology group is the (pr e-ordered) Grothendieck group of the C-algebra crossed product, and w e can decide when the pre-ordering is an ordering, in terms of dynamic al properties.