Orbit equivalence, flow equivalence and ordered cohomology

We study self-homeomorphisms of zero dimensional metrizable compact Hausdorff spaces by means of the ordered first cohomology group, particularly in the light of the recent work of Giordano, Putnam, and Skau on minimal homeomorphisms. We show that flow equivalence of systems is analogous to Morita equivalence between algebras, and this is reflected in the ordered cohomology group. We show that the ordered cohomology group 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 (pre-ordered) Grothendieck group of the C*-algebra crossed product, and we can decide when the pre-ordering is an ordering, in terms of dynamical properties. Let T be a homeomorphism of a compact metrizable zero-dimensional space X. Let C(X,Z) denote the continuous integer-valued functions on X. Denote the subgroup of coboundaries by cobdy (T ) = { f − (f ◦ T ) | f ∈ C(X,Z)} . The quotient group C(X,Z)/cobdy (T ) will be abbreviated G . Define G+ = { [f ] ∈ G ∣∣ there exists nonnegative f0 : X → Z such that [f0] = [f ]} . Let G denote the unital preordered group (G , G+, [1]). Building on earlier work [V, P, HPS], Giordano, Putnam, and Skau [GPS] recently proved that the groups G modulo their infinitesimals classify the minimal homeomorphisms of the Cantor set up to orbit equivalence. The possible G in this case are precisely the unital simple dimension groups [HPS]. This remarkable classification in the minimal case provides more than ample justification for a general investigation of G . This paper is devoted to laying down some early foundations for this investigation. Systems with a unique minimal set were studied in [HPS]. Poon [Po] showed G is an unperforated ordered group when T is topologically transitive (but not in general), pointed out that there one can recover the zeta function of T as an invariant of the abstract unital group G , and began a study of G when T is an irreducible shift of finite type. It should be emphasized that the order structure is crucial in all of this. This paper has five sections. In the first, we review some basics of preordered groups and suspensions, show the isomorphism class of (G , G+) is determined by the flow equivalence class of T , and in a refinement of Poon’s result, show that the set Z(T ) of zeta functions of homeomorphisms flow equivalent to T is an invariant of the abstract unital preordered group G modulo its infinitesimals.