Real Coboundaries for Minimal Cantor Systems

In this paper we investigate the role of real-valued coboundaries for classifying of minimal homeomorphisms of the Cantor set. This work follows the work of Giordano, Putnam, and Skau who showed that one can use integer-valued coboundaries to characterize minimal homeomorphisms up to strong orbit equivalence. First, we prove a rigidity result. We show that there is an orbit equivalence between minimal Cantor systems which preserves real-valued coboundaries if and only if the systems are flip conjugate. Second, we investigate a real analogue of the dynamical unital ordered cohomology group studied by Giordano, Putnam and Skau. We show that, in general, isomorphism of our unital ordered vector space determines a weaker relation than strong orbit equivalence and we characterize this relation in a certain finite dimensional case. Finally, we consider isomorphisms of this vector space which preserve the cohomology subgroup. We show that such an isomorphism gives rise to a strictly stronger relation than strong orbit equivalence. In particular, it determines topological discrete spectrum, but does not determine systems up to flip conjugacy.