Orbit equivalence for Cantor minimal systems and a Bratteli-Vershik model for Cantor minimal $Z^2$ actions

The talk will consist of two parts. In the first, I will give an overview of the work of myself with Thierry Giordano (Ottawa) and Christian Skau on orbit equivalence in the topological category of certain minimal dynamics on the Cantor set. This includes two important classes: the so-called approximately finite relations (AF) and free actions of $Z^d$. The interplay between these is a key feature. In the second part, I will describe more recent work to find a model for $Z^2$ actions on the Cantor set. The model for Z-actions given by Herman, Skau and myself, following ideas of Vershik, has been used extensively, but the generalization to $Z^2$ has been illusive. I will try to explain a new idea coming from cohomology which sheds light on the problem.