Thierry Giordano, Ian Putnam and Christian Skau
We prove several new results about $AF$-equivalence relations, and relate these to Cantor minimal systems (i.e. to minimal $Z$-actions). The results we obtain turn out to be crucial for the study of the topological orbit structure of more general countable group actions (as homeomorphisms) on Cantor sets, which will be the topic of a forthcoming paper. In all this, Bratteli diagrams and their dynamical interpretation, are indispensable tools.