Affable equivalence relations and orbit structure of Cantor dynamical systems

Abstract. 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 generalcountable 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.