A BRATTELI DIAGRAM FOR COMMUTING HOMEOMORPHISMS OF THE CANTOR SET

This paper presents a construction of a sequence of Kakutani-Rohlin Towers and a corresponding simple Bratteli Diagram, B, for a general minimal aperiodic Z action on a Cantor Set, X, with d 2. This is applied to the description of the orbit structure and to the calculation of the ordered group of coinvariants. For each minimal continuous Z action on X there is a minimal continuous Z action on X which has the same orbits on the set X nY , where Y is of the form [ v2Z , where Y is closed and nowhere dense, and Y is null with respect to every probability measure invariant under the action. Also the ordered top cohomology of (X; Z ), H , is a simple dimension group. There is a positive order embedding surjection K 0 (B) \\Gamma! H which becomes an order isomorphism upon quotienting by the infinitessimals. So there is no ordered cohomological obstruction to continuous orbit equivalence between minimal Z actions and Z actions in general.