Ordered Bratteli Diagrams, Dimension Groups and Topological Dynamics

Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations

“…We also state some basic facts about dimension groups and their relation to Bratteli diagrams and to Cantor minimal systems. For further details, the reader should consult the references [3], [4], [6], [7].…”

“…It is a theorem [7,Theorem 4.7] that any Cantor minimal system (Y, ψ) is conjugate to a Bratteli-Vershik system, which we will call a Bratteli-Vershik model for (Y, ψ) . We briefly recapitulate the construction.…”

“…By the above construction, the base point y will correspond to the unique minimal path x min of the resulting Bratteli-Vershik system. (Note: In the references [6], [7], the roles of the base and top floors are reversed, so that there y will correspond to x max .) It is important that the Bratteli-Vershik system we get is the same up to conjugacy regardless of the choice of base point and the choice of the sets shrinking down to the base point.…”

Augmenting dimension group invariants for substitution dynamics

“…Indeed, computing the K-groups of C * -algebras associated to certain shift spaces, we shall arrive at a flow (and hence conjugacy) invariant for these. This invariant is closely related to, but finer than, the dimension groups for substitutional shift spaces defined by Herman, Putnam and Skau in [18], and studied in this particular setting by Durand, Host and Skau in [14].…”

Cantor aperiodic systems and Bratteli diagrams

“…Every Cantor minimal system can be represented as the Vershik map acting on an ordered Bratteli diagram [6]. This representation turns out to be a powerful tool in the study of orbit equivalence of Cantor minimal systems [3,4,5,7].…”