K-groups associated with substitution minimal systems

Abstract

Two ordered Bratteli diagrams can be constructed from an aperiodic substitution minimal dynamical system. One, the proper diagram, has a single maximal path and a single minimal path and the Vershik map on the path space can be extended homeomorphically to a map conjugate to the substitution system. The other, the improper diagram, encodes the substitution more naturally but often has many maximal and minimal paths and no continuous compact dynamics. This paper connects the two diagrams by considering theirK 0-groups, obtaining the equation $$K_0 (Proper) = K_0 (Improper)/Q \oplus \mathbb{Z}^\nu $$ whereQ and ν can be determined from the combinatorial properties of the substitution. This allows several examples of substitution sequences to be distinguished at the level of strong orbit equivalence. A final section shows that every dimension group with unit which is a stationary limit of ℤ n groups can be represented as aK 0 group of some substitution minimal system. Also every stationary proper minimal ordered Bratteli diagram has a Vershik map which is either Kakutani equivalent to ad-adic system or is conjugate to a substitution minimal system. The equation above applies to a much wider class which includes those minimal transformations which can be represented as a path-sequence dynamical system on a Bratteli diagram with a uniformly bounded number of vertices in each level.