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It has been proved by Giordano, Putnam and Skau [4, Theorem 2.1] that a Krieger type theorem holds for the family of Cantor minimal systems, i.e. the (topological) orbit structure is related to the isomorphism class of the associated C -crossed products. It is natural to investigate whether a similar result is true for a larger class of minimal topological systems. Assuming the Elliott conjecture, namely that a complete isomorphism invariant for the associated (simple) C -crossed products is of (ordered) K-theoretic nature, we give a resounding counterexample by considering special non-homogeneous systems. The construction we present also has a purely dynamical aspect, which is of independent interest. If we restrict attention to the very special family of equicontinuous minimal systems, we show that a Krieger type theorem is true, extending a previous result by Riedel [13]. Finally, we investigate the notion of strong orbit equivalence for minimal topological systems in general. We present a rather surprising example in this connection.

$C^*$-algebras associated to non-homogeneous minimal systems and their K-theory