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Classifying crossed product C*-algebras

I combine recent results in the structure theory of nuclear C*-algebras and in topological dynamics to classify certain types of crossed products in terms of their Elliott invariants. In particular, transformation group C*-algebras associated to free minimal Z^d-actions on the Cantor set with compact space of ergodic measures are classified by their ordered K-theory. In fact, the respective statement holds for finite dimensional compact metrizable spaces, provided that projections of the crossed products separate tracial states. Moreover, C*-algebras associated to certain minimal homeomorphisms of odd dimensional spheres are only determined by their spaces of invariant Borel probability measures (without a condition on the space of ergodic measures). Finally, I show that for a large collection of classifiable C*-algebras, crossed products by Z^d-actions are generically again classifiable.