Crossed products from minimal dynamical systems

Let X be an infinite compact metric space with finite covering dimension and let α,β : X → X be two minimal homeomorphisms. We prove that the crossed productC∗-algebras C(X) ⋊αℤ and C(X) ⋊βℤ are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if X is an infinite compact metric space and if α : X → X is a minimal homeomorphism such that (X,α) has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This implies that the crossed product is in a classifiable class of amenable simple C∗-algebras.