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Crossed products and minimal dynamical systems

Let $X$ be an infinite compact metric space with finite covering dimension and let $α, β: X\to X$ be two minimal homeomorphisms. We prove that the crossed product $C^*$-algebras $C(X)\rtimes_α\Z$ and $C(X)\rtimes_\belta\Z$ 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 $α: X\to X$ 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.