Crossed products and minimal dynamical systems

Let $X$ be an infinite compact metric space with finite covering dimension and let $\alpha ,\beta :X\to X$ be two minimal homeomorphisms. We prove that the crossed product $C^{\ast }$-algebras $C(X){⋊}_{\alpha }\mathbb{Z}$ and $C(X){⋊}_{\text{belta}}\mathbb{Z}$ 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 $\alpha :X\to X$ is a minimal homeomorphism such that $(X,\alpha )$ 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 $C^{\ast }$-algebras.