Abstract
Let $X$ be an infinite compact metric space with finite covering dimension. Let $\af,\bt: X\to X$ be two minimal homeomorphisms. Suppose that the range of $K_0$-groups of both crossed products are dense in the space of real affine continuous functions on the tracial state space. We show that $\af$ and $\bt$ are approximately conjugate uniformly in measure if and only if they have affine homeomorphic invariant probability measure spaces.
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