Strong orbit realization for minimal homeomorphisms

Let X be a Canter set, S a minimal self-homeomorphism of X, and mu an S-invariant Borel probability. Let T be an ergodic automorphism of a non-atomic Lebesgue probability space (Y, v). Then there is a minimal homeomorphism S' with the same orbits as S such that (S', mu) is measurably conjugate to (T, v). Here S can be chosen strongly orbit equivalent to S if and only if the periodic spectrum of S is contained in the discrete spectrum of T. Corollaries of these results generalize Dye's Theorem and the Jewett-Krieger Theorem.