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The Dye Theorem states that any two free ergodic p.m.p automorphisms of a standard probability space are orbit-equivalent.

Question: Is there a version of the above theorem for non-ergodic automorphisms?

Specifically, I'd like a statement roughly as follows. Let $T\colon X\to X$ be a free ergodic pmp automorphism of a standard probability space, and let $S\colon X\to X$ be a free pmp automorphism. Then there exists a standard probability space $Z$ (possibly with atoms) such that $S$ is orbit equivalent to the product automorphism $T\times 1_Z\colon X\times Z \to X\times Z$.

I'd be grateful even for a reference which uses this sort of theorem and claims that it follows by inspecting the original Dye's argument.

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  • $\begingroup$ You'll need to assume that the powers of $S$ also act freely so that the orbits are infinite. But more to the point, I don't remember Dye assuming ergodicity, have you checked his original paper? $\endgroup$
    – Jesse Peterson
    Commented Apr 21, 2015 at 15:26

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