H. Dye’s Theorem
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Let σ and τ be Borel automorphisms on a standard Borel space (X, Ɓ). We would like to give a necessary and sufficient condition for σ and τ to be orbit equivalent, i.e., for there to exist a Borel isomorphism ϕ: X → X such that for all x, ϕ(orb (x, σ)) = orb (ϕ(x), τ). Let us observe that if σ and τ are orbit equivalent and if σ has an orbit of length n then so has τ and vice versa; moreover the cardinality of the set of orbits of length n for σ and τ is the same. If C k (σ) be the cardinality of the set of orbits of length k, then for each k ≤ ℵ0, C k (σ) = C k (τ) whenever σ and τ are orbit equivalent. We also know from Chapter 1 that if orbits of σ and τ admit Borel cross-sections and if C k (σ) = C k (τ) for all k ≤ ℵ0, then σ and τ are orbit equivalent. Thus the question as to when σ and τ are orbit equivalent needs to be settled only in the case when σ and τ are free and their orbit spaces do not admit Borel cross-sections. We will therefore assume in the rest of this chapter that σ and τ are free and their orbit spaces do not admit Borel cross-sections. The first important result on orbit equivalence was obtained by H. Dye  and the main aim of this chapter is to prove his theorem.
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