Basic Ergodic Theory pp 120-132 | Cite as

# H. Dye’s Theorem

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## Abstract

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* ≤ ℵ_{0}, *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* ≤ ℵ_{0}, 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 [2] and the main aim of this chapter is to prove his theorem.

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