Almost continuous orbit equivalence for non-singular homeomorphisms

Abstract

Let X and Y be Polish spaces with non-atomic Borel measures µ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, µ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III1 or that they are both of type IIIλ, 0 < λ < 1 and, in the IIIλ case, suppose in addition that both ‘topological asymptotic ranges’ (defined in the article) are log λ · ℤ. Then there exist invariant dense Gδ-subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a non-singular homeomorphism ϕ: X′ → Y′ which is an orbit equivalence between T|X′ and S|Y′, that is ϕ{Tix} = {Siϕx} for all x ∈ X′. Moreover, the Radon-Nikodym derivative dν ∘ ϕ/dµ is continuous on X′ and, letting S′ = ϕ−1Sϕ, we have Tx = S′n(x)x and S′x = Tm(x)x where n and m are continuous on X′.