Almost Continuous Orbit Equivalence for Non-singular Homeomorphisms

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{T x} = {Sx} 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′x and S = Tm(x)x where n and m are continuous on X.