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 $III_1$ or that they are both of type $III_λ$, $0<λ<1$ and, in the $III_λ$ case, suppose in addition that both `topological asymptotic ranges' (defined in the article) are $\logλ\cdot\Bbb Z$. Then there exist invariant dense $G_δ$-subsets $X'\subset X$ and $Y'\subset Y$ of full measure and a non-singular homeomorphism $ϕ: X' \to Y'$ which is an orbit equivalence between $T|_{X'}$ and $S|_{Y'}$, that is $ϕ\{T^{i}x\} = \{S^{i}x\}$ for all $x \in X'$. Moreover the Radon-Nikodym derivative $dν\circϕ/dμ$ is continuous on $X'$ and, letting $S' = ϕ^{-1}S ϕ$ we have $Tx= {S'}^{n(x)}x$ and $S' = T^{m(x)}x$ where $n$ and $m$ are continuous on $X'$.