Almost continuous orbit equivalence for non-singular homeomorphisms

Abstract

Let $X$ and $Y$ be Polish spaces with non-atomic Borel measures $\mu$ and $\nu$ of full support. Suppose that $T$ and $S$ are ergodic non-singular homeomorphisms of $(X,\mu )$ and $(Y,\nu )$ with continuous Radon-Nikodym derivatives. Suppose that either they are both of type $II{I}_1$ or that they are both of type $II{I}_{\lambda }$, $0<\lambda <1$ and, in the $II{I}_{\lambda }$ case, suppose in addition that both `topological asymptotic ranges' (defined in the article) are $\log \lambda \cdot \mathbb{Z}$. Then there exist invariant dense $G_{\delta }$-subsets $X^{\prime }\subset X$ and $Y^{\prime }\subset Y$ of full measure and a non-singular homeomorphism $\varphi :X^{\prime }\to Y^{\prime }$ which is an orbit equivalence between $T{|}_{X^{\prime }}$ and $S{|}_{Y^{\prime }}$, that is $\varphi \{T^{i}x\}=\{S^{i}x\}$ for all $x\in X^{\prime }$. Moreover the Radon-Nikodym derivative $d\nu \circ \varphi /d\mu$ is continuous on $X^{\prime }$ and, letting $S^{\prime }={\varphi }^{-1}S\varphi$ we have $Tx={S^{\prime }}^{n(x)}x$ and $S^{\prime }=T^{m(x)}x$ where $n$ and $m$ are continuous on $X^{\prime }$.

References in corpus (1)

• Orbit inequivalent actions of non-amenable groups