Irrational Rotation of the Circle and the Binary Odometer are Finitarily Orbit Equivalent

Abstract

Two invertible dynamical systems $(X,\mathfrak{A},\mu ,T)$ and $(Y,\mathfrak{B},\nu ,S)$, where $X,Y$ are metrizable spaces and $T,S$ are homeomorphisms on $X$ and $Y$, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping $\varphi$ from a subset $M$ of $X$ of measure one to a subset of $Y$ of full measure such that

(i) $\varphi {|}_M$ is continuous in the relative topology on $M$ and ${\varphi }^{-1}{|}_{\varphi (M)}$ is continuous in the relative topology on $\varphi (M)$,

(ii) $\varphi ({\mathrm{Orb}}_T(x))={\mathrm{Orb}}_S\varphi (x)$ for $\mu$-a.e. $x\in X$.

In this article a finitary orbit equivalence mapping has been constructed between an irrational rotation of the circle and the binary odometer.