## The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

### Volume 198 / 2008

#### Abstract

Two invertible dynamical systems $(X, {\mathfrak F} {A}, \mu, T)$ and $(Y, {\mathfrak F} {B}, \nu, S)$, where $X$ and $Y$ are Polish spaces and Borel probability spaces and $T$, $S$ are measure preserving homeomorphisms of $X$ and $Y$, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping $\phi$ from a subset $X_0$ of $X$ of measure one onto a subset $Y_0$ of $Y$ of full measure such that

(1) $\phi|_{X_0}$ is continuous in the relative topology on $X_0$ and $\phi^{-1}|_{Y_0}$ is continuous in the relative topology on $Y_0$,

(2) $\phi({\rm Orb}_T(x))={\rm Orb}_S(\phi(x))$ for $\mu$-a.e. $x\in X$.

$(X, {\mathfrak F} {A}, \mu, T)$ and $(Y, {\mathfrak F} {B}, \nu, S)$ are said to be finitarily evenly Kakutani equivalent if they are finitarily orbit equivalent by a mapping $\phi$ for which there are measurable subsets $A$ of $X$ and $B=\phi(A)$ of $Y$ with $\phi$ an isomorphism of $T_A$ and $T_B$.

It is shown here that the Morse minimal system and the binary odometer are finitarily evenly Kakutani equivalent.