The Morse minimal system is nearly continuously Kakutani equivalent to the binary odometer

Ergodic homeomorphisms $T$ and $S$ of Polish probability spaces $X$ and $Y$ are evenly Kakutani equivalent if there is an orbit equivalence $\phi: X_0 \rightarrow Y_0$ between full measure subsets of $X$ and $Y$ such that, for some $A \subset X_0$ of positive measure, $\phi$ restricts to a measurable isomorphism of the induced systems $T_A$ and $S_{\phi(A)}$. The study of even Kakutani equivalence dates back to the seventies, and it is well known that any two zero-entropy loosely Bernoulli systems are evenly Kakutani equivalent. But even Kakutani equivalence is a purely measurable relation, while systems such as the Morse minimal system are both measurable and topological. Recently del Junco, Rudolph and Weiss studied a new relation called nearly continuous Kakutani equivalence. A nearly continuous Kakutani equivalence is an even Kakutani equivalence where also $X_0$ and $Y_0$ are invariant $G_\delta$ sets, $A$ is within measure zero of both open and closed, and $\phi$ is a homeomorphism from $X_0$ to $Y_0$. It is known that nearly continuous Kakutani equivalence is strictly stronger than even Kakutani equivalence, and nearly continuous Kakutani equivalence is the natural strengthening of even Kakutani equivalence to the nearly continuous category---the category where maps are continuous after sets of measure zero are removed. In this paper we show that the Morse minimal substitution system is nearly continuously Kakutani equivalent to the binary odometer.