Two ergodic probability measure-preserving systems in ergodic theory, $T$ of $(X,\mu)$ and $S$ of $(Y,\nu)$, are said to be disjoint if the only joining (i.e. $T\times S$-invariant measure on $X\times Y$ factoring onto $\mu$ and $\nu$ under the coordinate maps) is the product joining.
I am interested in cases where $X$ and $Y$ are obtained by recursive cutting-and-stacking type constructions.
For concreteness, here is a particular problem. Let $U_0$ be the word consisting of a single 0, and define recursively $U_{n+1}=U_nU_n1U_n$. Then let $X$ be the set of sequences where each subword is a subword of some $U_n$. (The dynamical system $X$ (equipped with the shift map) is uniquely ergodic. This gives the Chacon system.)
We define $Y$ similarly. For example set $V_n$ to be the word consisting of a single 0, and define $V_{n+1}=V_nV_n0V_n1V_n1V_n$ and let $Y$ be the set of sequences whose subwords belong to one of the $V_n$'s. The dynamical system $Y$ is also uniquely ergodic.
Are $(X,\mu)$ and $(Y,\nu)$ disjoint?
