We consider a minimal compact metric flow $(X,T)$, where $T$ is a group, and a minimal idempotent $u^2=u\in E(X)$, where $E(X)$ is the Ellis semigroup (or enveloping semigroup) of the flow $(X,T)$. Using these, we define a subset of $X$ by $$ X_u:=\{x\in X: ux=x\}, $$ i.e. the set of all points in $X$ which are fixed by $u$.

Does $X_u$ contain any proximal points? (We say $x$ and $y$ are proximal points if $\inf_{t\in T} d(tx,ty)=0$.) My intuition would be 'no', since we have that a point is distal (i.e. proximal only to itself) if and only if all idempotents $u\in E(X)$ fix $x$, i.e. $ux=x \ \forall u^2=u\in E(X)$, and the condition used to define $X_u$ is related (though not the same). Any thoughts/suggestions/references?