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? 



Proximality in a special subset defined by an idempotent
The answer is no. According to <http://ac.els-cdn.com/S0166864107001538/1-s2.0-S0166864107001538-main.pdf?_tid=281576e6-9489-11e6-bb98-00000aab0f01&acdnat=1476722928_757fc4c08fcbc966d5afdf7e0c9e0477> for $x,y$ to be proximal there must be $s$ in the enveloping semigroup with $sx=sy$. Then since $u$ is minimal, $su$ and $u$ generate the same left ideal. So $tsu=u$ for some $t$ in the semigroup. Then $x=ux=tsux=tsx=tsy=tsuy=uy=y$. 


