Proposition 47.4.13.label Let $G$ be an infinite group, $G \curvearrowright X$ be a minimal $G$-flow, then there exists a minimal idempotent $\fU \in \beta G \setminus G$ such that $\fU x = x$.
Proof. Let $L \subset \beta G$ be a minimal left ideal, then the mapping $\fU \mapsto \fU x$ is a continuous surjection. As $x \in X = \bracsn{\fU x|x \in L}$, there exists $\fU \in L$ with $\fU x = x$. Let $F = \bracsn{\fU \in L|\fU x = x}$, then $F$ is a non-empty closed right topological subsemigroup of $L$. By Ellis’ Theorem, $F$ contains an idempotent.$\square$
Post a Comment