is also a $G$-flow, and minimal subflows of $E(X)$ are exactly minimal left ideals with respect to the semigroup structure on $E(X)$, in particular they are cosed and so compact. The following was proved by Ellis (e.g. see [@El Propositions 3.5, 3.6] and [@Gl Chapter 1, Propositions 2.3, 2.5]).

\[Ellis theorem\] Let ${{{\mathcal M}}}$ be a minimal left ideal in $E(X)$, and let $J({{{\mathcal M}}})$ be the set of all idempotents in ${{{\mathcal M}}}$. Then:\
i) For any $p \in {{{\mathcal M}}}$, $E(X)p={{{\mathcal M}}}p={{{\mathcal M}}}$.\
ii) ${{{\mathcal M}}}$ is the disjoint union of sets $u{{{\mathcal M}}}$ with $u$ ranging over $J({{{\mathcal M}}})$.\
iii) For each $u \in J({{{\mathcal M}}})$, $u{{{\mathcal M}}}$ is a group with the neutral element $u$, where the group operation is the restriction of the semigroup operation on $E(X)$.\
iv) All the groups $u{{{\mathcal M}}}$ (for $u \in J({{{\mathcal M}}})$) are isomorphic, even when we vary the minimal ideal ${{{\mathcal M}}}$.

For a given group $G$ we say that a $G$-ambit $(G,X,x_0)$ is [*universal*]{} if for every $G$-ambit $(G,Y,y_0)$ there exists a (unique) homomorphism $h \colon X \to Y$ of $G$-flows mapping $x_0$ to $y_0$. The following fact is fundamental [@Gl Chapter 1, Proposition 2.6].

\[universality of 