(see [@Gl Chapter 1, Proposition 2.6]), where $\beta G$ is the Stone-$\check{\mbox{C}}$ech compactification of $G$.

Now, we recall some flows which have been investigated in model theory.

Let $G$ be a group $\emptyset$-definable in a first order structure $M$. By $S_G(M)$ we denote the space of complete types over $M$ containing the formula defining $G$; equivalently, this is the space of ultrafilters in the Boolean algebra of all definable (with parameters from $M$) subsets of $G$, equipped with the Stone topology. By $S_{G,\operatorname{{ext}}}(M)$ we denote the space of all externally definable complete types over $M$ containing $G$, that is the space of ultrafilters in the Boolean algebra of all externally definable subsets of $G$ (i.e. subsets which are intersections with $G$ of sets definable in arbitrary elementary extensions of $M$).

Following [@GiPePi] and [@KrPi], we will use the notion of \[externally\] definable $G$-flows. Namely, let $C$ be a compact, Hausdorff space. A map $f\colon G \to C$ is said to be [*\[externally\] definable*]{} if for all disjoint, closed subsets $C_1$ and $C_2$ of $C$ the preimages $f^{-1}[C_1]$ and $f^{-1}[C_2]$ can be separated by an \[externally\] definable subset of $G$. An [*\[externally\] definable $G$-flow*]{} is a $G$-flow $(G,X)$ such that for 