every $x \in X$ the map $f_x\colon G \to X$ defined by $f_x(g)=gx$ is \[externally\] definable. An [*\[externally\] definable $G$-ambit*]{} is an \[externally\] definable $G$-flow $(G,X,x_0)$ with a distinguished point $x_0$ such that the orbit $Gx_0$ is dense in $X$.

Now, $G$ acts by translations as groups of homeomorphisms of the compact spaces $S_{G}(M)$, $S_{G,\operatorname{{ext}}}(M)$ and $\beta G$, turning them into $G$-flows. As mentioned before, $(G,\beta G, e)$ is the universal $G$-ambit. Similarly, $(G,S_{G,\operatorname{{ext}}}(M),\operatorname{{tp}}_{\operatorname{{ext}}}(e/M))$ is the universal externally definable $G$-ambit. In particular, by the universality, there is a left-continuous semigroup operation on $S_{G,\operatorname{{ext}}}(M)$ turning it into a semigroup isomorphic to the Ellis semigroup $E(S_{G,\operatorname{{ext}}}(M))$ (for the definitions of Ellis \[semi\]groups see Section \[section: preliminaries\]). So one can consider both the minimal ideals and the Ellis group inside $S_{G,\operatorname{{ext}}}(M)$ instead of in $E(S_{G,\operatorname{{ext}}}(M))$. However, the ambit $(G,S_G(M),\operatorname{{tp}}(e/M))$ is not necessarily definable (it is so if all types in $S_G(M)$ are definable in which case $S_G(M)=S_{G,\operatorname{{ext}}}(M)$), and we do not have a natural semigroup operation on $S_G(M)$. This makes $S_{G,\operatorname{{ext}}}(M)$ and the category of externally definable $G$-flows easier to work with, and that is why topological dynamics has been developed in this context. On the other hand, $S_G(M)$ and definable flows are 