In Section \[section: definable ambit\], we present the universal $\Delta$-definable $G$-ambit as the quotient of $S_{G,\Delta}(M)$ by some closed equivalence relation $E_\Delta$, we describe the semigroup operation on $S_{G,\Delta}(M)/E_\Delta$, and, using it, we give a description of the relation $E_\Delta$.

In Section \[section: connected components\], we define local versions of the connected components ${G^*}^{00}_M$ and ${G^*}^{000}_M$, namely ${G^*}^{00}_{\Delta,M}$ and ${G^*}^{000}_{\Delta,M}$, and we show that $G^*/{G^*}^{00}_{\Delta,M}$ is the $\Delta$-definable Bohr compactification of $G$. The proof follows the lines of the argument from [@GiPePi] showing that $G^*/{G^*}^{00}_{M}$ is the definable Bohr compactification, but it also requires some additional observations (in particular, the fact that ${G^*}^{00}_{\Delta,M}$ is a normal subgroup is not completely obvious, and we need Lemma \[lemma: mu is contained\] in order to show that the quotient map $G \to G^*/{G^*}^{00}_{\Delta,M}$ is $\Delta$-definable). Using results from Section \[section: definable ambit\], we find an explicitly given epimorphism $\theta$ from the universal $\Delta$-definable $G$-ambit $S_{G,\Delta}(M)/E_\Delta$ to $G^*/{G^*}^{00}_{\Delta,M}$, whose restriction to the Ellis group is also an epimorphism. We formulate interesting questions concerning an analogous statement for ${G^*}/{G^*}^{000}_{\Delta,M}$.

In Section \[section: results from KrPi\], we explain that some deeper arguments from [@KrPi] can be adapted to the $\Delta$-definable context, which results in Theorems \[theorem: 