Molino's description and foliated homogeneity

The topological Molino's description of equicontinuous foliated spaces, studied by the first author and Moreira Galicia, gives conditions to reduce their study to the particular case where the holonomy pseudogroup can be represented by a pseudogroup on some local group $G$ generated by some of its local left translations (a $G$-foliated space). That description is sharpened in this paper by introducing a foliated action of a compact topological group on the resulting $G$-foliated space, like in the case of Riemannian foliations. Moreover a $C^\infty$ version is also studied. The triviality of this compact group characterizes compact minimal $G$-foliated spaces, which are also characterized by their foliated homogeneity in the $C^\infty$ case. We also give an example where the projection of the Molino's description is not a principal bundle, and another example of positive topological codimension where the foliated homogeneity cannot be checked by only comparing pairs of leaves---in the case of zero topological codimension, weak solenoids with this property were given by Fokkink and Oversteegen, and later by Dyer, Hurder and Lukina.