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

Molino's description and foliated homogeneity