  We study locally constant skew-product maps over full shifts of finite
symbols with arbitrary compact metric spaces as fiber spaces. We introduce a
new criterion to determine the density of leaves of the strong unstable (and
strong stable) foliation, that is, for its minimality. When the fiber space is
a circle, we show that both strong foliations are minimal for an open and dense
set of robust transitive skew-products. We provide examples where either one
foliation is minimal or neither is minimal. Our approach involves investigating
the dynamics of the associated iterated function system (IFS). We establish the
asymptotic stability of the phase space of the IFS when it is a strict
attractor of the system. We also show that any transitive IFS consisting of
circle diffeomorphisms that preserve orientation can be approximated by a
robust forward and backward minimal, expanding, and ergodic (with respect to
Lebesgue) IFS. Lastly, we provide examples of smooth robust transitive IFSs
where either the forward or the backward minimal fails, or both.
