  Irrotational dust solutions of Einstein's equations are suitable models to
describe the general-relativistic aspects of the gravitational instability
mechanism for the formation of cosmic structures. In this paper we study their
state space by considering the local initial-value problem formulated in the
covariant fluid approach. We consider a wide range of models, from homogeneous
and isotropic to highly inhomogeneous irrotational dust models, showing how
they constitute equilibrium configurations (invariant sets) of the dynamics.
Moreover, we give the characterization of such configurations, which provides
an initial-data characterization of the models under consideration.
