We consider the scenario emerging from the dynamics of a generalized $d$-brane in a $(d+1, 1)$ spacetime. The equation of state describing this system is given in terms of the energy density, $ρ$, and pressure, $p$, by the relationship $p = - A/ρ^α$, where $A$ is a positive constant and $0 < α\le 1$. We discuss the conditions under which homogeneity arises and show that this equation of state describes the evolution of a universe evolving from a phase dominated by non-relativistic matter to a phase dominated by a cosmological constant via an intermediate period where the effective equation of state is given by $p = αρ$.

5 pages, 4 figures, revtex