An E_0-semigroup $\alpha = \{\alpha_t: t\geq 0\}$ acting on
$\Cal B(H)$ is called {\it pure} if its tail von Neumann
algebra is trivial in the sense that
$$
\cap_t\alpha_t(\Cal B(H)) = \Bbb C\bold 1.
$$
We determine all pure E_0-semigroups
which have a {\it weakly continuous}
invariant state $\omega$ and which are minimal in an appropriate
sense. In such cases the dynamics of the state space must
stabilize as follows: for every normal state
$\rho$ of $\Cal B(H)$ there is convergence to equilibrium
in the trace norm
$$
\lim_{t\to\infty}\|\rho\circ\alpha_t-\omega\|=0.
$$
A normal state $\omega$ with this property is
called an {\it absorbing} state for $\alpha$.
Such \esg s must be cocycle perturbations of
$CAR/CCR$ flows, and we develop systematic
methods for constructing those perturbations
which have absorbing states with prescribed
finite eigenvalue lists.