  We consider nonequilibrium systems such as the Edwards-Anderson Ising spin
glass at a temperature where, in equilibrium, there are presumed to be (two or
many) broken symmetry pure states. Following a deep quench, we argue that as
time goes to infinity, although the system is usually in some pure state
locally, either it never settles permanently on a fixed lengthscale into a
single pure state, or it does but then the pure state depends on both the
initial spin configuration and the realization of the stochastic dynamics. But
this latter case can occur only if there exists an uncountable number of pure
states (for each coupling realization) with almost every pair having zero
overlap. In both cases, almost no initial spin configuration is in the basin of
attraction of a single pure state; that is, the configuration space (resulting
from a deep quench) is all boundary (except for a set of measure zero). We
prove that the former case holds for deeply quenched two-dimensional
ferromagnets. Our results raise the possibility that even if more than one pure
state exists for an infinite system, time averages don't necessarily disagree
with Boltzmann averages.
