Equilibrium pure states and nonequilibrium chaos

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 th at as time t --> infinity, although the system is usually in some pure stal e locally, either it never settles permanently on a fixed length scale into a single pure state, or it does, but then the pure stale depends on both t he initial spin configuration and the realization of the stochastic dynamic s. But this latter case can occur only if there exists an uncountable numbe r of pure stales (for each coupling realization) with almost every pair hav ing 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 2D F erromagnets. Our results raise the possibility that even if more than one p ure slate exists for an infinite system, time averages do not necessarily d isagree with Boltzmann averages.