  Motivated by short-range Ising spin glasses, we review some rigorous results
and their consequences for the relation between the number/nature of
equilibrium pure states and nonequilibrium dynamics. Two of the consequences
for spin glass dynamics following a deep quench to a temperature with broken
spin flip symmetry are: (1) Almost all initial configurations lie on the
boundary between the basins of attraction of multiple pure states. (2) Unless
there are uncountably many pure states with almost all pairs having zero
overlap, there can be no equilibration to a pure state as time goes to
infinity. We discuss the relevance of these results to the difficulty of
equilibration of spin glasses. We also review some results concerning the
``nature vs. nurture'' problem of whether the large-time behavior of both
ferromagnets and spin glasses following a deep quench is determined more by the
initial configuration or by the dynamics realization.
