  We consider pure states in the SYK model. These are given by a simple local
condition on the Majorana fermions, evolved over an interval in Euclidean time
to project on to low energy states. We find that "diagonal" correlators are
exactly the same as thermal correlators at leading orders in the large $N$
expansion. We also describe "off diagonal" correlators that decay in time, and
are given simply in terms of thermal correlators. We also solved the model
numerically for low values of $N$ and noticed that subsystems become typically
entangled after an interaction time. In addition, we identified configurations
in two dimensional nearly-$AdS_2$ gravity with similar symmetries. These
gravity configurations correspond to states with regions behind horizons. The
region behind the horizon can be made accessible by modifying the Hamiltonian
of the boundary theory using the the knowledge of the particular microstate.
The set of microstates in the SYK theory with these properties generates the
full Hilbert space.
