  We study the transition to turbulence in the asymptotic suction boundary
layer (ASBL) by direct numerical simulation. Tracking the motion of
trajectories intermediate between laminar and turbulent states we can identify
the invariant object inside the laminar-turbulent boundary, the edge state. In
small domains, the flow behaves like a travelling wave over short time
intervals. On longer times one notes that the energy shows strong bursts at
regular time intervals. During the bursts the streak structure is lost, but it
reforms, translated in the spanwise direction by half the domain size. Varying
the suction velocity allows to embed the flow into a family of flows that
interpolate between plane Couette flow and the ASBL. Near the plane Couette
limit, the edge state is a travelling wave. Increasing the suction, the
travelling wave and a symmetry-related copy of it undergo a saddle-node
infinite-period (SNIPER) bifurcation that leads to bursting and
discrete-symmetry shifts. In wider domains, the structures localize in the
spanwise direction, and the flow in the active region is similar to the one in
small domains. There are still periodic bursts at which the flow structures are
shifted, but the shift-distance is no longer connected to a discrete symmetry
of the flow geometry. Two different states are found by edge tracking
techniques, one where structures are shifted to the same side at every burst
and one where they are alternatingly shifted to the left and to the right.
