  Parabolic geometric flows are smoothing for short time however, over long
time, singularities are typically unavoidable, can be very nasty and may be
impossible to classify. The idea of [CM6] and here is that, by bringing in the
dynamical properties of the flow, we obtain also smoothing for large time for
generic initial conditions. When combined with [CM1], this shows, in an
important special case, the singularities are the simplest possible.
  The question of the dynamics of a singularity has two parts. One is: What are
the dynamics near a singularity? The second is: What is the long time behavior?
That is, if the flow leaves a neighborhood of a singularity, can it return at a
much later time? The first question was addressed in [CM6] and the second here.
  Combined with [CM1], [CM6], we show that all other closed singularities than
the (round) sphere have a neighborhood where `nearly every' closed hypersurface
leaves under the flow and never returns, even to a dilated, rotated or
translated copy of the singularity. In other words, it wanders off. In
contrast, by Huisken, any closed hypersurface near a sphere remains close to a
dilated or translated copy of the sphere at each time.
