The forward (in blue) and backward (in red) limit sets of a semidiscrete semigroup. The intersection of the forward and backward limit sets is shown in purple, and in this case coincides with the limit set of a certain Kleinian group.

The forward (in green) and backward (in purple) limit sets of two semidiscrete semigroups. In each case the limit sets intersect on a finite set, each point of which is fixed by some parabolic element in the semigroup.