  This is a sequel to my paper "The Octagonal PET I: Renormalization and
Hyperbolic Symmetry". In this paper we use the renormalization scheme found in
the first paper to classify the limit sets of the systems according to their
topology. The main result is that the limit set is either a finite forest or a
Cantor set, with an explicit description of which cases occur for which
parameters. In one special case, the limit set is a disjoint union of 2 arcs if
and only if the continued fraction expansion of the parameter has the form
[a0:a1:a2:a3...] with a_k even for every odd k.
