  This paper is a continuation of our 2005 paper on complex topology and its
implication on invertibility (or non-invertibility). In this paper, we will try
to classify the complexity of inversion into 3 different classes. We will use
synthetic models based on well control to illustrate the different classes. The
first class is systems which have a group of symmetry. This class has clean
inversion. We will use two examples which include 2-term AVO on 1-D layers as
an example. The second class does not have a group support. It is in general
described by a semigroup which is a set of operators with or without inverses.
There is no guarantee in general of invertibility in a global sense. Even
though this class does not have invertibility in general, there could still be
local weak invertibility embedded in the semigroup. The last class is system
with complex topology where the underlying topology/geometry requires infinite
construction. In our previous 2005 paper, we gave the 1-D Cantor layers as the
forerunner of all complex topological interface. We will re-examine this last
class in detail. The idea of constructing the Cantor layers as inverse limit
was introduced in the previous paper. We will examine the finite approximation
of the Cantor layers and its implication on inversion.
