Our primary motivation in this research originates from the two dimensional Jacobian
Conjecture. The proposed program is to assume that the conjecture is false. This implies
the existence of a semigroup of \'etale mappings and we intend to study its structure.
A similar program in the past studied the structure of this semigroup with the structure
of algebraic degree filtration. Here we impose a totally different structure on the
semigroup. It is a fractal-like structure. We will describe our ideas for two different
\'etale semigroups. We will start with the semigroup of the entire local homeomorphisms in
one complex variable. After studying this classical case we will proceed to the case of
the two dimensional \'etale polynomial mappings. The two theories will turn out to be
different but still will share a few basic ingredients. We indicate how one might use the fractal structure we impose on the semigroup of \'etale mappings in order to solve the famous Jacobian Conjecture.