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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.We put a fractal-like structure on that semigroup.In fact we will describe our ideas for two different semigroups.The first is 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 described in order to prove the famous Jacobian Conjecture.