  We hope to be able (in the future) to carefully analyze this structure and to
tie the Jacobian Conjecture in dimension two to certain Zeta functions, thereby
invoking a powerful arithmetic machinery to handle the two dimensional Jacobian
Conjecture. Let us denote by ${\rm et}(\mathbb{C}^2)$ the semigroup of two
dimensional Keller mappings. We would like to prove something like the
following: a) That there exists an infinite index set $I$, and a family of
mappings indexed by $I$, $\{F_i\,|\,i\in I\} \subset {\rm et}(\mathbb{C}^2)$
such that $$ {\rm et}(\mathbb{C}^2)={\rm Aut}(\mathbb{C}^2)\cup\bigcup_{i\in I}
R_{F_i}({\rm et}(\mathbb{C}^2)), $$ where if $i\ne j$ then $R_{F_i}({\rm
et}(\mathbb{C}^2))\cap R_{F_j}({\rm et}(\mathbb{C}^2))=\emptyset$. b) That the
parallel representation to the representation described in (a) above holds
true, this time with respect to the left composition operators $L_{G_j}$.
  These two claims will be the basis for a fractal structure on ${\rm
et}(\mathbb{C}^2)$ because the pieces $R_{F_i}({\rm et}(\mathbb{C}^2))$ are
similar to each other in the sense that they are homeomorphic, and we further
have the scaling property of self-similarity, namely $R_F({\rm
et}(\mathbb{C}^2))$ is homeomorphic to its proper subspace $R_{G\circ F}({\rm
et}(\mathbb{C}^2))$ that is homeomorphic to its proper subspace $R_{H\circ
G\circ F}({\rm et}(\mathbb{C}^2))$ etc . This is the right place to remark that
the purpose of the current paper is to start and develop the parallel theory
for entire functions in one complex variable. Results in this setting will hint
that there are hopes to accomplish the above objective.
