On the structure of the semigroup of entire \'etale mappings

Abstract

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.