On the Structure of the Semigroup of Entire Étale Mappings

The motivation for this paper comes from new ideas for solving the two-dimensional Jacobian Conjecture. The Jacobian Conjecture is one of the most famous open problems in algebraic geometry. This long-standing conjecture is no doubt one of the central problems in this well developed field of mathematics and hence the importance of investigating it. We can consider a semigroup of local diffeomorphisms on the affine space with a composition of mappings as its binary operation. We put a geometric fractal-like structure on this semigroup after equipping it with a natural metric (this is heavily dependent on the fact that our mappings are local diffeomorphisms). This structure is much more general than the structure of the ind-variety suggested by Kambayashi for étale polynomial mappings in the algebraic context. Hence, it applies to other semigroups such as the semigroup of all the entire functions in one complex variable with a nonvanishing first order derivative. This last semigroup is the theme of the current paper. We hope that the corresponding Hausdorff measure and Hausdorff dimension will enable us to relate the structure of the semigroup with arithmetic machinery such as certain Zeta functions.