Injectivity of the composition operators of \'etale mappings

TL;DR

This paper proves that both left and right composition operators on the semigroup of étale polynomial mappings of c^2 are injective, providing a new approach to the two-dimensional Jacobian Conjecture by constructing a fractal structure.

Contribution

It establishes the injectivity of composition operators on the semigroup of étale polynomial mappings, a key step for a novel fractal-based approach to the Jacobian Conjecture.

Findings

Both left and right composition operators are injective on the semigroup of c9tale polynomial mappings.

Contrast with the non-injectivity of the left operator in the semigroup of entire functions.

Injectivity enables the construction of a fractal structure to address the Jacobian Conjecture.

Abstract

We consider the semigroup of \'etale polynomial mappings $\mathbb{C}^2\rightarrow\mathbb{C}^2$ where the binary operation is composition. We prove that both the right and the left composition operators on this semigroup are injective. This is in contrast to the situation in the semigroup of the entire functions $\mathbb{C}\rightarrow\mathbb{C}$ which are locally injective, where the left composition operator is not injective. Our interest in the injectivity (of the left composition operator) results from a new approach to deal with the two dimensional Jacobian Conjecture. In this approach we construct a fractal like structure on the above (first) semigroup in order to use it to settle the conjecture (in preparation). Injectivity is crucial for that construction.