Table of Contents
Algebra
Volume 2014 (2014), Article ID 782973, 11 pages
http://dx.doi.org/10.1155/2014/782973
Research Article

Injectivity of the Composition Operators of Étale Mappings

Department of Mathematics, Ben Gurion University of the Negev, 84105 Beer Sheva, Israel

Received 1 May 2014; Revised 10 November 2014; Accepted 19 November 2014; Published 10 December 2014

Academic Editor: Dae San Kim

Copyright © 2014 Ronen Peretz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a topological space. The semigroup of all the étale mappings of (the local homeomorphisms ) is denoted by . If , then the -right (left) composition operator on is defined by   , . When are the composition operators injective? The Problem originated in a new approach to study étale polynomial mappings and in particular the two-dimensional Jacobian conjecture. This approach constructs a fractal structure on the semigroup of the (normalized) Keller mappings and outlines a new method of a possible attack on this open problem (in preparation). The construction uses the left composition operator and the injectivity problem is essential. In this paper we will completely solve the injectivity problems of the two composition operators for (normalized) Keller mappings. We will also solve the much easier surjectivity problem of these composition operators.