Injectivity of the Composition Operators of Étale Mappings

Let X be a topological space. The semigroup of all the étale mappings of X (the local homeomorphisms X→X) is denoted by et(X). If G∈et(X), then the G-right (left) composition operator on et(X) is defined by RG LG:et(X)→et(X), RGF=F∘G (LGF=G∘F). When are the composition operators injective? The Problem originated in a new approach to study étale polynomial mappings C2→C2 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