Injectivity of the Composition Operators of étale Mappings

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. 1. Introduction Let be a topological space. A mapping is called a local homeomorphism of or an étale mapping of if for any point there exists a neighborhood of such that the restriction of to , denoted by , is an homeomorphism. The set of all the étale mappings of , denoted by , is a semigroup with a unit with the composition of mappings taken to be the binary operation. If , then the -right composition operator on is defined by The -left composition operator on is defined by We were interested in the injectivity of these two composition operators in two particular cases. The first is the case of entire functions that are étale (and normalized). The second case is that of the polynomial mappings with determinant of their Jacobian matrix equal (identically) to and whose -degrees equal their total degrees. For the first case we use the following. Definition 1. Consider the following Thus we use in this case the symbol instead of . Then we have the following. Proposition 2 (see [1]). Consider the following , is injective. Theorem 3 (see [1]). Let . Then is not injective if and only if This settled the first case. It should be noted (see [1]) that the proof for the left composition operator is much more involved than the proof for the right composition operator (which follows directly from the Picard little theorem). It is in fact the second case that initiated our interest in the injectivity of the composition operators. It results from a new approach to study étale polynomial mappings and in particular the two-dimensional Jacobian conjecture [2–4]. 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