Maps conjugating holomorphic maps in C^n

Abstract

If ψ is a bijection from C n onto a complex manifold M, which conjugates every holomorphic map in C n to an endomorphism in M, then we prove that ψ is necessarily biholomorphic or antibiholomorphic. This extends a result of A. Hinkkanen to higher dimensions. As a corollary, we prove that if there is an epimorphism from the semigroup of all holomorphic endomorphisms of C n to the semigroup of holomorphic endomorphisms of a complex manifold M consisting of more than one point, or an epimorphism in the opposite direction for a doubly-transitive M, then it is given by conjugation by some biholomorphic or antibiholomorphic map. We show also that there are two unbounded domains in C n with isomorphic endomorphism semigroups but which are neither biholomorphically nor antibiholomorphically equivalent.