Semigroup representations in holomorphic dynamics

We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.