Iterated function systems on hyperbolic Riemann surfaces

A classical result known as the Denjoy–Wolff theorem describes the dynamics of the iterates of a holomorphic self-map of the unit disc, and this theorem was generalised by Heins to all hyperbolic Riemann surfaces. Here we consider results of this type for left and right iterated function systems of holomorphic maps. We explore the geometric differences between these two types of systems and, using hyperbolic derivatives and topological arguments, we recover and improve on several recent results in complex dynamics. This is joint work with Marco Abate.