Abstract
An iterated function system is a sequence of self-maps of a space obtained by composing self-maps chosen from a given set of self-maps. If the set consists of a single map f, the iterated function system reduces to the sequence of iterates of f; so the study of the asymptotic behaviour of an iterated function system is a generalisation of the study of the dynamics
of a single self-map. In this talk we shall survey recent advances, obtained with Ian Short, in the study of iterated function systems generated by holomorphic self-maps of the unit disk of the complex plane or, more generally, of Riemann surfaces; in particular, we shall show how one can use the hyperbolic derivative to control the asymptotic behaviour of such systems. Finally, we shall recall how, thanks to the work of Benini, Evdoridou, Fagella, Ferreira, Rippon, Stallard and others, these results can be applied to the study of the dynamics of transcendental entire maps.