This paper is concerned with a generalisation of the classical theory of the dynamics associated to the iteration of a rational mapping of the Riemann sphere, to the more general setting of the dynamics associated to an arbitrary semigroup of rational mappings. We are partly motivated by results of Gehring and Martin which show that certain parameter spaces for Kleinian groups are essentially the stable basins of infinity for certain polynomial semigroups. Here we discuss the structure of the Fatou and Julia sets and their basic properties. We investigate to what extent Sullivan's 'no wandering domains' theorem remains valid. We obtain a complete generalisation of the classical results concerning classification of basins and their associated dynamics under an algebraic hypothesis analogous to the group-theoretical notion of 'virtually abelian'. We show that, in general, polynomial semigroups can have wandering domains. We put forward some conjectures regarding what we believe might be true. We also prove a theorem about the existence of filled in Julia sets for certain polynomial semigroups with specific applications to the theory of Kleinian groups in mind.
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