Iteration of functions which are meromorphic outside a small set

Abstract

In this paper, we investigate the dynamics of a broader class of functions which are meromorphic outside a compact totally disconnected set. We shall establish the connections between the Fatou components and the singularities of the inverse function and, accordingly, give sufficient conditions for the non-existence of wandering domains or Baker domains, and for the Julia set to be the Riemann sphere. Through the discussion of permutability of such functions, we shall construct several transcendental meromorphic functions which have Baker domains and wandering domains with special properties; for example, wandering and Baker domains with a critical value on the boundary and a wandering domain with the boundary being a Jordan curve (some such examples for entire functions were exhibited in other papers) and those of non-finite type which have no wandering domains.