Random iteration of analytic maps

We consider analytic maps Fj: D → D of a domain D into itself and ask when does the sequence f1 ο⋯ο fn converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence {w1, w2,...} in D whose values are not taken by any f j in D, and which is homogeneous in the sense that it comes within a fixed hyperbolic distance of any point of D. The situation for several complex variables is also discussed.published_or_final_versio