We consider analytic maps $f_j:D\to D$ of a domain D into itself and ask when does the sequence $f_1\circ\dotsb\circ f_n$ 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 $\{w_1,w_2,\dotsc\}$ in D whose values are not taken by any fj 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.