Accumulation constants of iterated function systems with Bloch target domains

The sequence {Fn} is called the iterated function system coming from the sequence f1, f2, f3, . . .; we abbreviate this to IFS. By Montel’s theorem (see for example [3]), the sequence Fn is a normal family, and every convergent subsequence converges uniformly on compact subsets of ∆ to a holomorphic function F . The limit functions F are called accumulation points. Therefore every accumulation point is either an open self map of ∆ or a constant map. The constant accumulation points may be located either inside X or on its boundary. Note that for the iterated systems we consider here, the compositions are taken in the reverse of the usual order; that is, backwards. There is a theory for forward iterated function systems that is somewhat simpler and is dealt with in [5]. For example, for forward iterated function systems, by using constant functions, it is easy to construct systems with non-unique limits. The first results for (backward) iterated function systems were found by Lorentzen and Gill ([8], [4]) who, independently proved that if X is relatively compact in ∆, the limit functions are always constant and each IFS has a unique limit. In [2] the authors considered iterated function systems for which the target domain is non-relatively compact. Using techniques from hyperbolic geometry, they defined a hyperbolic Bloch condition for the target domain and proved that any X satisfying this condition has only constant limit functions. In [6] we proved that this Bloch condition is also necessary. In [7] we turned to non-Bloch target domains. Using Blaschke products, we proved that any holomorphic map from ∆ to X can be realized as the limit