A convergence question inspired by Stieltjes and by value sets in continued fraction theory

Let V be a subset of the complex plane C. Let {fn}n=1∞ be a sequence of self-mappings of V; i.e., fn(V) ⊆ V. The question is then: Under what conditions will the sequence Fn(w):= f1 ∘ f2 … ∘ fn(w); n = 1,2,3, … of composite maps converge to a constant function in V? In this paper we give a survey of some of the answers and open problems connected with this question. Such answers have applications in dynamical systems, Schur analysis, continued fractions and other similar structures like infinite exponentials, infinite radicals.