Compositions of contractions

摘要

We prove that if D is a simply connected (open) domain in the complex plane C, E is a closed subset of D, and {fn∞n=1} are functions analytic in D such that fn(D)⊆ E for all n, then the compositions Fn(z) = f1ōf2ō⋯ōfn(z) for n = 1, 2, 3,…, converge uniformly in D to a constant function. Some generalizations are also presented.