International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 2, Pages 221-226
doi:10.1155/S0161171291000236

Inner composition of analytic mappings on the unit disk

Abstract

A basic theorem of iteration theory (Henrici [6]) states that f analytic on the interior of the closed unit disk D and continuous on D with Int(D)f(D) carries any point z ϵ D to the unique fixed point α ϵ D of f. That is to say, fn(z)→α as n→∞. In [3] and [5] the author generalized this result in the following way: Let Fn(z):=f1∘…∘fn(z). Then fn→f uniformly on D implies Fn(z)λ, a constant, for all z ϵ D. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structures f1∘…∘fn(z) where the fj's are analytic on Int(D) and continuous on D with Int(D)fj(D), but essentially random. Applications include analytic functions defined by this process.