Inner composition of analytic mappings on the unit disk

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, f n ( z ) → α as n → ∞ . In [3] and [5] the author generalized this result in the following way: Let F n ( z ) : = f 1 ∘ … ∘ f n ( z ) . Then f n → f uniformly on D implies F n ( 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 f 1 ∘ … ∘ f n ( z ) where the f j 's are analytic on Int ( D ) and continuous on D with Int ( D ) f j ( D ) , but essentially random. Applications include analytic functions defined by this process.