Inner composition of analytic mappings on the unit disk

John Gill

International Journal of Mathematics and Mathematical Sciences, 1991, vol. 14, 1-6

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, 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.

Date: 1991
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:509269

DOI: 10.1155/S0161171291000236

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