International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 2, Pages 221-226
Inner composition of analytic mappings on the unit disk
Department of Mathematics, University of Southern Colorado, Pueblo 81001-4901, CO, USA
Received 20 June 1989; Revised 29 June 1989
Copyright © 1991 John Gill. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A basic theorem of iteration theory (Henrici ) states that analytic on
the interior of the closed unit disk and continuous on with carries
any point to the unique fixed point of . That is to say, as
. In  and  the author generalized this result in the following way:
Let . Then uniformly on implies , a
constant, for all . 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 where the 's are
analytic on and continuous on with , but essentially random.
Applications include analytic functions defined by this process.