On the asymptotic convergence of sequences of analytic functions

We consider sequences {f(n)} of analytic self mappings of a domain Omega subset of C and the associated sequence {Theta(n)} of inner compositions given by Theta(n) = f(1 o) f(2) o center dot center dot center dot of(n), = 1, 2, center dot center dot center dot. The case of interest in this paper concerns sequences {f(n)} that converge assymptotically to a function f, in the sense that for any sequence of integers ink) with n(1) < n(2) < center dot center dot center dot one has that limk ->infinity(fn(k) o fn(k)+1 (o center dot center dot center dot o) fn(k+1)-1 - f(nk+1-nk)) = 0 locally uniformly in Omega. Most of the discussion concerns the case where the asymptotic limit f is the identity function in Omega.