On the asymptotic convergence of sequences of analytic functions

We consider sequences {fn} of analytic self mappings of a domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega\subset{\mathbb{C}}$$\end{document} and the associated sequence {Θn} of inner compositions given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta_n = f_1 \circ f_2 \circ \cdots\circ f_n, n = 1, 2, \cdots$$\end{document}. The case of interest in this paper concerns sequences {fn} that converge assymptotically to a function f, in the sense that for any sequence of integers {nk} with n1 < n2 < ... one has that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm lim}_{k\rightarrow\infty}}(f_{n_k}\circ f_{n_{k}+1}\circ\cdots\circ f_{n_{k+1}-1}-f^{n_{k+1}-n_k})=0$$\end{document} locally uniformly in Ω. Most of the discussion concerns the case where the asymptotic limit f is the identity function in Ω.