On the asymptotic convergence of sequences of analytic functions

Abstract

We consider sequences {f n } of analytic self mappings of a domain \(\Omega\subset{\mathbb{C}}\) and the associated sequence {Θ n } of inner compositions given by \(\Theta_n = f_1 \circ f_2 \circ \cdots\circ f_n, n = 1, 2, \cdots\). 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 {n k } with n 1 < n 2 < ... one has that \({{\rm lim}_{k\rightarrow\infty}}(f_{n_k}\circ f_{n_{k}+1}\circ\cdots\circ f_{n_{k+1}-1}-f^{n_{k+1}-n_k})=0\) locally uniformly in Ω. Most of the discussion concerns the case where the asymptotic limit f is the identity function in Ω.