On the asymptotic convergence of sequences of analytic functions

摘要

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