Continued compositions of linear fractional transformations (Q1916812)

Let \(V\) be an open subset of \(\widehat{\mathbb{C}}\) and \(\varepsilon> 0\). \(M(\varepsilon, V)\) denotes the class of Moebius transformations \(t(w)\) which map \(V\) into itself, omitting some circular disc with centre \(c_t\in \overline V\) and chordal radius \(\varepsilon\). Theorem 1. If \(t_n\in M(\varepsilon, V)\) for \(n\in \mathbb{N}\) and if \(\{t_n\}\) has a subsequence which converges in some open neighbourhood \(V_1\) of \(\overline V\) to a function \(t\) such that \(f(\overline V)\subseteq V\), then \(T_n= t_1\circ t_2\circ\cdots \circ t_n\) converges uniformly (in the chordal metric) in \(\overline V\) to a constant function. A corollary gives that if \(K\) is compact in \(V\) and if \(t_n\in M(\varepsilon, V)\), where for some subsequence \(n_k\to \infty\) and \(t_{n_k}(V)\subset K\) then \(T_n\) converges uniformly in \(V\) to a constant function. Some more complicated situations are considered and also some `symmetric' results involving conditions on \(t^{- 1}_n\).