Compositions of linear-fractional transformations (Q1274102)

Let \(S_n\) be a linear fractional transformation with fixed points \(p_{1,n}\) and \(p_{2,n}\) for each \(n\in\mathbb N\). The authors prove the following result for compositions of \(\{S_n\}\): Theorem: If \(\limsup_{n\to\infty}| S'_n(p_{2,n})| <1\), \( \lim_{n\to\infty} p_{2,n}=p_2\) and all \(p_{1,n}\) are contained in a closed set \(K \subseteq \overline{\mathbb C}\setminus\{p_2\}\), then \[ S_n\circ S_{n-1}\circ\dots\circ S_1(z)\quad\text{for} n=1,2,3,\dots \] converges to \(p_2\) for all \(z\in\overline{\mathbb C}\), except for some point \(z_0\). Since \((S_n\circ S_{n-1}\circ\dots\circ S_1)^{-1}=S_1^{-1} \circ S_2^{-1}\circ\dots\circ S_n^{-1}\), where also \(S_n^{-1}\) has fixed points \(p_{1,n}\) and \(p_{2,n}\), but \(| (S_n^{-1})'(p_{1,n})| =| S_n'(p_{2,n})| \) and \(| (S_n^{-1})'(p_{2,n})| =| S_n'(p_{1,n})| \), they also get: Corollary: If \(\liminf_{n\to\infty}| S_n'(p_{1,n})| >1\), \(\;\lim_{n\to\infty}p_{1,n} =p_1\) and all \(p_{2,n}\) are contained in a closed set \(K\subseteq \overline{\mathbb C}\setminus\{p_1\}\), then \[ S_1\circ S_2\circ\dots\circ S_n(z)\quad\text{for} n=1,2,3,\dots \] converges in \(\overline{\mathbb C}\setminus\{p_1\}\) to a constant.