Continued fractions and restrained sequences of Möbius maps
The convergence of continued fractions (CF) \[ \text{K}(a_n| b_n)={a_1\over b_1+}\,{a_2\over b_2+}\,\ldots,\quad a_n\not=0\;(n\geq 1) \tag{1} \] has been studied extensively using iterations of Möbius transforms \[ T_n=t_1\circ \cdots \circ t_n,\quad t_n(z)={a_n\over b_n+z}.\tag{2} \] Let \({\mathcal C}_{\infty}={\mathcal C}\cup\{\infty\}\), and introduce the chordal metric \[ \sigma(u,v)={2| u-v| \over \sqrt{1+| u| ^2}\sqrt{1+| v| ^2}}. \] Important concepts are then the following (1) The CF (1) converges strongly to \(\alpha\in\mathcal{C}\) if (a) There are two distinct points \(u,v\in{\mathcal C}\) with \[ \lim_{n\rightarrow\infty}\,T_n(u)=\lim_{n\rightarrow\infty}\,T_n(v)=\alpha. \] (b) The set \(\{S_n^{-1}(\infty):\;n=1,2,\ldots\} \) is not dense in \(\mathcal C\). (2) The sequence of transforms (2) is restrained if there exist two sequences \((u_n),(v_n)\in{\mathcal C}_{\infty}\) with \[ \lim_{n\rightarrow\infty}\,\sigma(T_n(u_n),T_n(v_n))=0\quad\text{and}\quad \liminf_{n\rightarrow\infty}\,\sigma(u_n,v_n)>0. \] (3) Two sequences \((u_n),(v_n)\) are separated if \[ \liminf_{n\rightarrow\infty}\,\sigma(u_n,v_n)>0. \] (4) Two sequences \((u_n),(v_n)\) are asymptotic if \[ \lim_{n\rightarrow\infty}\,\sigma(u_n,v_n)=0. \] (5) A sequence \((u_n)\in{\mathcal C}_{\infty}\) is \(T_n\)-admissable if there exists a sequence \((v_n)\) such that the two sequences \((u_n),(v_n)\) are separated and the sequences \((T_n(u_n)),(T_n(v_n))\) are asymptotic. (6) A sequence \((z_n)\) is strongly exceptional if there exists an admissable sequence \((u_n)\) such that \((T_n(z_n))\) and \((T_n(u_n))\) are separated. The authors then give equivalent characterizations of restrained, admissable and strongly exceptional sequences using chordal disks, chordal distortion, chordal derivatives, Euclidean geometry and hyperbolic geometry. This is an extremely well-written paper which can be seen as a decisive publication. The authors use their pen as the surgeon its scalpel: each line unravels more about the intricacies of convergence properties of iterated Möbius transforms.
scientific article; zbMATH DE number 2104645
scientific article