Outer compositions of hyperbolic/loxodromic linear fractional transformations
The author looks at outer compositions \(F_ n(z)=f_ n\circ\dots\circ f_ 1(z)\) for \(n=1,2, \dots\), where \(\{f_ n\}_{n=1}^ \infty\) is a sequence of linear fractional transformations such that the limit \(\lim_{n\to\infty}f_ n=f\) exists. He treats the case where \(f\) is a linear fractional transformation of hyperbolic or loxodromic type. He proves that then \(F_ n(z)\to \alpha\) for all \(z\in\hat{\mathbf C}\) except possibly one, \(z_ 0\), where \(\alpha\) is the attractive fixed point of \(f\). In the exceptional case he proves that \(F_ n(z_ 0)\) converges to the repulsive fixed point \(\beta\) of \(f\). Examples of such outer compositions are the so--called tail sequences of continued fractions. The result obtained is closely related to previous work in the field of linear fractional transformations.
scientific article; zbMATH DE number 147593