Outer compositions of hyperbolic/loxodromic linear fractional transformations
type: Item
instance of: scholarly article
title: Outer compositions of hyperbolic/loxodromic linear fractional transformations
zbMATH Open document ID: 0768.30002
DOI: 10.1155/S016117129200108X
author: John Gill
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1993-04-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46827
review text: 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.
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 40A15
Mathematics Subject Classification ID: 40A30
zbMATH DE Number: 147593
zbMATH Keywords: convergence
zbMATH Keywords: linear fractional transformations
reviewed by: Lisa Lorentzen
MaRDI profile type: Publication
OpenAlex ID: W2076900016
description: scientific article; zbMATH DE number 147593