Iteration of Möbius transforms and continued fractions
type: Item
instance of: scholarly article
title: Iteration of Möbius transforms and continued fractions
zbMATH Open document ID: 0734.30006
author: Hans Wallin
author: Jan Gelfgren
author: Johan Karlsson
published in: Rocky Mountain Journal of Mathematics
publication date: 1991-01-01T00:00:00Z
review text: Continued fractions are traditionally studied through their convergents \(S_ n(0)=s_ 1\circ s_ 2\circ...\circ s_ n(0)\) where \(s_ k(z)=a_ k/(1+z)\). In particular in connection with convergence acceleration it has become popular to study so called modifications \(S_ n(w_ n)\) with \(w_ n\) not necessarily 0. From the point of view of discrete dynamical systems it seems natural to study \(\tilde S_ n(z)=s_ n\circ...\circ s_ 2\circ s_ 1(z)\) instead, perhaps in particular when all \(s_ k\) are equal so we are studying iteration. In this paper we show that the behaviour of \(S_ n\) can be seen in a study of \(\tilde S_ n\) if \(s_ k\) are constant in k (which is of course trivial \((S_ n=\tilde S_ n))\), \(s_ k\) form a periodic sequence, or \(s_ k\) form a convergent sequence. We also think the paper is useful as a primer of continued fractions.
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 4214309
reviewed by: Johan Karlsson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181073017
OpenAlex ID: W2051944191
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DOI: 10.1216/RMJM/1181073017
description: scientific article; zbMATH DE number 4214309