Divergent sequences satisfying the linear fractional transformations
type: Item
instance of: scholarly article
title: Divergent sequences satisfying the linear fractional transformations
zbMATH Open document ID: 0774.40002
DOI: 10.1155/S0161171293000353
author: A. McD. Mercer
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1993-11-14T00:00:00Z
full work available at URL: https://eudml.org/doc/46868
review text: The linear fractional transformation \(T(x)={ax+b\over cx+d}\) with \(a,b,c,d\) real, \(ad-bc=1\) and \(c\neq 0\) maps the extended real axis one- one onto itself. If the fixed points of \(T\) are real, every real sequence satisfying \(x_{n+1}=T(x_ n)\) will converge to one of the fixed points. If \(\lambda\) is this fixed point and \(f\in C(-\infty,\infty)\), then \({1\over N}\sum^ N_ 1f(x_ n)\to f(\lambda)\) as \(N\to\infty\). In this paper the corresponding limit is examined when the fixed points of \(T\) are nonreal (and therefore complex conjugates). If the fixed points are \(\alpha\) and \(\overline\alpha\) then the transformation \(y=T(x)\) may be rewritten \({y-\alpha\over y-\overline\alpha}=e^{i\beta x}{x-\alpha\over x-\overline\alpha}\) for some real \(\beta\) in \((0,2\pi)\). It is shown that if \(\beta\) is not a rational multiple of \(\pi\), then for almost all \(x_ 1\) (in the sense of Lebesgue measure), \({1\over N}\sum^ N_ 1f(x_ n)\) converges to \(\int^ \infty_{- \infty}{Kf(x)dx\over cx^ 2+(d-a)x-b}\), where \(K\) is a certain normalizing constant.
Mathematics Subject Classification ID: 40A30
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 179248
zbMATH Keywords: divergent series
zbMATH Keywords: ergodic theory
zbMATH Keywords: Cesàro means
zbMATH Keywords: linear fractional transformation
zbMATH Keywords: Lebesgue measure
reviewed by: Gerald A. Heuer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 179248