A convergence question inspired by Stieltjes and by value sets in continued fraction theory
type: Item
instance of: scholarly article
title: A convergence question inspired by Stieltjes and by value sets in continued fraction theory
zbMATH Open document ID: 0870.30021
DOI: 10.1016/0377-0427(95)00113-1
published in: Journal of Computational and Applied Mathematics
publication date: 1997-09-07T00:00:00Z
review text: Let \(V\subset\mathbb{C}\) and \(\{f_n\}\) a sequence of functions analytic in \(V\) with \(f_n(V)\subset V\). The general subject of this survey article is to discuss conditions under which the sequence \(F_n(w):= f_1\circ f_2\circ\cdots\circ f_n(w)\) converges (locally uniformly) to a constant function in \(V\). A rather special case of this question is the subject of the Denjoy-Wolff theorem, where \(V\) is the unit disk, and all \(f_n\) are equal. Already this classical theorem indicates, that really interesting questions arise if the \(f_n\) are linear fractions, which immediately, in the present context, leads to the study of continued fractions as a major case, both for developing a feeling of what can happen, as well as for concrete applications. This is discussed in greater detail in Sections 2, 3, where, starting out from the fundamental findings of Stieltjes, the famous parabola theorem of W. J. Thron, and a number of more recent results and generalizations (of L. J. Lange, A. Córdova, among others) are presented and their interaction is demonstrated. In Section 4 the case of more general \(f_n\) is studied, and the results of I. N. Baker and P. J. Rippon, and A. Beardon are put in perspective, also with their potential applicability to special cases. In Section 5 the general case of linear fractions \(f_n\) is considered, where extensions of the early and fundamental theorem of K. L. Hillam and W. J. Thron, obtained by the author and others, are given. Section 6 deals with the case of the special continued fractions \(K(a_n/1)\), while in Section 7 some new ideas and approaches for further research are discussed. This survey article is beautifully written by one of the foremost experts in the subject. It is a nice addition to (parts of) the book [the author and \textit{H. Waadeland}, Continued fractions and applications (1992; Zbl 0782.40001)], and excellent reading for everyone who wants to become more acquainted with this fascinating area of complex an…
reviewed by: Stephan Ruscheweyh
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30B70
zbMATH DE Number: 903577
zbMATH Keywords: continued fractions
author: Lisa Lorentzen
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00113-1
OpenAlex ID: W2071656798
cites work: Towers of exponents and other composite maps
cites work: On compositions of analytic self-mappings of a convex domain
cites work: Iteration of Contractions and Analytic Maps
cites work: Q4341343
cites work: Q4285796
cites work: Compositions of analytic functions of the form \(F_ n(z)=F_{n-1}(f_ n(z))\), \(f_ n(z)\to f(z)\)
cites work: Inner composition of analytic mappings on the unit disk
cites work: Outer compositions of hyperbolic/loxodromic linear fractional transformations
cites work: Q4295465
cites work: Truncation error estimates for Stieltjes fractions
cites work: A General Convergence Criterion for Continued Fractions K(a n /b n )
cites work: Q3724619
cites work: Q3961723
cites work: General Convergence of Continued Fractions
cites work: Q3724880
cites work: Q3724878
cites work: Twin-Convergence Regions for Continued Fractions K(a n /1)
cites work: Q4295468
cites work: On a family of twin convergence regions for continued fractions
cites work: Q4295469
cites work: A two-parameter family of best twin convergence regions for continued fractions
cites work: Continued fractions with complex elements
cites work: On the Transformation and Convergence of Continued Fractions
cites work: Compositions of contractions
cites work: Bestness of the parabola theorem for continued fractions
cites work: Properties of limit sets and convergence of continued fractions
cites work: Q4295471
cites work: Continued compositions of linear fractional transformations
cites work: Simple convergence sets for continued fractions \(K(a_ n/1)\)
cites work: Continued fractions with applications
cites work: The continued fraction as a sequence of linear transformations
cites work: On sequences of Moebius transformations
cites work: Boundedness of Value Regions and Convergence of Continued Fractions
cites work: Reflections on value regions, limit regions and truncation errors for continued fractions
cites work: Infinite Radicals in the Complex Plane
cites work: A Convergence Theorem for Continued Fractions
cites work: Value regions for continued fractions
cites work: A Family of Best Twin Convergence Regions for Continued Fractions
cites work: On parabolic convergence regions for continued fractions
cites work: Two families of twin convergence regions for continued fractions
cites work: Convergence of Infinite Exponentials with Complex Elements
cites work: Zwillingskonvergenzgebiete für Kettenbrüche \(1+K (a_n/1)\), deren eines die Kreisscheibe \(| a_{2n-1} | \leqq \varrho^2\) ist
cites work: Convergence Regions for Continued Fractions and Other Infinite Processes
cites work: Q3797644
cites work: Where do all the values go? Playing with two-element continued fractions
description: scientific article; zbMATH DE number 903577