A convergence question inspired by Stieltjes and by value sets in continued fraction theory
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 analysis.
scientific article; zbMATH DE number 903577
scientific article