Generalized iteration
Let \((f_n)\) be a sequence of rational functions and let \(F_n:=f_n \circ f_{n-1}\circ \dots\circ f_1\). The study of the limiting behavior of the sequence \((F_n)\) is a generalization of iteration. This article is a survey of results in this area that have been obtained by the authors and others. As in the case of iteration the Fatou set is the set where the sequence \((F_n)\) is normal, and its complement is the Julia set. After introducing the basic properties of Fatou and Julia sets, the authors investigate these sets under various conditions on the sequence \((f_n)\). For example, they consider the case where the \(f_n\) are chosen at random from a certain set of functions and the case where the sequence \((f_n)\) converges. Next they consider the case where the \(f_n\) are polynomials satisfying certain conditions which ensure that there exists \(R>0\) such that \(| f_n(z)| >R\) for \(| z| \geq R\) and all \(n\). Special attention is given to the case where the \(f_n\) are of the form \(f_n(z)=z^2+c_n\). Finally the case of entire and meromorphic \(f_n\) is considered. Instead of the functions \(F_n=f_n \circ \dots \circ f_1\) one may also consider the functions \(\Phi_n=f_1\circ \dots\circ f_n\). These functions are considered in particular in the final section on infinite exponentials, but also e.g. in the section on random iteration. The article surveys many results that have been obtained about generalized iteration and it serves as a very useful introduction to this subject.
scientific article; zbMATH DE number 2112936
scientific article