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The main purpose of the Fatou-Julia theory is to study the global behaviour of the sequence (fn) of iterates of a rational function f. In this survey article we consider generalized iteration which means that the iterated function f may vary from step to step. More precisely, let (fn) be a sequence of rational functions, and let \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$ F_{n}: = f_{n}\circ \cdots \circ f_{1}$\end{document} be the sequence of forward compositions, and let the Fatou set and Julia set of (Fn) be defined as usual. Then, in general, most of the results of the Fatou-Julia theory fail to hold. On the other hand, under appropriate restrictions on the sequence (fn) many results can be carried over to this more general situation, but the proofs are often completely different.We also consider compositions of holomorphic self-maps fn of the unit disk. In this case there is no need to deal with Fatou and Julia sets, and the main interest lies in the dynamics of (Fn). It also makes sense to consider the sequence of backward compositions \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Phi_{n}:=f_{1}\circ \cdots \circ f_{n}$\end{document}, because such sequences arise, for example, in continued fraction expansions.
Computational Methods and Function Theory – Springer Journals
Published: Mar 1, 2004
Keywords: Iteration; random iteration; forward composition; backward composition; Fatou set; Julia set; dynamics; 30D05; 37F10; 30D45
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