Computational Methods and Function Theory 3 (2003), No. 1, 201--252
Copyright Heldermann Verlag 2003
Rainer Brück
rainer.brueck@math.uni-dortmund.de, Universität Dortmund, Fachbereich Mathematik, 44221 Dortmund, Germany.
Matthias Büger
matthias.bueger@math.uni-giessen.de, Helgebachstraße 3, 35578 Wetzlar, Germany.
The main purpose of the Fatou-Julia theory is to study the global behaviour of the sequence $(f^n)$ 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 $(f_n)$ be a sequence of rational functions, and let $F_n := f_n \circ\dotsb\circ f_1$ be the sequence of forward compositions, and let the Fatou set and Julia set of $(F_n)$ 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 $(f_n)$ 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 $f_n$ 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 $(F_n)$. It also makes sense to consider the sequence of backward compositions $\Phi_n := f_1 \circ\dotsb\circ f_n$, because such sequences arise, for example, in continued fraction expansions.
Keywords: Iteration, random iteration, forward composition, backward composition, Fatou set, Julia set, dynamics.
MSC 2000: 30D05, 37F10, 30D45.
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