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On the structure of self-similar sets. (English) Zbl 0608.28003

Let \(\{f_ 1,...,f_ r\}\) be a family of contractions of a separable, complete metric space. A set K is called self-similar if it satisfies \(K=f_ 1(K)\cup...\cup f_ r(K).\) It is known that this equation has a unique non-empty compact solution. In this paper this result is extended to a slightly larger class of maps, and a systematic treatment of the topological structure of K is given. Considered are for example connectedness and ”treelikeness” of K, parametrization of K as a curve, functional equations satisfied by the curve, and the spacefilling property of the curve. Very useful are the many examples and the extensive list of references.
Reviewer: F.M.Dekking

MSC:

28A75 Length, area, volume, other geometric measure theory
26A30 Singular functions, Cantor functions, functions with other special properties
54F99 Special properties of topological spaces
58C30 Fixed-point theorems on manifolds
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