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Self-similar sets as Tarski’s fixed points. (English) Zbl 0618.54030
J. E. Hutchinson [Indiana Univ. Math. J. 30, 713-747 (1981; Zbl 0598.28011)] showed that the unique non-empty compact solution X of the equation \(K=F_ 1(X)\cup...\cup F_ n(X)\), where \(F_ 1,...,F_ n\) are contractions of \({\mathbb{R}}^ N\), can be considered as fixed point of a certain contraction on the complete metric space of compact subsets of \({\mathbb{R}}^ N\). The author considers the same problem from the viewpoint of Tarski’s fixed point theorem for ordered spaces.
Reviewer: J.Appell

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
28A75 Length, area, volume, other geometric measure theory
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[1] H.P. Barendregt, The Lambda Calculus, Its Syntax and Semantics, North-Holland, Amsterdam, 1981. · Zbl 0467.03010
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[3] J.E. Hutchinson, Fractals and Self-similarity, Indiana Univ. Math. /., 30, pp. 713-747, 1981. · Zbl 0598.28011
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[9] R. F. Williams, Composition of contractions, Bol. Soc. Brasil. Math. 2, pp. 55-59, 1971. P. S. Prof. Dana Scott pointed out that our COP was well known to the authors of ”A COMPENDIUM OF CONTINUOUS LATTICES, G. Gierz. et. al., Springer-Verlag, Berlin, 1980”. They consider the open-set lattice of a locally compact space, say 0(X) for a locally compact space X, instead of closed sets (see Chapter I, 1.7, (5) in the Compendium). Our CPO is equi- valent to 0(X)-{X}, and Proposition 1.4 of Chapter I says that 0 ( X ) - { X ] is a CPO. I would like to thank Prof. Dana Scott for this and other valu- able comments on the paper. I also would like to thank Prof. Gordon Plotkin for pointing our a terminological error.}
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