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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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##### References:
 [1] H.P. Barendregt, The Lambda Calculus, Its Syntax and Semantics, North-Holland, Amsterdam, 1981. · Zbl 0467.03010 [2] M. Hata, On the Structure of Self-similar Sets, to appear in Japan J. Appl. Math.. [ 3 ] M. Hata, On Some Properties of Set-dynamical Systems, to appear in Proc. Japan Acad.. · Zbl 0608.28003 [3] J.E. Hutchinson, Fractals and Self-similarity, Indiana Univ. Math. /., 30, pp. 713-747, 1981. · Zbl 0598.28011 [4] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. · Zbl 0504.28001 [5] E. Michael, Topologies on Spaces of Subsets, Trans. Amer. Math. Soc., 71, pp. 152-182, 1951. · Zbl 0043.37902 [6] D. S. Scott, Continuous Lattices, in Toposes, Algebraic Geometry and Logic, F. W. Lawvere (ed.), LNM 274, Springer Verlag, Berlin, 1972. · Zbl 0239.54006 [7] M.B. Smyth, G.D. Plotkin, The Category-theoretic Solution of Recursive Domain Equations, Internal Report of Department of Computer Science CSR-102-82, Uni- versity of Edinburgh, Feb. 1982. [8] J. Soto-Andrade, F. J. Varela, Self-Reference and Fixed Points: A Discussion and an Extension of Lawvere’s Theorem, Ada Applicandae Mathematicae, 2, pp. 1-19, 1984. · Zbl 0538.03052 [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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