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The Tarski-Kantorovitch principle and the theory of iterated function systems. (English) Zbl 0952.54029

An iterated function system (IFS) is a set \(X\) together with self maps \(f_1,f_2,\dots, f_n\) of \(X\) and the Hutchinson-Barnsley operator \(F(A)= \bigcup \{f_i(A)\mid i=1, 2,\dots, n\}\) for \(A \subseteq X\). A set \(A_0 \subseteq X\) is invariant with respect to the IFS provided \(F(A_0)= A_0\). The authors apply the Tarski-Kantorovitch fixed point theorem [J. Dugundji and A. Granas, Fixed point theory, Vol. I, Monogr. Mat., Warszawa, Tom 61 (1982; Zbl 0483.47038)] for maps on partially ordered sets to iterated function systems on the following families partially ordered by \(\supseteq\): (1) the family \(2^X\) of all subsets of a set \(X\); (2) the family \(C(X)\) of all non-empty closed subsets of a Hausdorff topological space \(X\); (3) the family \(K(X)\) of all non-empty compact subsets of a Hausdorff topological space \(X\).
The following result is typical:
Let \(X\) be a compact Hausdorff space, let \(f_1, f_2,\dots, f_n\) be continuous self-maps of \(X\) and let \(F\) be the associated Hutchinson-Barnsley operator. Then \(A_0= \bigcap \{F^n(X)\mid n=1,2, 3\dots\}\) is a non-empty compact invariant set, and, in fact, \(A_0\) is the greatest invariant set with respect to the IFS \(\{f_1, f_2,\dots, f_n\}\).
The authors also obtain some new characterizations of continuity of maps on countably compact and sequential spaces.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
06A06 Partial orders, general
54C05 Continuous maps
54D55 Sequential spaces
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)

Citations:

Zbl 0483.47038
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References:

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