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Extremal sets as fractals. (English) Zbl 1161.54301
Despite the title fractals do not feature in this work whose purpose is to find conditions under which a map \(F:\mathcal K (X)\to\mathcal K(X)\) from the family of compact sets in a Hausdorff space to itself has invariant sets.
If there is an \(A\in\mathcal K(X)\) with \(F(A)\subseteq A\) and \(F\) is monotone then \(A\) contains a minimal (via Zorn’s Lemma) and a maximal set with the same property (the first transfinite iterate \(F^\gamma(A)\) with \(F^\gamma(A)=F^{\gamma+1}(A))\). This is applied to (compact-valued) multifunctions.

54C60 Set-valued maps in general topology
28A80 Fractals
37B99 Topological dynamics
54B20 Hyperspaces in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)