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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.

##### MSC:
 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)