Session 9. General forms of self-similarity in algebra and topology

Topological Fractals and Multifractals: their properties and metrization

Taras Banakh, Jan Kochanowski University in Kielce, Poland
The talk is based on joint work with Filii Strobic
In this talk we shall discuss some topological and metric properties of topological fractals and multifractals. A topological space \(X\) is called a {\em topological fractal} if \(X=\bigcup_{f\in\mathcal F} f(X)\) for a finite family \(\mathcal{ F}\) of continuous self-maps of \(X\) such that for every open cover \(\mathcal{U}\) of \(X\) there is a number \(n\) such that for every choice of maps \(f_1,\dots,f_n\in \mathcal{F}\) the set \(f_1\circ\cdots\circ f_n(X)\) is contained in some set \(U\in\mathcal{U}\). We shall prove that each Hausdorff topological fractal is compact and metrizable. Moreover, its topology is generated by a metric \(d\) making all maps \(f\in\mathcal{F}\) Edelstein contractive in the sense that \(d(f(x),f(y))<d(x,y)\) for any distinct points \(x,y\in X\). Topological fractals are partial cases of multifractals. A topological space \(X\) is called a {\em multifractal} if there is a continuous finitely-valued map \(\Phi:X\multimap X\) such that \(X=\lim_{n\to\infty}\Phi^n(x)\) for every point \(x\in X\). The class of multifractals is much wider than the class of topological fractals. For example, each compact Hausdorff space admitting a minimal action of a finitely generated group is a multifractal. This implies that there are compact connected multifractals which are not locally connected, there is a first countable compact multifractal, which is not metrizable, etc.
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