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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