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A topological space $X$ is called a topological fractal if$X=\bigcup_{f\in\mathcal F}f(X)$ for a finite system $\mathcal F$ of continuousself-maps of $X$, which is topologically contracting in the sense that forevery open cover $\mathcal U$ of $X$ there is a number $n\in\mathbb N$ suchthat for any functions $f_1,\dots,f_n\in \mathcal F$, the set$f_1\circ\dots\circ f_n(X)$ is contained in some set $U\in\mathcal U$. If, inaddition, all functions $f\in\mathcal F$ have Lipschitz constant $<1$ withrespect to some metric generating the topology of $X$, then the space $X$ iscalled a Banach fractal. It is known that each topological fractal is compactand metrizable. We prove that a zero-dimensional compact metrizable space $X$is a topological fractal if and only if $X$ is a Banach fractal if and only if$X$ is either uncountable or $X$ is countable and its scattered height$\hbar(X)$ is a successor ordinal. For countable compact spaces thisclassification was recently proved by M.Nowak.

Detecting topological and Banach fractals among zero-dimensional spaces