Detecting topological and Banach fractals among zero-dimensional spaces

Abstract

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 continuous self-maps of $X$, which is topologically contracting in the sense that for every open cover $\mathcal{U}$ of $X$ there is a number $n\in \mathbb{N}$ such that for any functions $f_1,\dots ,f_n\in \mathcal{F}$, the set $f_1\circ \cdots \circ f_n(X)$ is contained in some set $U\in \mathcal{U}$. If, in addition, all functions $f\in \mathcal{F}$ have Lipschitz constant $<1$ with respect to some metric generating the topology of $X$, then the space $X$ is called a Banach fractal. It is known that each topological fractal is compact and 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 $\hslash (X)$ is a successor ordinal. For countable compact spaces this classification was recently proved by M.Nowak.

7 pages

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