Detecting topological and Banach fractals among zero-dimensional spaces

TL;DR

This paper characterizes zero-dimensional compact metrizable spaces as topological fractals and Banach fractals, establishing conditions based on countability and scattered height, thus unifying their classification.

Contribution

It provides a complete characterization of zero-dimensional compact metrizable spaces as topological and Banach fractals, linking fractal properties to countability and scattered height.

Findings

A zero-dimensional compact metrizable space is a topological fractal if and only if it is a Banach fractal.

Such spaces are either uncountable or countable with scattered height as a successor ordinal.

The classification for countable compact spaces was recently established by M. Nowak.

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 $f_1\circ\dots\circ f_n(X)$ 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 $\hbar(X)$ is a successor ordinal. For countable compact spaces this classification was recently proved by M.Nowak.