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 $f_1\circ\dots\circ f_n(X)$ 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.Comment: 7 page

Counterexamples for IFS-attractors

“…Moreover, in zero-dimensional sets both notions of Banach and topological fractals are equivalent. This was shown in [4].…”

Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

“…e.g., Reference [22]). Let us note that our terminology differs from the one in e.g., References [58][59][60] where the notion of a topological fractal is understood in a different way.…”

“…Remark 13. The IFS-attractors B, C in Theorem 5 can be called topological fractals in the sense of References [58][59][60] and, jointly with a metric fractal A in the sense of References [29,53], they can be called more precisely Banach fractals in the sense of References [58,59]. Remark 14.…”

Peano continua with self regenerating fractals

“…Topological fractals have been studied in various contexts, eg. among countable spaces [6] or zero-dimensional spaces [2]. Now we are interested in topological fractals in the class of Peano continua.…”