Detecting topological and Banach fractals among zero-dimensional spaces

Abstract

A topological space X is called a topological fractal if X=⋃f∈Ff(X) for a finite system F of continuous self-maps of X, which is topologically contracting in the sense that for every open cover U of X there is a number n∈N such that for any functions f1,…,fn∈F, the set f1∘…∘fn(X) is contained in some set U∈U. If, in addition, all functions f∈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 ħ(X) is a successor ordinal. For countable compact spaces this classification was recently proved by M. Nowak.