Magdalena Nowak: Zero-dimensional spaces as topological and Banach fractals

Tuesday, June 16, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Magdalena Nowak (Jan Kochanowski University in Kielce)

Title: Zero-dimensional spaces as topological and Banach fractals

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.

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