June 2015

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

Tuesday, June 16, 2015 17:15

Room: D1-215

Speaker:
Magdalena Nowak

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.