Magdalena Nowak: Zero-dimensional spaces as topological and Banach fractals
08/06/15 22:02
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.
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.