Journal of Fractal Geometry

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Volume 2, Issue 4, 2015, pp. 377–388
DOI: 10.4171/JFG/25

Embedding topological fractals in universal spaces

Taras Banakh[1] and Filip Strobin[2]

(1) Faculty of Mechanics and Mathematics Lviv, Ivan Franko National University, ul. Universytets'ka 1, 79000, Lviv, Ukraine
(2) Institute of Mathematics, Lodz University of Technology, Wólczańska 215, 93-005, Lodz, Poland

A compact metric space $X$ is called a Rakotch (Banach) fractal if\linebreak $X=\bigcup_{f\in\mathcal F}f(X)$ for some finite system $\mathcal F$ of Rakotch (Banach) contracting self-maps of $X$. A Hausdorff topological space $X$ is called a topological fractal if $X=\bigcup_{f\in\F}f(X)$ for some finite system $\mathcal F$ of continuous self-maps, which is topologically contracting in the sense that for any sequence $(f_n)_{n\in\w}\in\F^\w$ the intersection $\bigcap_{n\in\w}f_0\circ\dots\circ f_n(X)$ is a singleton. It is known that each topological fractal is homeomorphic to a Rakotch fractal. We prove that each Rakotch (Banach) fractal is isometric to the attractor of a Rakotch (Banach) contracting function system on the universal Urysohn space $\mathbb U$. Also we prove that each topological fractal is homemorphic to the attractor $A_\mathcal F$ of a topologically contracting function system $\mathcal F$ on an arbitrary Tychonoff space $U$, which contains a topological copy of the Hilbert cube. If the space $U$ is metrizable, then its topology can be generated by a bounded metric making all maps $f\in\mathcal F$ Rakotch contracting.

Keywords: Topological fractal, Rakotch fractal, Banach fractal, Rakotch contraction, Banach contraction, topologically contracting function system, universal Urysohn space

Banakh Taras, Strobin Filip: Embedding topological fractals in universal spaces. J. Fractal Geom. 2 (2015), 377-388. doi: 10.4171/JFG/25