Embedding topological fractals in universal spaces

Abstract

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 $X={\bigcup }_{f\in \mathcal{F}}f(X)$ for some finite system $\mathcal{F}$ of continuous self-maps, which is topologically contracting in the sense that for any sequence $(f_n{)}_{n\in \omega }\in {\mathcal{F}}^{\omega }$ the intersection ${\bigcap }_{n\in \omega }{f}_0\circ \cdots \circ f_n(X)$ 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.