JournalsjfgVol. 2 , No. 4pp. 377–388

Embedding topological fractals in universal spaces

  • Taras Banakh

    Ivan Franko National University, Lviv, Ukraine
  • Filip Strobin

    Lodz University of Technology, Poland
Embedding topological fractals in universal spaces cover
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Abstract

A compact metric space XX is called a Rakotch (Banach) fractal if\linebreak X=fFf(X)X=\bigcup_{f\in\mathcal F}f(X) for some finite system F\mathcal F of Rakotch (Banach) contracting self-maps of XX. A Hausdorff topological space XX is called a topological fractal if X=f\Ff(X)X=\bigcup_{f\in\F}f(X) for some finite system F\mathcal F of continuous self-maps, which is topologically contracting in the sense that for any sequence (fn)n\w\F\w(f_n)_{n\in\w}\in\F^\w the intersection n\wf0fn(X)\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 U\mathbb U. Also we prove that each topological fractal is homemorphic to the attractor AFA_\mathcal F of a topologically contracting function system F\mathcal F on an arbitrary Tychonoff space UU, which contains a topological copy of the Hilbert cube. If the space UU is metrizable, then its topology can be generated by a bounded metric making all maps fFf\in\mathcal F Rakotch contracting.

Cite this article

Taras Banakh, Filip Strobin, Embedding topological fractals in universal spaces. J. Fractal Geom. 2 (2015), no. 4 pp. 377–388

DOI 10.4171/JFG/25