Embedding topological fractals in universal spaces

  • Taras Banakh

    Ivan Franko National University, Lviv, Ukraine
  • Filip Strobin

    Lodz University of Technology, Poland

Abstract

A compact metric space is called a Rakotch (Banach) fractal if\linebreak for some finite system of Rakotch (Banach) contracting self-maps of . A Hausdorff topological space is called a topological fractal if for some finite system of continuous self-maps, which is topologically contracting in the sense that for any sequence the intersection 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 . Also we prove that each topological fractal is homemorphic to the attractor of a topologically contracting function system on an arbitrary Tychonoff space , which contains a topological copy of the Hilbert cube. If the space is metrizable, then its topology can be generated by a bounded metric making all maps 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