- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:11
Journal of Fractal Geometry
J. Fractal Geom.
JFG
2308-1309
2308-1317
Measure and integration
General
10.4171/JFG
http://www.ems-ph.org/doi/10.4171/JFG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
2
2015
4
Embedding topological fractals in universal spaces
Taras
Banakh
Ivan Franko National University, LVIV, UKRAINE
Filip
Strobin
Lodz University of Technology, LODZ, POLAND
Topological fractal, Rakotch fractal, Banach fractal, Rakotch contraction, Banach contraction, topologically contracting function system, universal Urysohn space
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.
Measure and integration
Dynamical systems and ergodic theory
General topology
377
388
10.4171/JFG/25
http://www.ems-ph.org/doi/10.4171/JFG/25