Embedding topological fractals in universal spaces

A compact metric space X is called a Rakotch (Banach) fractal if X = boolean OR(f is an element of) (F) f(X) for some F nite system F of Rakotch (Banach) contracting self-maps of X. A Hausdorff topological space X is called a topological fractal if X = boolean OR(f is an element of) (F) f(X) for some finite system F of continuous self-maps, which is topologically contracting in the sense that for any sequence (f(n))n is an element of omega is an element of F-omega the intersection boolean AND(n is an element of omega)f(0) circle...circle 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. Also we prove that each topological fractal is homemorphic to the attractor A(F) of a topologically contracting function system F on an arbitrary Tychono F 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 is an element of F Rakotch contracting.