Embedding topological fractals in universal spaces

TL;DR

This paper demonstrates that any topological fractal can be embedded into a universal space as an attractor of a system of Rakotch contractions, highlighting the universality of such spaces for fractal structures.

Contribution

It proves that all topological fractals can be realized as attractors within a universal space using Rakotch contractions, extending the understanding of fractal embeddings.

Findings

Topological fractals can be embedded in universal spaces.

Every topological fractal is homeomorphic to an attractor of Rakotch contractions.

Universal spaces can host a wide class of fractal structures.

Abstract

Let $X$ be a universal (Urysohn) space. We prove that every topological fractal is homeomorphic (isometric) to the attractor $A_{\mathcal F}$ of a function system ${\mathcal F}$ on $X$ consisting of Rakotch contractions.