Self-Similar Topological Fractals

TL;DR

This paper introduces a new constructive framework called similarity schemes for modeling topological self-similar fractals, generalizing existing notions and providing a method to produce and analyze such fractals in a topological setting.

Contribution

The paper develops the concept of similarity schemes as a constructive approach to topological fractals, extending the Kigami-Kameyama framework and allowing for more general parametrizations.

Findings

Similarity schemes produce self-similar topological fractals.

Many Kigami-Kameyama topological fractals can be constructed via similarity schemes.

The framework generalizes existing models by not requiring the finiteness of the parameter space Y.

Abstract

We introduce the notion of (abelian) similarity scheme, as a constructive model for topological self-similar fractals, in the same way in which the notion of iterated function system furnishes a constructive notion of self-similar fractals in a metric environment. At the same time, our notion gives a constructive approach to the Kigami-Kameyama notion of topological fractals, since a similarity scheme produces a topological fractal a la Kigami-Kameyama, and many Kigami-Kameyama topological fractals may be constructed via similarity schemes. Our scheme consists of objects $X_0\stackrel{\varphi}{\rightarrow}X_1\stackrel{\pi}{\leftarrow} Y\times X_0$, where $X_0,X_1$ and $Y$ are compact Hausdorff spaces, the map $\varphi$ is continuous injective and the map $\pi$ is continuous surjective. This scheme produces a sequence $X_n$, $n\in\mathbb{N}$, of compact Hausdorff spaces, $X_n$ embedded in $X_{n+1}$, and a compact Hausdorff space $X_\infty$ giving a sort of injective limit space, which turns out to be self-similar. We observe that the space $Y$ parametrizes the generalized similarity maps, and finiteness of $Y$ is not required.