On the topology of the limit set of non-autonomous IFS

TL;DR

This paper introduces a homological framework to analyze the topological structure of limit sets in non-autonomous iterated function systems, providing new insights into fractal topology and addressing a variant of Mandelbrot's percolation problem.

Contribution

It develops a novel homological approach for studying non-autonomous IFS limit sets and applies it to fractal squares, advancing understanding of fractal topology.

Findings

Homological framework effectively captures fractal limit set structure

Applied theory to fractal square, revealing new topological insights

Provided solutions to a variant of Mandelbrot's percolation problem

Abstract

Fractals are ubiquitous in nature, and since Mandelbrot's seminal insight into their structure, there has been growing interest in them. While the topological properties of the limit sets of IFSs have been studied -- notably in the pioneering work of Hata -- many aspects remain poorly understood, especially in the non-autonomous setting. In this paper, we present a homological framework which captures the structure of the limit set. We apply our novel abstract theory to the concrete analysis of the so-called fractal square, and provide an answer to a variant of Mandelbrot's percolation problem. This work offers new insights into the topology of fractals.