On the topology of the limit sets of non-autonomous iterated function systems

Abstract

Since Mandelbrot's seminal work, there has been growing interest in the geometric nature of fractals. 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, specially in the non-autonomous setting. In this paper, we investigate the topology of limit sets arising from randomly generated non-autonomous IFSs. To this end, we develop a simplicial-homological framework that makes their topological structure accessible to rigorous analysis. We apply our abstract theory to the concrete analysis of the so-called fractal squares, and provide an answer to a variant of Mandelbrot's percolation problem. Moreover, for the non-autonomous fractal squares considered here, we prove that the Betti numbers of the finite-stage approximations grow exponentially at a rate equal to the natural symbolic entropy of the system. This reveals a quantitative link between topology across scales and dynamical complexity.