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.
From: Takayuki Watanabe [view email]
[v1]
Mon, 27 Oct 2025 12:16:50 UTC (530 KB)
[v2]
Mon, 20 Jul 2026 08:38:00 UTC (538 KB)
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