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arXiv · 2510.23255

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

Yuto Nakajima·Takayuki Watanabe
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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.

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Yuto Nakajima, Takayuki Watanabe. 2025-10-27. On the topology of the limit sets of non-autonomous iterated function systems. https://arxiv.org/abs/2510.23255

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