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

Yuto Nakajima, Takayuki Watanabe

TL;DR

The paper develops a Čech-Sumi homology framework for limit sets of non-autonomous IFS, establishing an isomorphism between the inverse-limit (co)homology of a nested nerve system and the Čech (co)homology of the limit set $J$. It introduces a subcomplex-based long exact sequence to enable recursive computation of homology groups and proves a probabilistic total-disconnectedness result for random non-autonomous IFSs starting from a post-critically finite base. The authors apply the theory to non-autonomous fractal squares and their higher-dimensional generalizations, deriving precise connectivity criteria, percolation-type phenomena, and detailed 2D homology outcomes, including explicit ranks in various regimes. This work links fractal topology with random processes and dynamical systems, offering new tools to quantify topological complexity via homology and illuminating how overlap-driven structure shapes global connectivity. Overall, it extends fractal topology to non-autonomous and random settings, providing computationally tractable invariants and deepening connections between topology, dynamics, and fractal geometry.

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.

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

TL;DR

The paper develops a Čech-Sumi homology framework for limit sets of non-autonomous IFS, establishing an isomorphism between the inverse-limit (co)homology of a nested nerve system and the Čech (co)homology of the limit set . It introduces a subcomplex-based long exact sequence to enable recursive computation of homology groups and proves a probabilistic total-disconnectedness result for random non-autonomous IFSs starting from a post-critically finite base. The authors apply the theory to non-autonomous fractal squares and their higher-dimensional generalizations, deriving precise connectivity criteria, percolation-type phenomena, and detailed 2D homology outcomes, including explicit ranks in various regimes. This work links fractal topology with random processes and dynamical systems, offering new tools to quantify topological complexity via homology and illuminating how overlap-driven structure shapes global connectivity. Overall, it extends fractal topology to non-autonomous and random settings, providing computationally tractable invariants and deepening connections between topology, dynamics, and fractal geometry.

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.
Paper Structure (18 sections, 37 theorems, 61 equations, 3 figures)

This paper contains 18 sections, 37 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction and the main theorems
  2. Background
  3. Main theorems
  4. Related work from various fields
  5. Fractal Dimensions
  6. Mandelbrot fractal percolation
  7. Dynamical Systems
  8. Topology of wild spaces
  9. Organization of the paper
  10. Preliminary remarks on random IFSs
  11. Definitions of the simplicial and homological framework
  12. Connectedness, total disconnectedness, and local connectedness
  13. The subcomplex and the exact sequence
  14. Non-autonomous fractal squares and their generalization
  15. Connectedness
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\{f_i\}_{i \in I}$ be an autonomous IFS which is post-critically finite (see Definition def:pcf). We choose the index sets $I^{(j)} \subset I$ independently and according to a fixed distribution such that the probability of $i \notin I^{(1)}$ is positive for every $i \in I$. Then almost surely,

Figures (3)

  • Figure 1: Samples of non-autonomous fractal squares. These are randomly constructed as in Theorem \ref{['th:randomSqDim2']}, with $r = 1$, $2$, and $3$ from left to right.
  • Figure 2: Samples of Mandelbrot percolation fractals with $p = 6/9$ and $p = 8/9$.
  • Figure 3: An example of fractal square (above); and the nerves and $(1, 2, 3)$-subcomplex (below) where $d =2$, $n_1 = 2$, $n_2 = 2$, and $r=1$. In the bottom-right figure, the dashed lines are elements of $\mathcal{N}_{1, 3}\setminus\mathcal{M}_{1, 2, 3}$ while the solid lines are elements of $\mathcal{M}_{1, 2, 3}$. Observe that each dashed line represents a relative homology class of $H _1(\mathcal{N}_{1, 3}, \mathcal{M}_{1, 2, 3})$.

Theorems & Definitions (87)

  • Definition 1.1
  • Example 1.2
  • Theorem 1: Theorem \ref{['th:A']}
  • Theorem 2: Theorem \ref{['th:cechsumi']} and Remark \ref{['rem:CohomCoeff']}
  • Theorem 3: Corollary \ref{['cor:Hata']}
  • Theorem 4: Corollary \ref{['cor:totDisc']}
  • Theorem 5: Theorem \ref{['th:exact']}
  • Theorem 6: Theorems \ref{['th:randomSq']} and \ref{['th:randomSqDim2']} and Remark \ref{['rem:tosionFree']}
  • Remark 1.3
  • Example 1.4: Fractal percolation, which is not the same as Example \ref{['ex:fracSqSimple']}
  • ...and 77 more