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The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems

Huo-Jun Ruan, Jian-Ci Xiao

TL;DR

The paper addresses the connectedness of sponge-like Cantor-type graph-directed attractors in low- and high-dimensional spaces when rotations and reflections are allowed. It reduces the problem to a coordinate-wise intersection test using auxiliary graphs $G_P$ for faces of the unit cube and a global intersection graph with solid and dashed edges, showing that $K_i ∩ K_j$ is nonempty iff there exists an infinite solid walk or a terminated finite walk starting from $(i,j)$. A finite-iteration criterion is established with a computable depth $c(n,d)$ so that $K_i ∩ K_j ≠ ∅$ is detected by level-$c(n,d)$ approximations $Q_{i,c(n,d)}$, enabling practical connectivity checks via Hata graphs of symmetry orbits. The authors further relate sponge connectivity to Hata-graphs of the symmetry family $(O(F))_{O∈ ext{O}_d}$, and provide improvements on iteration bounds (e.g., $F_{d+1}$ suffices when all rotations are identity) with extensions to broader graph-directed systems. Overall, the work furnishes actionable, finite criteria for testing connectedness of complex sponge-like fractals with rotational/reflectional components and generalizes to broader graph-directed frameworks."

Abstract

We provide two methods to characterize the connectedness of all $d$-dimensional generalized Sierpiński sponges whose corresponding IFSs are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.

The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems

TL;DR

The paper addresses the connectedness of sponge-like Cantor-type graph-directed attractors in low- and high-dimensional spaces when rotations and reflections are allowed. It reduces the problem to a coordinate-wise intersection test using auxiliary graphs for faces of the unit cube and a global intersection graph with solid and dashed edges, showing that is nonempty iff there exists an infinite solid walk or a terminated finite walk starting from . A finite-iteration criterion is established with a computable depth so that is detected by level- approximations , enabling practical connectivity checks via Hata graphs of symmetry orbits. The authors further relate sponge connectivity to Hata-graphs of the symmetry family , and provide improvements on iteration bounds (e.g., suffices when all rotations are identity) with extensions to broader graph-directed systems. Overall, the work furnishes actionable, finite criteria for testing connectedness of complex sponge-like fractals with rotational/reflectional components and generalizes to broader graph-directed frameworks."

Abstract

We provide two methods to characterize the connectedness of all -dimensional generalized Sierpiński sponges whose corresponding IFSs are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let be a Cantor-type graph-directed attractor in . By creating an auxiliary graph, we provide an effective criterion for whether is empty for every pair of . Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.
Paper Structure (10 sections, 30 theorems, 82 equations, 4 figures)

This paper contains 10 sections, 30 theorems, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The intersection problem I: Creating auxiliary graphs
  3. Examination of faces of the unit cube
  4. The intersection problem
  5. An induction process to determine terminated edges
  6. The intersection problem II: The finite-iteration approach
  7. Connectedness of sponge-like sets
  8. Hata graphs and the proof of Theorem \ref{['thm:main3']}
  9. Improvements on the constant $C(d)$
  10. Further remarks

Key Result

Theorem 1.1

Let $\Phi=\{\varphi_i\}_{i=1}^n$ be an IFS on $\mathbb R^d$ and let $K$ be its attractor. Then $K$ is connected if and only if the associated Hata graph is connected, where the Hata graph is defined as follows.

Figures (4)

  • Figure 1: Three sponge-like sets in $\mathbb R^2$
  • Figure 2: Directed graph in Example \ref{['exa:1']}
  • Figure 3: Auxiliary graphs in Example \ref{['exa:1']}
  • Figure 4: Intersection graph in Example \ref{['exa:1']}

Theorems & Definitions (66)

  • Theorem 1.1: Hata Hat85
  • Example 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Example 2.1
  • Lemma 2.2
  • ...and 56 more