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Elementary fractal geometry. 5. Weak separation is strong separation

Christoph Bandt, Michael F. Barnsley

TL;DR

This work shows that self-similar sets with the weak separation condition (WSC) can be recast as graph-directed iterated function systems (GIFS) that satisfy the open set condition (OSC) by systematically cutting overlaps via an overlap graph. The authors introduce a practical combinatorial algorithm to transform a finite-overlap IFS into a GIFS with attractors { $B_k$ } obeying GIFS equations, thereby unifying WSC and OSC through a modular, automata-inspired structure. They develop a robust framework for constructing GIFS from overlap graphs, prove OSC for the resulting system, and demonstrate the approach with several 2D examples, including complex Pisot cases and self-replicating tilings. The method yields explicit dimension information from the GIFS (via spectral radii) and produces new tiling constructions beyond classical substitution schemes, with substantial computational validation and discussion of practical issues in larger systems.

Abstract

For self-similar sets, there are two important separation properties: the open set condition and the weak separation condition introduced by Zerner, which may be replaced by the formally stronger finite type property of Ngai and Wang. We show that any finite type self-similar set can be represented as a graph-directed construction obeying the open set condition. The proof is based on a combinatorial algorithm which performed well in computer experiments.

Elementary fractal geometry. 5. Weak separation is strong separation

TL;DR

This work shows that self-similar sets with the weak separation condition (WSC) can be recast as graph-directed iterated function systems (GIFS) that satisfy the open set condition (OSC) by systematically cutting overlaps via an overlap graph. The authors introduce a practical combinatorial algorithm to transform a finite-overlap IFS into a GIFS with attractors { } obeying GIFS equations, thereby unifying WSC and OSC through a modular, automata-inspired structure. They develop a robust framework for constructing GIFS from overlap graphs, prove OSC for the resulting system, and demonstrate the approach with several 2D examples, including complex Pisot cases and self-replicating tilings. The method yields explicit dimension information from the GIFS (via spectral radii) and produces new tiling constructions beyond classical substitution schemes, with substantial computational validation and discussion of practical issues in larger systems.

Abstract

For self-similar sets, there are two important separation properties: the open set condition and the weak separation condition introduced by Zerner, which may be replaced by the formally stronger finite type property of Ngai and Wang. We show that any finite type self-similar set can be represented as a graph-directed construction obeying the open set condition. The proof is based on a combinatorial algorithm which performed well in computer experiments.
Paper Structure (10 sections, 4 theorems, 48 equations, 12 figures, 1 table)

This paper contains 10 sections, 4 theorems, 48 equations, 12 figures, 1 table.

Table of Contents

  1. Overview
  2. The basic idea
  3. The overlap graph
  4. Cut overlaps and neighborhoods
  5. Combinatorial construction of the GIFS
  6. Proof of the open set condition
  7. A small example
  8. Another detailed example
  9. Computational problems
  10. Self-replicating tilings

Key Result

Theorem 1

Let $F$ be an IFS of similitudes on $\mathbb{R}^d,$ each with the same scaling factor, with the finite overlap type property. Let the attractor (self-similar set) be $A.$ Then $F$ can be extended to a GIFS of the form gifs with the OSC and $B_k\subseteq A.$

Figures (12)

  • Figure 1: The example of Ngai and Wang and an associated non-overlapping GIFS.
  • Figure 2: The golden triangle with overlaps, and as a non-overlapping GIFS on level 3.
  • Figure 3: The overlap graph for the golden triangle. To minimize intersections of edges, the initial vertex 0 was drawn three times, and once as terminal vertex in the middle. For the construction of the GIFS, it is sufficient to consider the small subgraph on the right.
  • Figure 4: Left: The initial overlaps for the golden triangle. Right: The six attractors $B_k.$ Cut overlaps are shaded.
  • Figure 5: The six nontrivial neighborhood types of pieces in the golden triangle. Reflected and rotated neighborhoods have the same type. The brown pieces represent complete overlaps and cause rational entries in the substitution matrix.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3