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Rational Gluing in Edge Replacement Systems

Davide Perego, Matteo Tarocchi

TL;DR

The paper addresses the rationality of the gluing relation arising from edge replacement systems and constructs a finite-state automaton $\mathrm{Gl}_{\mathcal{R}}$ that recognises pairs of infinite words that are glued in the limit space. By encoding pairs of edges across expansion levels with transitions of types 0, 1, and 2, the automaton captures when two words are equivalent under the gluing relation $\sim$ on the Cantor-type symbol space $\Omega_{\mathcal{R}}$ and the limit space $X_{\mathcal{R}} = \Omega_{\mathcal{R}} / \sim$. The main result proves that the gluing relation is rational: $(x_0x_1\ldots, y_0y_1\ldots)$ is in $\sim$ if and only if the run of $(x_0,y_0)(x_1,y_1)\ldots$ in $\mathrm{Gl}_{\mathcal{R}}$ is accepting, with a careful delineation of how Type 0–Type 2 transitions reflect edge adjacency across expansion levels. The framework yields explicit automata for classical examples such as interval, Ważewski dendrite, and Basilica replacement systems, providing a computable finite-state model for the gluing relation and enabling further analysis of limit spaces and their rearrangement groups in totally disconnected settings.

Abstract

In this paper, we prove the rationality of the gluing relation of edge replacement systems, which were introduced for studying rearrangement groups of fractals. More precisely, we describe an algorithmic procedure for building a finite state automaton that recognizes pairs of equivalent sequences that are glued in the fractal. This fits in recent interest towards the rationality of gluing relations on totally disconnected compact metrizable spaces.

Rational Gluing in Edge Replacement Systems

TL;DR

The paper addresses the rationality of the gluing relation arising from edge replacement systems and constructs a finite-state automaton that recognises pairs of infinite words that are glued in the limit space. By encoding pairs of edges across expansion levels with transitions of types 0, 1, and 2, the automaton captures when two words are equivalent under the gluing relation on the Cantor-type symbol space and the limit space . The main result proves that the gluing relation is rational: is in if and only if the run of in is accepting, with a careful delineation of how Type 0–Type 2 transitions reflect edge adjacency across expansion levels. The framework yields explicit automata for classical examples such as interval, Ważewski dendrite, and Basilica replacement systems, providing a computable finite-state model for the gluing relation and enabling further analysis of limit spaces and their rearrangement groups in totally disconnected settings.

Abstract

In this paper, we prove the rationality of the gluing relation of edge replacement systems, which were introduced for studying rearrangement groups of fractals. More precisely, we describe an algorithmic procedure for building a finite state automaton that recognizes pairs of equivalent sequences that are glued in the fractal. This fits in recent interest towards the rationality of gluing relations on totally disconnected compact metrizable spaces.
Paper Structure (9 sections, 6 theorems, 8 equations, 8 figures, 2 tables)

This paper contains 9 sections, 6 theorems, 8 equations, 8 figures, 2 tables.

Table of Contents

  1. Background on Edge Replacement Systems
  2. Edge Replacement Systems
  3. Colors and Loops
  4. Graph Expansions
  5. Limit Spaces
  6. The Color Graph
  7. The Limit Space
  8. The Gluing Automaton
  9. Rationality of the Gluing Relation

Key Result

Lemma 3.1

The full sub-automaton $\mathrm{T}_\mathcal{R} = (\Sigma^2, Q_0, \rightarrow, q_0)$ induced by the states $Q_0$ is naturally isomorphic to the color graph. More precisely, a finite word $(x_0, y_0) \dots (x_m, y_m)$ in the alphabet $\Sigma^2$ is recognized by $\mathrm{T}_\mathcal{R}$ if and only if

Figures (8)

  • Figure 1: The interval replacement system $\mathcal{I}$.
  • Figure 2: The Dendrite replacement systems $\mathcal{D}_n$.
  • Figure 3: The original Basilica replacement system.
  • Figure 4: The modified Basilica replacement system $\mathcal{B}$.
  • Figure 5: Examples of graph expansions.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Definition 1.1
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 2.1
  • Definition 2.2
  • ...and 12 more