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Graph Iterated Function Systems and Fractal Tops

Grover Lancaster-Cole, Georgiana Lyall, Thomas Malcolm, Qiyu Zhou

TL;DR

For the simplest overlapping interval IFS, a sufficient condition is found for the closure of its tops code space to be a shift space of finite type and it is found that shift invariance properties do not directly extend to the graph-directed setting.

Abstract

Following the work of Louisa and Michael Barnsley on results in tops of iterated function systems, we extend their work to graph-directed iterated function systems by investigating the relationship between top addresses and shift spaces. For the simplest overlapping interval IFS, we find a sufficient condition for the closure of its tops code space to be a shift space of finite type. Likewise, we find that shift invariance properties do not directly extend to the graph-directed setting.

Graph Iterated Function Systems and Fractal Tops

TL;DR

For the simplest overlapping interval IFS, a sufficient condition is found for the closure of its tops code space to be a shift space of finite type and it is found that shift invariance properties do not directly extend to the graph-directed setting.

Abstract

Following the work of Louisa and Michael Barnsley on results in tops of iterated function systems, we extend their work to graph-directed iterated function systems by investigating the relationship between top addresses and shift spaces. For the simplest overlapping interval IFS, we find a sufficient condition for the closure of its tops code space to be a shift space of finite type. Likewise, we find that shift invariance properties do not directly extend to the graph-directed setting.
Paper Structure (10 sections, 22 theorems, 22 equations, 6 figures)

This paper contains 10 sections, 22 theorems, 22 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Shift spaces
  3. Graph-directed iterated function systems
  4. Fractal tops and shifts
  5. The fractal top
  6. Banned words in top shift spaces
  7. Shift invariance of the top
  8. A range of behaviours
  9. Points not in the shift
  10. Diagrams

Key Result

Proposition 2.2

For a subset $X$ of $\mathcal{A}^\mathbb{N}$, the following are equivalent:

Figures (6)

  • Figure 1: The overlapping IFS with two maps
  • Figure 2: A graph IFS such that $\Sigma_\mathrm{top}$ is not shift invariant
  • Figure 3: A relabelling of the graph IFS in figure \ref{['fig:notinvariant']} such that $\Sigma_\mathrm{top}$ is shift invariant
  • Figure 4: A graph IFS for which $\Sigma_\mathrm{top}$ is shift invariant for every ordering of the edges
  • Figure 5: A graph IFS with no self-referencing maps for which $\Sigma_\mathrm{top}$ is shift invariant for every ordering
  • ...and 1 more figures

Theorems & Definitions (60)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 50 more