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Tilings from Tops of Overlapping Iterated Function Systems

Michael F. Barnsley, Corey de Wit

Abstract

The top of the attractor $A$ of a hyperbolic iterated function system $\left\{ f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}|i=1,2,\dots,M\right\} $ is defined and used to extend self-similar tilings to overlapping systems. The theory provides sequences of approximate supertiles that converge to tilings. Individual tiles in a tiling are limits of nested decreasing sequences of approximate tiles. Examples include systems of finite type, tilings related to aperiodic monotiles, and ones where there are infinitely many distinct but related prototiles.

Tilings from Tops of Overlapping Iterated Function Systems

Abstract

The top of the attractor of a hyperbolic iterated function system is defined and used to extend self-similar tilings to overlapping systems. The theory provides sequences of approximate supertiles that converge to tilings. Individual tiles in a tiling are limits of nested decreasing sequences of approximate tiles. Examples include systems of finite type, tilings related to aperiodic monotiles, and ones where there are infinitely many distinct but related prototiles.

Paper Structure

This paper contains 14 sections, 5 theorems, 44 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Iterated Function Systems, Tops, Blowups, and Tilings
  3. Iterated function systems
  4. Tops
  5. Tops at finite depth
  6. Top blowups and tilings
  7. NOTION OF PRE-TILES AND TOPS
  8. Basic example of top tiling when the OSC holds
  9. Illustrations of pre-tiles when the OSC does not hold
  10. EXAMPLES OF TOPS TILINGS
  11. Leaf example of a two-dimensional top tiling
  12. Top tilings related to hat tilings
  13. Generalization
  14. Acknowledgement

Key Result

Lemma 1

$\Sigma=\sigma\left( \Sigma\right)$.

Figures (11)

  • Figure 1: The right hand panel represents the attractor of the IFS in Equation \ref{['eqifs1']} partitioned at depth one into four pre-tiles, indicated by different colours. Dotted lines indicate open boundaries. The left hand panel is similar, but partitioned to depth two.
  • Figure 2: The maps $f_{4}^{-1}$ (left) and $f_{1}^{-1}$ (right) have been applied to the right-hand partition in Figure \ref{['ownbound']}. Notice the different positionings of the closed and open edges of the pre-tiles that cover $\left( 0,1\right) \times\left(0,1\right).$
  • Figure 3: This picture relates to the IFS in Equation \ref{['ex2formula']} and shows successive tilings of the form $f_{-\mathbf{k}|n}(\{\overline{A_{\mathbf{i}|(n+1)}}:\mathbf{i\in\Sigma\})}$ where $\mathbf{k}|4=1414.$ In successive panels the bounding box shown in red remains constant.
  • Figure 4: This shows the converged tiling, within the red bounding box, following on from Figure \ref{['plaintiles']}. See text. As the tiling develops further, parts within the bounding box cease to change.
  • Figure 5: The overlapping attractor of an IFS of two similitudes, each with the same scaling factor. The figure has been rotated ninety degrees anticlockwise relative to the usual presentation of the x,y coordinate system.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • Example 1
  • ...and 2 more