PPubMed9 Jul 2020Cited 3×

Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction Inflation versus projection sets in aperiodic systems

Michael Baake, Uwe Grimm

Abstract

Averaged quantities such as mean shelling numbers, scaling behaviour or diffraction for cut-and-project sets can conveniently be computed in internal space, also for systems with fractally bounded windows.

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The paper shows how internal-space methods, especially the internal cocycle approach, enable efficient diffraction and averaging calculations for aperiodic inflation and projection sets, even with fractal windows.

What’s new

i.Averaged quantities for cut‑and‑project sets can be computed in internal space, including systems with fractally bounded windows
ii.Development of the internal cocycle approach to efficiently compute diffraction for systems with both inflation and projection descriptions
iii.Application of the method to planar Fibonacci‑based examples, revealing diffraction patterns of square Fibonacci tilings and DPV variants
iv.Analysis of hyperuniformity, showing distinct scaling behaviours for inflation‑invariant versus generic windows

How they did it

The authors review the theory of cut‑and‑project and inflation sets, using the Fibonacci tiling as a guiding example. They lift the inflation cocycle to internal space to form an internal Fourier matrix and iterate it, allowing diffraction amplitudes to be obtained from matrix products. Numerical calculations based on this cocycle are then performed for various planar examples.

What they found

The internal cocycle approach yields fast and accurate diffraction calculations for fractal windows, outperforming standard Fourier‑window methods
48 distinct inflation rules were identified for the square Fibonacci tiling variants
Hyperuniformity analysis shows diffraction intensity near the origin scales as k² for inflation‑invariant windows and as k for generic windows

Caveats

✕The Pisot substitution conjecture remains open, limiting understanding of pure‑point spectra for many substitutions
✕Many averaged‑quantity problems for aperiodic tilings still lack general tools and remain open questions
✕Standard diffraction formulas become impractical for fractal windows, requiring the new cocycle approach
✕Scaling behaviour does not reveal the distribution of different scatterers in the structure

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