Pair correlations of one-dimensional model sets and monstrous covariograms of Rauzy fractals

TL;DR

The paper addresses the problem of understanding the averaged distance statistics of one-dimensional regular model sets through covariograms of Rauzy fractal windows. It introduces an exact renormalisation framework for pair correlation functions in inflation tilings, enabling computation of cross-covariograms via the $\star$-map and window IFS. Two explicit binary inflation examples based on the Pisot silver mean inflation are analyzed, revealing that the resulting covariogram functions are continuous yet exhibit highly fractal and slow-converging behaviour, including a split covariogram in the second example. The results highlight rich, fractal-like structures in aperiodic order and suggest further study of covariogram-based invariants for aperiodic spectra.

Abstract

The averaged distance structure of one-dimensional regular model sets is determined via their pair correlation functions. The latter lead to covariograms and cross covariograms of the windows, which give continuous functions in internal space. While they are simple tent-shaped, piecewise linear functions for intervals, the typical case for inflation systems leads to convolutions of Rauzy fractals, which are difficult to compute. In the presence of an inflation structure, an alternative path is possible via the exact renormalisation structures of the pair correlation functions. We introduce this approach and derive two concrete examples, which display an unexpectedly complex and wild behaviour.

Figures (6)

• Figure 1: Geometric inflation rule for the self-similar SSM tiling.
• Figure 2: The windows of the SSM inflation tiling, which are the fixed point of the IFS \ref{['eq:IFS_SSM']}; $W^{}_{a}$ is red (top) and $W^{}_{b}$ is blue (bottom). The windows are one-dimensional, but we assign some fixed arbitrary height to the points for illustration. The windows are measure-theoretically disjoint, but the resolution is limited by the large Hausdorff dimension of the window boundaries.
• Figure 3: Representation of the window covariograms of the SSM system. The red function is $\nu^{}_{aa}$, $\nu^{}_{bb}$ is blue, $\nu^{}_{ab}$ is black, $\nu^{}_{ba}$ is grey, and the total covariogram of the window $\nu$ is pink. The plot of $\nu$ consists of $35{\space}323$ points; we also include a closer view of a small section to illustrate the irregular behaviour.
• Figure 4: Illustration of the windows for the substitution $\sigma$; $W^{}_{a}$ is red (top) and $W^{}_{b}$ is blue (bottom). As before, the windows are one-dimensional, but we assign some fixed arbitrary height to the points for illustration. Furthermore, by Proposition \ref{['prop:IFS_window']}, the windows are measure-theoretically disjoint, but we are limited by the resolution of the plot and the large Hausdorff dimension of the window boundaries.
• Figure 5: Point plot, with $188{\space\space}214$ points, of the covariogram of the window of Figure \ref{['fig:Example-window']}. Here, the splitting behaviour is highlighted; the distances involving an even number of $b$'s are blue, and those with an odd number are red.

Theorems & Definitions (14)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5: SingThesis
• Remark 2.6
• Proposition 2.7: ManiboThesis
• Proposition 2.8
• proof
• Theorem 3.1