Distribution of Patches in Tilings and Spectral Properties of Corresponding Dynamical Systems

TL;DR

This paper explores the relationship between patch distribution in tilings and the spectral properties of the associated dynamical systems, revealing hidden periodic structures and implications for diffraction analysis.

Contribution

It introduces a novel approach to infer tiling properties from dynamical system spectra, including the detection of hidden periodicities in non-periodic tilings.

Findings

Periodic structures can be hidden in non-periodic tilings.

Results provide insights into inverse problems related to diffraction measures.

The study connects patch distribution with spectral properties of dynamical systems.

Abstract

A tiling is a cover of R^d by tiles such as polygons that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in tilings. From a tiling, we can construct a dynamical system which encodes the nature of the tiling. In the literature, properties of this dynamical system were investigated by studying how patches distribute in each tiling. In this article we conversely research distribution of patches from properties of the corresponding dynamical systems. We show periodic structures are hidden in tilings which are not necessarily periodic. Our results throw light on inverse problem of deducing information of tilings from information of diffraction measures, in a quite general setting.