Planar dynamical systems with pure Lebesgue diffraction spectrum

TL;DR

This paper investigates the diffraction spectra of algebraic lattice dynamical systems, revealing that diverse dynamical behaviors produce essentially indistinguishable pure Lebesgue diffraction spectra, complicating inverse structure determination.

Contribution

It demonstrates that various algebraic dynamical systems with different properties share similar diffraction spectra, highlighting challenges in inferring structure from diffraction data.

Findings

Different dynamical properties lead to indistinguishable diffraction spectra

Higher-order correlations can help differentiate systems

Diffraction spectra alone are insufficient for structure determination

Abstract

We examine the diffraction properties of lattice dynamical systems of algebraic origin. It is well-known that diverse dynamical properties occur within this class. These include different orders of mixing (or higher-order correlations), the presence or absence of measure rigidity (restrictions on the set of possible shift-invariant ergodic measures to being those of algebraic origin), and different entropy ranks (which may be viewed as the maximal spatial dimension in which the system resembles an i.i.d.\ process). Despite these differences, it is shown that the resulting diffraction spectra are essentially indistinguishable, thus raising further difficulties for the inverse problem of structure determination from diffraction spectra. Some of them may be resolved on the level of higher-order correlation functions, which we also briefly compare.