Close-packed dimers on the line: diffraction versus dynamical spectrum

TL;DR

This paper investigates the relationship between diffraction and dynamical spectra in a random system of close-packed dimers on the line, revealing cases where the diffraction spectrum is a proper subset of the dynamical spectrum, especially with absolutely continuous spectra.

Contribution

It constructs a novel random dimer system with underlying order, demonstrating the complex relationship between diffraction and dynamical spectra, including cases with absolutely continuous spectra.

Findings

Diffraction spectrum can be a proper subset of the dynamical spectrum.

Constructed a random dimer system with long-range order.

Demonstrated phenomena with absolutely continuous spectra.

Abstract

The translation action of $\RR^{d}$ on a translation bounded measure $\omega$ leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of $\omega$, which is the carrier of the diffraction measure, live on a subset of the dynamical spectrum. It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point dynamical spectrum (the opposite implication always being true). For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with singular continuous diffraction) in \cite{EM}. Here, we construct a random system of close-packed dimers on the line that have some underlying long-range periodic order as well, and display the same type of phenomenon for a system with absolutely continuous spectrum. An interpretation in terms of `atomic' versus `molecular' spectrum suggests a way to come to a more general correspondence between these two types of spectra.