Dynamical versus diffraction spectrum for structures with finite local complexity

TL;DR

This paper explores the relationship between dynamical and diffraction spectra in systems with finite local complexity, establishing an equivalence that simplifies spectral analysis.

Contribution

It proves the equivalence of the dynamical spectrum with diffraction spectra for ergodic systems of finite local complexity, enabling easier spectral determination.

Findings

Dynamical spectrum is equivalent to diffraction spectra in these systems.

The equivalence allows spectral analysis via simpler diffraction spectra.

Few diffraction spectra are needed to understand the dynamical spectrum.

Abstract

It is well-known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.