Spectral theory of dynamical systems as diffraction theory of sampling functions

TL;DR

This paper establishes a connection between spectral theory and diffraction theory in dynamical systems, showing how diffraction measures serve as spectral invariants and can be derived from sampling along orbits.

Contribution

It introduces a framework linking diffraction measures to spectral theory in dynamical systems over abelian groups, including sampling methods for autocorrelation.

Findings

Diffraction measures are spectral measures of functions.

Sampling along orbits can recover autocorrelation for groups with a countable basis.

Diffraction measures of factors serve as complete spectral invariants.

Abstract

We consider topological dynamical systems over $\ZZ$ and, more generally, locally compact, $\sigma$-compact abelian groups. We relate spectral theory and diffraction theory. We first use a a recently developed general framework of diffraction theory to associate an autocorrelation and a diffraction measure to any $L^2$-function over such a dynamical system. This diffraction measure is shown to be the spectral measure of the function. If the group has a countable basis of the topology one can also exhibit the underlying autocorrelation by sampling along the orbits. Building on these considerations we then show how the spectral theory of dynamical systems can be reformulated via diffraction theory of function dynamical systems. In particular, we show that the diffraction measures of suitable factors provide a complete spectral invariant.