An autocorrelation and discrete spectrum for dynamical systems on metric spaces

TL;DR

This paper establishes a characterization of discrete spectrum in dynamical systems on metric spaces using a space average over the metric, linking it to Bohr almost periodic functions, and drawing parallels to autocorrelation measures in aperiodic order.

Contribution

It introduces a novel criterion for discrete spectrum in general dynamical systems based on metric averages, extending concepts from aperiodic order.

Findings

Discrete spectrum characterized by metric averages being Bohr almost periodic

Links between metric averages and autocorrelation measures in aperiodic order

Provides a new perspective on spectral analysis of dynamical systems

Abstract

We study dynamical systems $(X,G,m)$ with a compact metric space $X$ and a locally compact, $\sigma$-compact, abelian group $G$. We show that such a system has discrete spectrum if and only if a certain space average over the metric is a Bohr almost periodic function. In this way, this average over the metric plays for general dynamical systems a similar role as the autocorrelation measure plays in the study of aperiodic order for special dynamical systems based on point sets.