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An autocorrelation and discrete spectrum for dynamical systems on metric spaces

We study dynamical systems $(X,G,m)$ with a compact metric space $X$ and a locally compact, $σ$-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.