An autocorrelation and a discrete spectrum for dynamical systems on metric spaces

We study dynamical systems $(X,G,m)$ with a compact metric space $X$ , a locally compact, $\unicode[STIX]{x1D70E}$ -compact, abelian group $G$ and an invariant Borel probability measure $m$ on $X$ . We show that such a system has a 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 to that of the autocorrelation measure in the study of aperiodic order for special dynamical systems based on point sets.

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