On the spectrum of non-ergodic measures

TL;DR

This paper investigates the spectral properties of invariant measures in abelian topological dynamical systems, showing that spectral characteristics are preserved across ergodic decompositions and applying this to almost periodic measures.

Contribution

It establishes the preservation of spectral properties under ergodic decomposition for invariant measures in abelian systems, extending understanding of spectral types in dynamical systems.

Findings

Invariant measures with spectrum in a Borel set have ergodic measures with the same property.

Pure point spectrum in an invariant measure implies almost all ergodic measures share this spectrum.

Existence of Besicovitch almost periodic measures within the hull of any mean almost periodic measure.

Abstract

Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel subset of the dual group then the same holds almost surely for all ergodic measures arising via the Choquet theorem. In particular, if the invariant measure has pure point dynamical spectrum, so do almost all the ergodic measures. As an application, we show that given any mean almost periodic measure, in its hull there exists a Besicovitch almost periodic measure.