Spectrum of invariant measures via generic points

TL;DR

This paper investigates the spectrum of ergodic invariant measures by analyzing generic points and their limits, introducing new concepts like Wiener--Wintner generic points and studying measure limits under specific pseudometrics.

Contribution

It generalizes the characterization of generic points for discrete spectrum measures and studies the behavior of spectra under measure limits using novel pseudometric frameworks.

Findings

Characterization of generic points for discrete spectrum measures.

Closure properties of measures with specific spectral types under pseudometric limits.

New proof of rational discrete spectrum for Mirsky measures associated with b-free numbers.

Abstract

We describe the spectrum of an ergodic invariant measure by examining the behaviour of its generic points. We define regular Wiener--Wintner generic points for a measure to generalise the characterisation of generic points for discrete spectrum measure from Lenz et al. [Ergodic Theory and Dynamical Systems vol. \textbf{44} (2024), no. 2, 524--568]. We also study limits of sequences of generic points with respect to the Besicovitch pseudometric. This translates to results about limits of measures with respect to the metric rho-bar $\bar{\rho}$ generalising Ornstein's d-bar metric. We study how the spectrum behaves when passing to the limit and we prove that points generic for discrete spectrum, totally ergodic, or (weakly) mixing measures, property K, zero entropy measures form a closed set with respect to the Besicovitch pseudometric. Hence, the same holds for corresponding measures with respect to the rho-bar metric. Our methods have already been used to prove existence of ergodic measures with desired properties, in particular with discrete spectrum. They also lead to a new proof of rational discrete spectrum of the Mirsky measure associated with a given set of $\mathscr B$-free numbers.