Discrete spectrum for amenable group actions

TL;DR

This paper characterizes discrete spectrum for invariant measures in amenable group actions, linking it to measure complexity, mean equicontinuity, and providing new criteria for spectral properties.

Contribution

It introduces a measure-theoretic complexity characterization of discrete spectrum, connecting it with mean equicontinuity and providing a new perspective on spectral analysis in group actions.

Findings

Invariant measure has discrete spectrum iff it has bounded measure complexity.

Discrete spectrum characterized via measure-theoretic complexity using partitions and Hamming distance.

Discrete spectrum is equivalent to mean equicontinuity and equicontinuity in the mean.

Abstract

In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions. We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.