Discrete spectrum of probability measures for locally compact group actions

TL;DR

This paper characterizes when probability measures for locally compact group actions have a discrete spectrum, linking it to bounded measure-max-mean-complexity and mean equicontinuity, with implications for amenable groups.

Contribution

It establishes a precise criterion for discrete spectrum in terms of measure-max-mean-complexity and mean equicontinuity for locally compact group actions.

Findings

Discrete spectrum characterized by bounded measure-max-mean-complexity.

Invariant measures with discrete spectrum are mean equicontinuous along F exto lner sequences.

Provides equivalence conditions for discrete spectrum in amenable group actions.

Abstract

In this paper, we investigate the discrete spectrum of probability measures for actions of locally compact groups. We establish that a probability measure has a discrete spectrum if and only if it has bounded measure-max-mean-complexity. As applications: 1) An invariant measure for a locally compact amenable group action has a discrete spectrum if and only if it has bounded mean-complexity along F\o lner sequences; 2) An invariant measure for a locally compact amenable group action has a discrete spectrum if and only if it is mean equicontinuous along a tempered F\o lner sequence, or equicontinuous in the mean along a tempered F\o lner sequence.