Discrete spectrum of probability measures for locally compact group actions

TL;DR

This paper characterizes discrete spectrum for probability measures under locally compact group actions by linking it to measure- and mean-complexity notions. It proves that a measure has discrete spectrum iff it has bounded measure-max-mean-complexity and bounded max-mean-complexity, and extends these equivalences to amenable locally compact groups where discrete spectrum is also equivalent to bounded mean-complexity along Følner sequences and to mean equicontinuity along tempered Følner sequences. The results generalize prior discrete-spectrum characterizations from discrete or countable groups to the locally compact setting, providing practical criteria for detecting pure-point dynamical behavior. These connections illuminate how dynamical complexity controls spectral types and diffraction properties in 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 Følner 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 Følner sequence, or equicontinuous in the mean along a tempered Følner sequence.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm-2']}
• proof : Proof of Theorem \ref{['thm-1']}
• Lemma 4.1
• proof
• proof : Proof of Theorem \ref{['thm-3']}