Discrete spectrum for group actions

Abstract

In this paper, we study discrete spectrum of invariant measures for group actions. We show that an invariant measure has discrete spectrum if and only if it has finite max-mean-measure-complexity. As an application, we show that for countable discrete amenable group actions an invariant measure has discrete spectrum if and only if it has bounded mean-measure-complexity along any Følner sequence.

Keywords:

• Group actions
• discrete spectrum
• bounded complexity

2020 Mathematics Subject Classification:

• 37A15

Consent for publication

This manuscript has not been published and is not under consideration for publication elsewhere. All the authors have approved the manuscript and agree with this submission.

Ethics approval

The submitted work is original and have not been published elsewhere in any form or language.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Fang Xu was supported by Natural Science Research Project of colleges and universities in Anhui Province [grant number KJ2020A1176]. Leiye Xu was partially supported by NNSF of China [grant number 12031019].