Discrete spectrum for amenable group actions

Tao Yu; Guohua Zhang; Ruifeng Zhang

doi:10.3934/dcds.2021099

Article Contents

2021, Volume 41, Issue 12: 5871-5886. Doi: 10.3934/dcds.2021099 open access

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Discrete spectrum for amenable group actions

1.
Department of Mathematics, Shantou University, Shantou 515063, Guangdong, China
2.
School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China
3.
School of Mathematics, Hefei University of Technology, Hefei 230009, Anhui, China

^* Corresponding author: Guohua Zhang
^* Corresponding author: Guohua Zhang

Received: December 2020

Revised: May 2021

Early access: July 2021

Published: December 2021

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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.

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Mathematics Subject Classification: Primary: 37A15, 37A35.

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