# American Institute of Mathematical Sciences

December  2021, 41(12): 5871-5886. doi: 10.3934/dcds.2021099

## 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

Received  December 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

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.

Citation: Tao Yu, Guohua Zhang, Ruifeng Zhang. Discrete spectrum for amenable group actions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5871-5886. doi: 10.3934/dcds.2021099
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