Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (2): 769-773.doi: 10.1007/s10473-022-0221-x

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A NOTE ON MEASURE-THEORETIC EQUICONTINUITY AND RIGIDITY

Chiyi LUO, Yun ZHAO   

  1. School of Mathematical Sciences and Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, China
  • Received:2020-11-30 Revised:2021-03-01 Online:2022-04-25 Published:2022-04-22
  • Supported by:
    Supported by the National Natural Science Foundation of China (11790274 and 11871361). The second author is partially supported by Qinglan project of Jiangsu Province.

Abstract: Given a topological dynamical system $(X,T)$ and a $T$-invariant measure $\mu$, let $\mathcal{B}$ denote the Borel $\sigma$-algebra on $X$. This paper proves that $(X,\mathcal{B},\mu,T)$ is rigid if and only if $(X,T)$ is $\mu$-$A$-equicontinuous in the mean for some subsequence $A$ of $\mathbb{N}$, and a function $f\in L^2(\mu)$ is rigid if and only if $f$ is $\mu$-$A$-equicontinuous in the mean for some subsequence $A$ of $\mathbb{N}$. In particular, this gives a positive answer to Question 4.11 in [1].}

Key words: Measure-theoretic equicontinuity, rigidity, mean metric

CLC Number: 

  • 37A05
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