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A Note on Measure-Theoretic Equicontinuity and Rigidity

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Abstract

Given a topological dynamical system (X, T) and a T-invariant measure μ, let \({\cal B}\) denote the Borel σ-algebra on X. This paper proves that (X, \({\cal B}\), μ, T) is rigid if and only if (X, T) is μ-A-equicontinuous in the mean for some subsequence A of ℕ, and a function f ∈ L2 (μ) is rigid if and only if f is μ-A-equicontinuous in the mean for some subsequence A of ℕ. In particular, this gives a positive answer to Question 4.11 in [1].

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Authors and Affiliations

  1. School of Mathematical Sciences and Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou, 215006, China

    Chiyi Luo  (罗炽逸) & Yun Zhao  (赵云)

Authors
  1. Chiyi Luo  (罗炽逸)
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  2. Yun Zhao  (赵云)
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Corresponding author

Correspondence to Yun Zhao  (赵云).

Additional information

Supported by the National Natural Science Foundation of China (11790274 and 11871361). The second author is partially supported by Qinglan project of Jiangsu Province.

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Luo, C., Zhao, Y. A Note on Measure-Theoretic Equicontinuity and Rigidity. Acta Math Sci 42, 769–773 (2022). https://doi.org/10.1007/s10473-022-0221-x

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  • DOI: https://doi.org/10.1007/s10473-022-0221-x

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