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Measure Complexity and Rigid Systems

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Abstract

In this paper we introduce two metrics: the max metric dn,q and the mean metric \({\bar d_{n,q}}\). We give an equivalent characterization of rigid measure preserving systems by the two metrics. It turns out that an invariant measure μ on a topological dynamical system (X, T) has bounded complexity with respect to dn,q if and only if μ has bounded complexity with respect to \({\bar d_{n,q}}\) if and only if (X, \({{\cal B}_X}\), μ, T) is rigid. We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system (resp. the topological entropy of a topological dynamical system) by the two metrics dn,q and \({\bar d_{n,q}}\).

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Acknowledgements

We would like to thank Leiye Xu for valuable remarks and discussions. We also thank the anonymous referees for their careful reading and useful suggestions that greatly improved the manuscript.

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Correspondence to Run Ju Wei.

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Supported by NNSF of China (Grant Nos. 11971455, 11801538, 11801193, 11871188, 11731003 and 12090012); Tao Yu is supported by STU Scientific Research Foundation for Talents (Grant No. NTF19047)

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Huang, W., Wei, R.J., Yu, T. et al. Measure Complexity and Rigid Systems. Acta. Math. Sin.-English Ser. 38, 68–84 (2022). https://doi.org/10.1007/s10114-021-0179-y

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  • DOI: https://doi.org/10.1007/s10114-021-0179-y

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