Measure Complexity and Rigid Systems


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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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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Corresponding author

Correspondence to Run Ju Wei.

Additional information

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. (2021).

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  • Rigid system
  • entropy
  • bounded complexity

MR(2010) Subject Classification

  • 37B05
  • 54H20