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<p>We introduce directional complexities for ℤ<sup><i>q</i></sup>-measure preserving dynamical systems via a collection of new metrics along non-zero directions in ℝ<sup><i>q</i></sup>. It turns out that a ℤ<sup><i>q</i></sup>-measure preserving dynamical system is rigid if and only if the invariant measure has bounded directional complexity. We also obtain ergodic decomposition formula for the measure-theoretic directional entropy of a ℤ<sup><i>q</i></sup>-measure preserving dynamical system.</p>

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Bounded Directional Complexity and Rigidity for ℤq-actions

  • 期刊论文

  • 发表日期：

    2026年1月15日

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  • Runju Wei,
  • Leiye Xu,
  • Liqi Zheng,
  • Xiaomin Zhou

摘要

We introduce directional complexities for ℤq-measure preserving dynamical systems via a collection of new metrics along non-zero directions in ℝq. It turns out that a ℤq-measure preserving dynamical system is rigid if and only if the invariant measure has bounded directional complexity. We also obtain ergodic decomposition formula for the measure-theoretic directional entropy of a ℤq-measure preserving dynamical system.