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<p>In this paper, we study the weak mean metric and give some properties by replacing the Besicovitch pseudometric with weak mean metric in the definition of mean equicontinuity and mean sensitivity. We study an opposite side of weak mean equicontinuity, namely strong mean sensitivity and we obtain some dichotomies: minimal topological dynamical systems are either weakly mean equicontinuous or strongly mean sensitive and transitive topological dynamical systems are either almost weakly mean equicontinuous or strongly mean sensitive. Furthermore, motivated by the localized idea of sensitivity, we introduce some notions of new version sensitive tuples and study the properties of these sensitive tuples, we show that a transitive dynamical system is strongly mean sensitive if and only if it admits a strongly mean sensitive tuple. Finally, we introduce the notion of weak mean equicontinuity of a topological dynamical system with respect to a given continuous function <i>f</i>, and we show that a topological dynamical system is weakly mean equicontinuous then it is weakly mean equicontinuous with respect to every continuous function.</p>

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Equicontinuity and Sensitivity Under Weak Mean Metric

  • 期刊论文

  • 发表日期：

    2026年4月15日

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  • Zhongxuan Yang,
  • Xiaojun Huang

摘要

In this paper, we study the weak mean metric and give some properties by replacing the Besicovitch pseudometric with weak mean metric in the definition of mean equicontinuity and mean sensitivity. We study an opposite side of weak mean equicontinuity, namely strong mean sensitivity and we obtain some dichotomies: minimal topological dynamical systems are either weakly mean equicontinuous or strongly mean sensitive and transitive topological dynamical systems are either almost weakly mean equicontinuous or strongly mean sensitive. Furthermore, motivated by the localized idea of sensitivity, we introduce some notions of new version sensitive tuples and study the properties of these sensitive tuples, we show that a transitive dynamical system is strongly mean sensitive if and only if it admits a strongly mean sensitive tuple. Finally, we introduce the notion of weak mean equicontinuity of a topological dynamical system with respect to a given continuous function f, and we show that a topological dynamical system is weakly mean equicontinuous then it is weakly mean equicontinuous with respect to every continuous function.