In this paper we introduce two metrics: the max metric d_n,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 d_n,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 d_n,q and \({\bar d_{n,q}}\).

在本文中，我们介绍了两个指标：最大指标d _n，q和平均指标\（{\ bar d_ {n，q}} \）。我们通过这两个指标对刚性度量保存系统进行了等效表征。事实证明，一个不变测度μ拓扑动力系统（上X，T）具有有界的复杂相对于d _N，Q当且仅当μ具有有界的复杂相对于\（{\巴D_ {N，Q} } \）当且仅当（X，\（{\ cal B} x \），μ，T）是刚性的。我们还通过两个度量值d _n，q和\（{\ bar d_ {n，q}}获得遍历测度保存系统的测度理论熵（分别是拓扑动力系统的拓扑熵）的计算公式。\）。