We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

中文翻译：

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平均尺寸的双重变分原理

我们在平均维数理论和速率失真理论之间发展了变分原理。我们考虑关于两个变量（度量和度量）的速率失真维的极小极大问题。我们证明了minimax值等于具有标记属性的动力学系统的平均尺寸。该证明展现了遍历理论，速率失真理论和几何度量理论的新组合。在证明的过程中，我们还表明，如果动力学系统具有标记属性，则它具有一个度量，其上度量的平均维等于该平均维。