We consider continuous free semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$ , subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$ , where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the semigroup action with the corresponding notions for the associated skew product and the shift map $\sigma $ on $Y^{\mathbb {N}}$ , and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever $\nu $ is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure $\nu $ , and to test the scope of our results.

中文翻译：

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自由半群动作度量平均维的变分原理

我们考虑由一个家庭产生的连续自由半群动作$(g_y)_{y \,\in \, Y}$紧度量空间的连续自同态$(X,d)$, 随机游走$\mathbb P_\nu =\nu ^{\mathbb N}$在移位空间上定义$Y^{\mathbb N}$， 在哪里$(Y, d_Y)$是具有有限上盒维数的紧度量空间，并且$\n$是一个 Borel 概率测度是. 为了阐明随机游走对度量平均维度的影响，我们证明了一个变分原理，它将半群动作的度量平均维度与相关斜积和移位映射的相应概念联系起来$\西格玛$在$Y^{\mathbb {N}}$，并将它们与是. 特别是，我们在任何时候都能得到精确的公式$\n$是同质的并且有充分的支持。我们还讨论了几个例子来启发度量的同质性、支持和上框维度的作用$\n$，并测试我们结果的范围。