The authors aim to investigate the concept of metric mean dimension concerning both a compactly generated semigroup action and a compactly generated free semigroup action. Let \(\overline{\text{mdim}}_M\left(X, \mathbb{S},d, \mathbb{P}\right)\) be the upper metric mean dimensions of the free semigroup action \(\mathbb{S}\) on \((X, d)\) with respect to a fixed set of generators \(G_1\) and a random walk \(\mathbb{P}\) in \(Y^{\mathbb{N}}\). The initial finding presented in the paper demonstrates that it is possible to calculate the metric mean dimension of a semigroup action using the entropy function.
Theorem. Let \((X, d)\) be a compact metric space and \(\mathbb{S}\) be the free semigroup action induced on \((X, d)\) by a family of continuous maps \((g_y : X\to X)_{y\in Y}\). Then: \[ \overline{\text{mdim}}_M\left(X, \mathbb{S},d, \mathbb{P}\right) =\lim\sup _{\varepsilon\to 0^+} \frac{\sup\limits_{x\in X}h_d(x,\varepsilon)}{- \log\varepsilon}, \] for every \(\mathbb{P}\in \mathcal{M}\left(Y^{\mathbb{N}}\right)\).
The authors then obtain new results regarding U. Shapira’s entropy [Isr. J. Math. 158, 225–247 (2007; Zbl 1121.37008)] of a semigroup action. In another theorem, the authors extend the upper metric mean dimensions for compactly generated free semigroup actions. As a logical continuation of the metric mean dimension applied to a singular dynamic system, the authors investigate the upper metric mean dimension by employing the notion of entropy discussed in [E. Ghys et al., Acta Math. 160, No. 1–2, 105–142 (1988; Zbl 0666.57021)]. In the subsequent theorem, they present a form of partial variational principle concerning the metric mean dimension of the action, specifically in relation to the volume measure, when \(G\) constitutes a group of homeomorphisms.