Metric mean dimension of amenable group actions: localization and non-uniformity

Abstract

In this paper, we extend Tsukamoto's recent localization formula for metric mean dimension to actions of countable discrete amenable groups, which previously applied only to $\mathbb{R}^k$- and $\mathbb{Z}^k$-actions -- by proving that the global metric mean dimension is characterized by the asymptotic entropy of pointwise $\varepsilon$-stable sets (Theorem 2.3). To achieve this generalization, we introduce equivalent definitions of the invariant using topological entropy, packing topological entropy, and Bowen's dimensional entropy, respectively. A key technical contribution is our replacement of tiling arguments with Lindenstrauss's combinatorial covering lemma, which enables us to handle the general structure of amenable groups. Furthermore, we resolve all three questions regarding uniformity raised in Section 6 of a recent paper by Yang, Chen, and Zhou by constructing counterexamples (Theorem 2.6 and Proposition 5.8), which demonstrates that the supremum and limit superior in the localization formula cannot generally be interchanged, thereby highlighting the heterogeneous nature of the convergence. These results clarify the uniformity issue and offer insights into the link between local dynamics and global invariants, while our equivalent definitions provide flexible tools for computing metric mean dimension in concrete settings.