| Abstract: |
| We investigates the theory of metric mean dimension for actions of countable discrete amenable groups on compact metric spaces. We introduce the metric mean dimension, packing metric mean dimension, and Bowen metric mean dimension, respectively, using the topological entropy, packing topological entropy and Bowen`s dimensional entropy, and establish the equivalence among these three metric mean dimensions. Our main results demonstrate that the global metric mean dimension of the system can be completely characterized by the asymptotic behavior of various entropies of local sets, particularly the $\epsilon$-stable sets at individual points. These results generalize previous work on single transformations, and provide new tools for studying dynamical systems with infinite topological entropy. This is a joint work with Xinyao He and Guohua Zhang. |
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