Amenable Metric Mean Dimension and Amenable Mean Hausdorff Dimension of Product Sets and Metric Varying

Metric mean dimension and mean Hausdorff dimension depend on metrics. In this paper, we investigate the continuity of the metric mean dimension and mean Hausdorff dimension concerning the metrics for amenable group actions, which extends recent results by Muentes, Becker, Baraviera et al… Moreover, we give proof of the product formulas for the mean Hausdorff dimension and the metric mean dimension for amenable group actions.