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 prove the product formulas for the mean Hausdorff dimension and the metric mean dimension for amenable group actions. •A new proof of product formula for the metric mean dimension for amenable group actions.•Analysis of the continuity of the metric mean dimension for amenable group actions.•Analysis of the continuity of the mean Hausdorff dimension for amenable group actions.•Study of relationship between metric mean dimension and Minkowski dimension.