Some Variational Principles for the Metric Mean Dimension of a Semigroup Action

In this manuscript, we show that the metric mean dimension of a semigroup action satisfies three variational principles: (a) in our first result, consider local entropy function for free and principle terms such function; (b) second one is about definition Katok’s introduced Carvalho et al. (Ergod, 42, 65–85, 2022); (c) third based on Shapira’s entropy, Shapira (Israel J Math, 158, 225–247, 2007) single dynamic, extend action. We also obtain formula which relates induced skew product; (d) fourth involving action; (e) last two theorems, topological when have finitely generated inspired Ghys (Acta 160, 105–142, 1988). context, partial dimension. Our results are ones obtained by (Lindenstrauss 2019) Velozo (2017) Gutman Sṕiewak (Stud 261, 345–360, 2021) Shi (IEEE Trans Inf Theory, 68, 4282–4288, 2022).