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

Abstract

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