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

In this manuscript we show that the metric mean dimension of a free semigroup action satisfies three variational principles: (a) the first one is based on a definition of Shapira’s entropy, introduced in [22] for a singles dynamics and extended for a semigroup action in this note; (b) the second one treats about a definition of Katok’s entropy for a free semigroup action introduced in [8]; (c) lastly 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. Our results are inspired in the ones obtained by [19], [28], [24] and [23].

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