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In this paper, we study metric mean dimension with potential and set up a variational principle for it, which connects rate distortion function to metric mean dimension with potential. Moreover, we give alternative definitions of metric mean dimensions with potential and show that classic metric mean dimension with potential is equal to Bowen upper and lower capacity metric mean dimensions with potential. [ABSTRACT FROM AUTHOR]

The aim of this paper is to reveal the relationship between the topological entropy of a free semigroup action and that of its induced transformations. More precisely, we prove that the topological entropy of a free semigroup action (X , F) vanishes if and only if the topological entropy of its induced system (M (X) , F) is zero; if the topological entropy of (X , F) is positive, then that of its induced systems (M (X) , F) , (K (X) , F) are infinite. This in particular extends the results for classical topological entropy done by Bauer and Sigmund (1975 Monatsh. Math.79 81–92), Glasner and Weiss (1995 J. Amer. Math. Soc.8 665–686). [ABSTRACT FROM AUTHOR]

The aim of this paper is to examine the topological entropy for a free semigroup action defined by Bufetov using separated and spanning sets. First, this study reveals that such entropy is a topological conjugacy invariant and also can be equivalently defined using open covers. Furthermore, a quantitative analogue of Bowen's theorem for semiconjugacy is provided and we compared the topological entropies presented by Bufetov and Biś. Finally, a formula for the entropy of skew-product transformation with respect to the subshift is obtained. [ABSTRACT FROM PUBLISHER]

We introduce a notion of shadowing property for actions of finitely generated groups and study its basic properties. We formulate and prove a shadowing lemma for actions of nilpotent groups. We construct an example of a faithful linear action of a solvable Baumslag–Solitar group and show that the shadowing property depends on quantitative characteristics of hyperbolicity. Finally, we show that any linear action of a non-abelian free group does not have the shadowing property. [ABSTRACT FROM AUTHOR]