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In this paper, we develop a variational relation between the metric mean dimension and several types of measure-theoretic ε-entropies. Inspired by dimension theory, we introduce the Bowen metric mean dimension for random dynamical systems and establish a variational principle. Furthermore, we provide a new characterization of the metric mean dimension via the local ε-entropy function. Finally, we study the Bowen metric mean dimension for the set of generic points.
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Fan et al. (2026) studied this question.
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