15 research outputs found

    Variational principle for mean dimension with potential of Rd\mathbb{R}^d-actions: I

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    We develop a variational principle for mean dimension with potential of Rd\mathbb{R}^d-actions. We prove that mean dimension with potential is bounded from above by the supremum of the sum of rate distortion dimension and a potential term. A basic strategy of the proof is the same as the case of Z\mathbb{Z}-actions. However measure theoretic details are more involved because Rd\mathbb{R}^d is a continuous group. We also establish several basic properties of metric mean dimension with potential and mean Hausdorff dimension with potential for Rd\mathbb{R}^d-actions.Comment: 56 pages, 3 figure

    Variational principles of metric mean dimension for random dynamical systems

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    It is well-known that the relativized variational principle established by Bogenschutz and Kifer connects the fiber topological entropy and fiber measure-theoretic entropy. In context of random dynamical systems, metric mean dimension was introduced to characterize infinite fiber entropy systems. We give four types of measure-theoretic ϵ\epsilon-entropies, called measure-theoretic entropy of partitions decreasing in diameter, Shapira's entropy, Katok's entropy and Brin-Katok local entropy, and establish four variational principles for metric mean dimension
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