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arXiv · 2310.03989

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

Masaki Tsukamoto
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Abstract

We develop a variational principle for mean dimension with potential of $\mathbb{R}^d$-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 $\mathbb{Z}$-actions. However measure theoretic details are more involved because $\mathbb{R}^d$ is a continuous group. We also establish several basic properties of metric mean dimension with potential and mean Hausdorff dimension with potential for $\mathbb{R}^d$-actions.

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Masaki Tsukamoto. 2023-10-06. Variational principle for mean dimension with potential of $\mathbb{R}^d$-actions: I. https://arxiv.org/abs/2310.03989

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