Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
Variational principle for mean dimension with potential of ℝ𝑑-actions. I
Masaki Tsukamoto
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2025 Volume 77 Issue 2 Pages 391-447

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Abstract

We develop a variational principle for mean dimension with potential of ℝ𝑑-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-actions. However measure theoretic details are more involved because ℝ𝑑 is a continuous group. We also establish several basic properties of metric mean dimension with potential and mean Hausdorff dimension with potential for ℝ𝑑-actions.

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