Variational principle for mean dimension with potential of ℝ^d-actions: I

2023/10/06 by Masaki Tsukamoto, Tsukamoto, Masaki
Computer Science · Economics, Econometrics and Finance · Mathematics · #37B99 #54F45 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2310.03989

openalex publication_date 2023/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop a variational principle for mean dimension with potential of 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-actions. However measure theoretic details are more involved because 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 ℝ^d-actions.

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