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

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.