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

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.