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

TL;DR

This paper establishes a variational principle for mean dimension with potential in $R^d$-actions, linking it to rate distortion dimension and potential, and extends key properties known from $Z$-actions to continuous group actions.

Contribution

It introduces a variational principle for mean dimension with potential for $R^d$-actions, expanding the theoretical framework to continuous group actions.

Findings

Mean dimension with potential is bounded by the supremum of rate distortion dimension plus potential.

Basic properties of metric mean dimension with potential are established for $R^d$-actions.

Mean Hausdorff dimension with potential is also analyzed for $R^d$-actions.

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.