Double variational principle for mean dimension of $\mathbb{Z}^{K}$-actions

TL;DR

This paper extends the double variational principle to $bZ^k$-actions, linking mean dimension with rate distortion dimension through a minimax variational approach, under the marker property.

Contribution

It generalizes the double variational principle from $bZ$-actions to $bZ^k$-actions for dynamical systems with the marker property.

Findings

Established a minimax variational principle for mean dimension of $bZ^k$-actions.

Extended the double variational principle from $bZ$-actions to higher-dimensional actions.

Connected mean dimension with rate distortion dimension via a variational framework.

Abstract

In this paper, we introduce mean dimension and rate distortion dimension for $\mathbb{Z}^{k}$-actions dynamical system $(\mathcal{X},\mathbb{Z}^k,T)$. Suppose $(\mathcal{X},\mathbb{Z}^k,T)$ has the marker property. Taking these two variables, the metric $d$ on $\mathcal{X}$ and $\mathbb{Z}^{k}$-invariant measure $\mu$, into consideration, a minimax-type variational principle for mean dimension of $\mathbb{Z}^{k}$-actions is established. This result extends the double variational principle obtained recently by Lindenstrauss and Tsukamoto \cite{LT19} from $\mathbb{Z}$-actions dynamical systems to $\mathbb{Z}^{k}$-actions dynamical systems.