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

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.