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

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.

Dr Yunping Wang told us the main result in this manuscript has been obtained in her PhD thesis