Equivalence of Sofic $p$-Metric Mean Dimensions and a Tame-Metric Variational Formula

Abstract

Let $\Gamma$ be a countable discrete sofic group acting by homeomorphisms on a compact metrizable space $X$ and $\Sigma$ a sofic approximation of $\Gamma$. We prove that for every $1\le p<\infty$, the sofic $p$-metric mean dimension is equivalent to the sofic metric mean dimension, i.e there exists a common value $D_{\Sigma }(X,\Gamma )\in \{-\infty \}\cup [0,+\infty ]$ such that, $$D_Σ(X,Γ)=\mdim_{Σ,\mathrm M,p}(X,Γ) =\mdim_{Σ,\mathrm M,\infty}(X,Γ),$$ which answers a question of Hayes in \cite[Question 3]{Hayes}. Moreover, a tame-metric variational formula is established. That is for every $1\le q\le \infty$, $$D_Σ(X,Γ) =\inf_{ρ\in\mathcal T(X)} \underline{\mdim}_{Σ,q}(X,ρ), $$ where $\mathcal{T}(X)$ is the set of all compatible metrics on $X$ having tame growth of covering numbers.

14 pages