2026/07/23 by Xianqiang Li, Zhuowei Liu
#math.DS
Let Γ be a countable discrete sofic group acting by homeomorphisms on a compact metrizable space X and Σ a sofic approximation of Γ. We prove that for every 1≤ p<∞, the sofic p-metric mean dimension is equivalent to the sofic metric mean dimension, i.e there exists a common value DΣ(X,Γ)∈\-∞\∪[0,+∞] such that, DΣ(X,Γ)=\mdimΣ,\mathrm M,p(X,Γ) =\mdimΣ,\mathrm M,∞(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≤ q≤∞, DΣ(X,Γ) =infρ∈\mathcal T(X) \underline\mdimΣ,q(X,ρ), where \mathcal T(X) is the set of all compatible metrics on X having tame growth of covering numbers.