A Hard-Core Subshift Whose Sofic Mean Dimension Depends on the Sofic Approximation

We exhibit a topologically mixing continuous-alphabet subshift of the free group $F_2$ whose sofic mean dimension depends on the chosen homomorphic sofic approximation, which gives an answer of Li \cite[Remark 2.7]{Li13}. The system is the hard-core subshift
\[
X_{\rm hc}=\{x\in [0,1]^{F_2}:x_gx_{gs}=0\text{ for every }g\in F_2\text{ and }s\in\{a,b\}\}.
\]
We construct one sofic approximation from finite quotients compatible with the parity homomorphism $F_2\to\Z/2\Z$; all of its action graphs are bipartite and give sofic mean dimension exactly $1/2$. A second approximation is selected from two independent uniform random permutations and gives a value in $[1/5,9/20]$. Consequently a mixing action can have two distinct positive sofic mean dimensions.