2026/07/23 by Xianqiang Li, Zhuowei Liu
#math.DS
We exhibit a topologically mixing continuous-alphabet subshift of the free group F2 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∈ [0,1]F2:xgxgs=0 for every g∈ F2 and s∈\a,b\\. We construct one sofic approximation from finite quotients compatible with the parity homomorphism F2→\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.