Local P Entropy and Stabilized Automorphism Groups of Subshifts

The paper proves, for a non-trivial mixing SFT (XA,σA), that hPS_{C,D,r^2}((Aut(∞)(σA),σA^k)) = htop(σA^k) under the A-admissibility hypothesis and the mild condition that A^{|k|} has an entry > 2 (Theorem 24). The upper bound comes from a general inequality (Theorem 19) relating local PS entropy to periodic-point growth, hence to topological entropy in this setting. The lower bound is obtained by taking H = Simp(∞)_ev(ΓA) and showing H ∩ C(σA^{kn}) = Simpev(ΓA^{(kn)}) with factorial growth controlled by Perron–Frobenius asymptotics and Stirling’s formula, yielding lim (1/n) log log |H ∩ C(σA^{kn})| = log λ_A^{k} = htop(σA^k) . The candidate solution establishes the same identity via (i) the same canonical subgroup H built from even simple higher-block symmetries and (ii) a direct combinatorial upper bound using a synchronizing-marker construction and an embedding into products of symmetric groups. This differs from the paper’s use of Theorem 19 for the upper bound but reaches the same conclusion. One technical oversight in the candidate solution is that H was defined using only levels ℓ ≥ L (where A-admissibility guarantees PS-membership); this does not ensure H ∩ C_G(σA^{|k|}) ≠ {e} when L > |k|, and the claimed equality H ∩ C_G(g^n) = Simpev(ΓA^{(mn)}) then holds only for mn ≥ L. This is easily fixed by defining H as the union over all levels (as in the paper) and then invoking A-admissibility only for large n. With this adjustment, the model’s argument aligns with the paper’s result and is correct in substance. Key steps and objects (Simp(∞)_ev, the inclusion im,k, and the equality H ∩ C(σA^{kn}) = Simpev(ΓA^{(kn)})) appear explicitly in the paper’s Section 4.1 and the proof of Theorem 24 . The paper’s general upper bound via periodic points (Theorem 19) is also clearly stated and used .