Local $${\mathcal {P}}$$ entropy and stabilized automorphism groups of subshifts

For a homeomorphism $$T :X \rightarrow X$$ of compact metric space X, the stabilized automorphism group $${\text {Aut}}^{(\infty )}(T)$$ consists all self-homeomorphisms X which commute with some power T. Motivated by study these groups in context shifts finite type, we introduce certain entropy for called local $${\mathcal {P}}$$ entropy. We show that when (X, T) is non-trivial mixing shift determined topological T). use this to give complete classification isomorphism type full shifts.