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

Abstract

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