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

Abstract

For a homeomorphism $T:X\to X$ 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.

29 pages