Local P entropy and stabilized automorphism groups of subshifts

For a homeomorphism T: X→ X of a compact metric space X, the stabilized automorphism group Aut ^(∞)(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 P entropy. We show that when (X, T) is a non-trivial mixing shift of finite type, the local P entropy of the group Aut ^(∞)(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.