Hyperspatiality for isomorphisms of stabilized automorphism groups of shifts of finite type

Given a homeomorphism T X X of a compact metric space X, the stabilized automorphism group Aut^ (T) of the system (X, T) is the group of self-homeomorphisms of X which commute with some power of T. We study the question of spatiality for stabilized automorphism groups of shifts of finite type. We prove that any isomorphism Aut^ (₌) Aut^ (₍) between stabilized automorphism groups of full shifts is spatially induced by a homeomorphism between respective stabilized spaces of chain recurrent subshifts. This spatialization in particular gives a bijection between the sets of periodic points which intertwines some powers of the shifts, and this bijection recovers the isomorphism at the level of the faithful actions on the sets of periodic points. We also prove that the outer automorphism group of Aut^ (₍) is uncountable, and deduce several other properties of Aut^ (₍) using the spatiality results.