Automorphisms of Bowen–Franks groups of shifts of finite type

Abstract

There are four Bowen–Franks groups associated to each shift of finite type. For an irreducible shift of finite type, we show that a 4-tuple of automorphisms corresponding to the four Bowen–Franks groups can be induced simultaneously by a specific path of flow equivalence from the shift to itself, if and only if it is F-compatible. The F-compatibleness defined in this paper describes completely the intrinsic relations among the four automorphisms induced by a flow equivalence. This result is one of the key ingredients in classifying reducible shifts of finite type up to flow equivalence. In the mean time, it also discloses a new and sharp difference between the invariants of flow equivalence for an irreducible shift of finite type, and the invariants of stable isomorphism for the associated simple Cuntz–Krieger algebra.