Automorphisms of the Dimension Group and Gyration Numbers

Let Aut(O"A) denote the group of automorphisms of a subshift of finite type (XA'O"A) built from a primitive matrix A. We show that the sign-gyrationcompatibility-condition homomorphism SGCC A, m defined on Aut( 0" A) factors through the group Aut(s A) of automorphisms of the dimension group. This is used to find a mixing subshift of finite type with a permutation of fixed points that cannot be lifted to an automorphism of the shift. We also give an example of a mixing subshift of finite type where the dimension group representation is not surjective. This example is used in [KR3] to give examples of subshifts of finite type (reducible, with two mixing components) that are shift equivalent but not strong shift equivalent over the nonnegative integers.