Automorphisms of the dimension group and gyration numbers

Abstract

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. O. INTRODUCTIONMethods of symbolic dynamics arise in such fields as information theory, ergodic theory and dynamical systems, cellular automata theory, and statistical mechanics.Among the simplest symbolic systems are the subshifts of finite type O"A: X A -. X A coming from square matrices A with entries lying in the nonnegative integers A+.Consult, for example, [DGS, E, Fr, PT].In the setting of smooth dynamical systems, the matrix A typically comes from a Markov partition on the zero-dimensional part of the nonwandering set of a diffeomorphism.There are many ways of choosing Markov partitions, and this produces different matrices A so that the various shifts O"A: X A -. X A are topologically