Flow Equivalence of Reducible Shifts of Finite Type and Cuntz-krieger Algebras

Using ltered Bowen-Franks group BF ? (A), we classify reducible shifts of nite type A with nite BF(A) up to ow equivalence. Using this result and some new development in C-algebras due to RRrdam and Cuntz, we classify non-simple Cuntz-Krieger algebras O A with nite K 0 (O A) up to stable isomorphism by the ltered K 0-group K ? 0 (O A); up to unital isomorphism by K ? 0 (O A) together with the distinguished element 1 A ] in K 0 (O A). These complete invariants require that BF(A) or K 0 (O A) be nite. 0. Introduction Shifts of nite type (SFT's), also called topological Markov chains, are the fundamental building blocks of symbolic dynamics. Up to topological conjugacy, a shift of nite type can always be represented as a graph shift A deened by a non-negative integral square matrix A 2 M n (Z +). Two SFT's A and B (or their adjacency matrices A and B) are said to be ow equivalent (FE) if A and B have topologically equivalent suspension ows. The suspension ows of SFT's have important applications in studying hyperbolic ows (see B]). Parry and Sullivan PS] began the study of the classiication of SFT's up to ow equivalence. They showed that a ow equivalence of SFT's is generated by topological conjugacy and a sort of \time delay," or equivalently, by strong shift equivalence and the \Parry-Sullivan expansion" in a matrix version. From this they obtained the rst computable matrix invariant det(I?A) of ow equivalence. Bowen and Franks showed in BF] another important FE-invariant called the Bowen-Franks group (BF-group) of A. Franks showed later in F] that two matrix invariants above completely characterize ow equivalence of irreducible SFT's whose adjacency matrices are not permutations. Notice that BF(A) actually determines det(I ? A) up to its sign: 0; +; or ?, which we will denote by sgn(A).