WITH AN APPENDIX ON ALMOST FLOW EQUIVALENCE JOINTLY WITH

Abstract. We survey recent results in classifying embeddings of suspensions of shifts of nite type as hyperbolic basic sets of ows up to twistwise ow equivalence. The problem has been reduced to matrix equivalence over SL(ZZ=2). But more is true: G- ow equivalence of G-SFTs, for any nite group G is has a similar algebraic formalization. The appendix proves a new result: all nontrivial faithful irreducible G-SFTs are almost ow equivalent. 1. introduction Square matrices of nonnegative integers are ow equivalent if the suspensions of their corresponding shifts of nite type (SFTs) are topo-logically equivalent. (Denitions are in Section 2.) A complete set of easily computed invariants determines ow equivalence of nontrivial irreducible square nonnegative matrices [PS, BF, F2]. When the as-sumption of irreducibility is dropped the classication of matrices up to ow equivalence becomes harder but has been solved; see [H1, H2, H3] or [H4, BH]. In [Su2] the concept of twistwise ow equivalence was introduced to describe the orientability of the stable manifolds of the orbits of a suspended and embedded SFT. The twist matrices are square matrices over the semi-group ring