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Together with (BH), this paper gives an alternate development of the Huang classiflcation of shifts of flnite type up to ∞ow equivalence, and provides additional functorial information, used to analyze the action of the mapping class group of the mapping torus of a shift of flnite type on the \isotopy futures" group, which is introduced here. For a shift of flnite type aeA, this group is isomorphic to the Bowen-Franks group cok(IiA). The action on the isotopy futures group of a subshift is the ∞ow equivalence analogue of the dimension group representation.

Flow equivalence of shifts of finite type via positive factorizations