A DIMENSION GROUP FOR LOCAL HOMEOMORPHISMS AND ENDOMORPHISMS OF ONESIDED SHIFTS OF FINITE-TYPE

We attach to a local homeomorphism S of a compact zero dimensional met rizable space X two abstract ordered groups, the images group G(S) and the preimages group if(S). The images group is a dimension group with a distinguished order unit. We use these groups to study commuting lo cal homeomorphisms. When S is a onesided mixing shift of finite type ( SFT), G(S) is isomorphic to the direct limit group of a defining matri x, and a onesided SFT commuting with S is determined up to eventual co njugacy (but not conjugacy) by its action on G(S). We obtain complete and computable invariants for the eventual conjugacy classes of onesid ed SFTs, and show commuting onesided SFTs have a common measure of max imal entropy. This places severe constraints on the existence of commu ting presentations of onesided SFTs. There are applications to cellula r automata and links to Ruelle's thermodynamic formalism.