Inverse semigroups associated to subshifts

The dynamics of a one-sided subshift XX can be modeled by a set of partially defined bijections. From this data we define an inverse semigroup SXSX and show that it has many interesting properties. We prove that the Carlsen–Matsumoto C*-algebra OXOX associated to XX is canonically isomorphic to Exel's tight C*-algebra of SXSX. As one consequence, we obtain that OXOX can be written as a partial crossed product of a commutative C*-algebra by a countable group.