Inverse semigroups associated to subshifts

The dynamics of a one-sided subshift $\mathsf{X}$ can be modeled by a set of partially defined bijections. From this data we define an inverse semigroup ${\mathcal{S}}_{\mathsf{X}}$ and show that it has many interesting properties. We prove that the Carlsen-Matsumoto C*-algebra ${\mathcal{O}}_{\mathsf{X}}$ associated to $\mathsf{X}$ is canonically isomorphic to Exel’s tight C*-algebra of ${\mathcal{S}}_{\mathsf{X}}$. As one consequence, we obtain that ${\mathcal{O}}_{\mathsf{X}}$ can be written as a partial crossed product of a commutative C*-algebra by a countable group.