  To every one-sided shift space $\mathsf{X}$ we associate a cover
$\tilde{\mathsf{X}}$, a groupoid $\mathcal{G}_{\mathsf{X}}$ and a
$\mathrm{C^*}$-algebra $\mathcal{O}_{\mathsf{X}}$. We characterize one-sided
conjugacy, eventual conjugacy and (stabilizer preserving) continuous orbit
equivalence between $\mathsf{X}$ and $\mathsf{Y}$ in terms of isomorphism of
$\mathcal{G}_{\mathsf{X}}$ and $\mathcal{G}_{\mathsf{Y}}$, and diagonal
preserving $^*$-isomorphism of $\mathcal{O}_{\mathsf{X}}$ and
$\mathcal{O}_{\mathsf{Y}}$. We also characterize two-sided conjugacy and flow
equivalence of the associated two-sided shift spaces $\Lambda_{\mathsf{X}}$ and
$\Lambda_{\mathsf{Y}}$ in terms of isomorphism of the stabilized groupoids
$\mathcal{G}_{\mathsf{X}}\times \mathcal{R}$ and
$\mathcal{G}_{\mathsf{Y}}\times \mathcal{R}$, and diagonal preserving
$^*$-isomorphism of the stabilized $\mathrm{C^*}$-algebras
$\mathcal{O}_{\mathsf{X}}\otimes \mathbb{K}$ and
$\mathcal{O}_{\mathsf{Y}}\otimes \mathbb{K}$. Our strategy is to lift relations
on the shift spaces to similar relations on the covers. Restricting to the
class of sofic shifts whose groupoids are effective, we show that it is
possible to recover the continuous orbit equivalence class of $\mathsf{X}$ from
the pair $(\mathcal{O}_{\mathsf{X}}, C(\mathsf{X}))$, and the flow equivalence
class of $\Lambda_{\mathsf{X}}$ from the pair $(\mathcal{O}_{\mathsf{X}}\otimes
\mathbb{K}, C(\mathsf{X})\otimes c_0)$. In particular, continuous orbit
equivalence implies flow equivalence for this class of shift spaces.

Operator Algebras