C*-algebras, groupoids and covers of shift spaces

To every one-sided shift space we associate a cover, a groupoid and a C*-algebra. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between shift spaces in terms of isomorphism of the associated groupoids, and diagonal-preserving *-isomorphism of the associated C*-algebras. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces in terms of isomorphism of the associated stabilized groupoids, and diagonal-preserving *-isomorphism of the associated stabilized C*-algebras. 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 such a one-sided shift space X from the pair consisting of it C*-algebra together with the canonical copy of C(X), and the flow equivalence class of the two-sided shift space of X from the pair consisting of the stabilization of the C*-algebra of X and the tensor product of C(X) and the algebra of diagonal compact operators. In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.