Asymptotic continuous orbit equivalence of expansive systems

We introduce notions of asymptotic continuous orbit equivalence and (strongly) asymptotic conjugacy for expansive systems, and characterize them in terms of the transformation groupoids, the principal groupoids coming from the local conjugacy relations and the semi-direct product groupoids of the principal groupoids by the canonical group actions, together with their associated reduced groupoid C*-algebras. In particular, we show that two asymptotically essentially free expansive systems are asymptotically continuous orbit equivalent if and only if the associated semi-direct product groupoids are topologically isomorphic if and only if there exists a C*-algebra isomorphism preserving the canonical Cartan subalgebras between the corresponding reduced C*-algebras of these semi-direct product groupoids.