Continuous orbit equivalence of groupoid actions

In this paper, I studied continuous actions of étale groupoids on compact Hausdorff spaces and the associated operator algebras. I introduced notions of conjugacy and continuous orbit equivalence for such groupoid actions and characterized them in terms of the corresponding transformation groupoids and their reduced $C^{\ast }$-algebras. In particular, I proved that two topologically free actions are continuously orbit equivalent if and only if their associated transformation groupoid $C^{\ast }$-algebras are isomorphic, and if and only if there exists a $C^{\ast }$-algebra isomorphism preserving the canonical Cartan subalgebras between the corresponding reduced $C^{\ast }$-algebras of these transformation groupoids.