Topological rigidity of automorphism systems

This paper studies the automorphism actions of countable discrete groups on the étale equivalence relations on compact metric spaces. First, the notion of continuous strong orbit equivalence for automorphism systems is introduced and it is proved that two conjugate automorphism systems are continuously strong orbit equivalent. Conversely,under the conditions of essentially freeness and discrete groups being amenable and torsion-free, two continuously strong orbit equivalent automorphism systems satisfying the rigid condition are conjugate.