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 intro-duced 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.
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