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 Orbit equivalence on self-similar groups and their $C^*$-algebras J. Korean Math. Soc.Published online July 17, 2019 Inhyeop Yi Ewha Womans University Abstract : Following Matsumoto's definition of continuous orbit equivalence for one-sided subshifts of finite type, we introduce the notion of orbit equivalence to canonically associated dynamical systems, called the limit dynamical systems, of self-similar groups. We show that the limit dynamical systems of two self-similar groups are orbit equivalent if and only if their associated Deaconu groupoids are isomorphic as topological groupoids. We also show that the Cuntz-Pimsner groupoids and the Cuntz-Pimsner algebras of self-similar groups are invariants for orbit equivalence of limit dynamical systems. Keywords : self-similar group, limit dynamical system, orbit equivalence MSC numbers : 37B10, 46L55, 37A55 Full-Text :