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
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