Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2020; 57(2): 383-399

Online first article July 17, 2019      Printed March 1, 2020

https://doi.org/10.4134/JKMS.j190090

Copyright © The Korean Mathematical Society.

Orbit equivalence on self-similar groups and their $C^*$-algebras

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 equivalence class of Cuntz-Pimsner groupoids and the stably isomorphism class of 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

Supported by: This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07048313).

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