Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ORBIT EQUIVALENCE ON SELF-SIMILAR GROUPS AND THEIR C∗-ALGEBRAS

  • Yi, Inhyeop (Department of Mathematics Education Ewha Womans University)
  • Received : 2019.01.30
  • Accepted : 2019.05.23
  • Published : 2020.03.01
Download PDF
⟨ Previous Next ⟩

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

References

  1. M. Boyle and J. Tomiyama, Bounded topological orbit equivalence and C*-algebras, J. Math. Soc. Japan 50 (1998), no. 2, 317-329. https://doi.org/10.2969/jmsj/05020317
  2. N. Brownlowe, T. M. Carlsen, and M. F. Whittaker, Graph algebras and orbit equiva-lence, Ergodic Theory Dynam. Systems 37 (2017), no. 2, 389-417. https://doi.org/10.1017/etds.2015.52
  3. T. Carlsen, S. Eilers, E. Ortega, and G. Restor, Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids, J. Math. Anal. Appl. 469 (2019), no. 2, 1088-1110. https://doi.org/10.1016/j.jmaa.2018.09.056
  4. T. M. Carlsen and M. L. Winger, Orbit equivalence of graphs and isomorphism of graph groupoids, Math. Scand. 123 (2018), no. 2, 239-248. https://doi.org/10.7146/math.scand.a-105087
  5. V. Deaconu, Groupoids associated with endomorphisms, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1779-1786. https://doi.org/10.2307/2154972
  6. T. Giordano, I. F. Putnam, and C. F. Skau, Topological orbit equivalence and C*-crossed products, J. Reine Angew. Math. 469 (1995), 51-111.
  7. A. Kumjian and D. Pask, Actions of Zk associated to higher rank graphs, Ergodic Theory Dynam. Systems 23 (2003), no. 4, 1153-1172. https://doi.org/10.1017/S0143385702001670
  8. X. Li, Continuous orbit equivalence rigidity, Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1543-1563. https://doi.org/10.1017/etds.2016.98
  9. K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger alge-bras, Pacific J. Math. 246 (2010), no. 1, 199-225. https://doi.org/10.2140/pjm.2010.246.199
  10. K. Matsumoto, Asymptotic continuous orbit equivalence of Smale spaces and Ruelle algebras, to appear in Canad. J. Math.
  11. K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Kyoto J. Math. 54 (2014), no. 4, 863-877. https://doi.org/10.1215/21562261-2801849
  12. H. Matui, Homology and topological full groups of etale groupoids on totally disconnected spaces, Proc. Lond. Math. Soc. (3) 104 (2012), no. 1, 27-56. https://doi.org/10.1112/plms/pdr029
  13. P. S. Muhly, J. N. Renault, and D. P. Williams, Equivalence and isomorphism for groupoid C*-algebras, J. Operator Theory 17 (1987), no. 1, 3-22.
  14. V. Nekrashevych, Cuntz-Pimsner algebras of group actions, J. Operator Theory 52 (2004), no. 2, 223-249.
  15. V. Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005. https://doi.org/10.1090/surv/117
  16. V. Nekrashevych, C*-algebras and self-similar groups, J. Reine Angew. Math. 630 (2009), 59-123. https://doi.org/10.1515/CRELLE.2009.035
  17. A. L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics 170, Birkhauser Boston, Inc., Boston, MA, 1999. https://doi.org/10.1007/978-1-4612-1774-9
  18. I. F. Putnam and J. Spielberg, The structure of C*-algebras associated with hyperbolic dynamical systems, J. Funct. Anal. 163 (1999), no. 2, 279-299. https://doi.org/10.1006/jfan.1998.3379
  19. J. Renault, A groupoid approach to C*-algebras, Lecture Notes in Mathematics 793, Springer, Berlin, 1980.
  20. I. Yi, Smale spaces from self-similar graph actions, Rocky Mountain J. Math. 48 (2018), no. 4, 1359-1384. https://doi.org/10.1216/RMJ-2018-48-4-1359