Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CONTINUOUS ORBIT EQUIVALENCES ON SELF-SIMILAR GROUPS

  • Yi, Inhyeop (Department of Mathematics Education Ewha Womans University)
  • Received : 2020.01.18
  • Accepted : 2020.10.16
  • Published : 2021.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

For pseudo-free and recurrent self-similar groups, we show that continuous orbit equivalence of inverse semigroup partial actions implies continuous orbit equivalence of group actions. Conversely, if group actions are continuous orbit equivalent, and the induced homeomorphism commutes with the shift maps on their groupoids, we obtain continuous orbit equivalence of inverse semigroup partial actions.

Keywords

References

  1. N. Brownlowe, T. M. Carlsen, and M. F. Whittaker, Graph algebras and orbit equivalence, Ergodic Theory Dynam. Systems 37 (2017), no. 2, 389-417. https://doi.org/10.1017/etds.2015.52
  2. A. Buss and R. Exel, Inverse semigroup expansions and their actions on C*-algebras, Illinois J. Math. 56 (2012), no. 4, 1185-1212. http://projecteuclid.org/euclid.ijm/1399395828 https://doi.org/10.1215/ijm/1399395828
  3. 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
  4. L. G. Cordeiro and V. Beuter, The dynamics of partial inverse semigroup actions, J. Pure Appl. Algebra 224 (2020), no. 3, 917-957. https://doi.org/10.1016/j.jpaa.2019.06.001
  5. M. Dokuchaev and R. Exel, Partial actions and subshifts, J. Funct. Anal. 272 (2017), no. 12, 5038-5106. https://doi.org/10.1016/j.jfa.2017.02.020
  6. R. Exel, Inverse semigroups and combinatorial C*-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191-313. https://doi.org/10.1007/s00574-008-0080-7
  7. R. Exel, Partial dynamical systems, Fell bundles and applications, Mathematical Surveys and Monographs, 224, American Mathematical Society, Providence, RI, 2017. https://doi.org/10.1090/surv/224
  8. R. Exel, Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras, Adv. Math. 306 (2017), 1046-1129. https://doi.org/10.1016/j.aim.2016.10.030
  9. M. V. Lawson, Inverse Semigroups, the Theory of Partial Symmetries, World Scientific, 1998.
  10. X. Li, Partial transformation groupoids attached to graphs and semigroups, Int. Math. Res. Not. IMRN 2017 (2017), no. 17, 5233-5259. https://doi.org/10.1093/imrn/rnw166
  11. X. Li, Continuous orbit equivalence rigidity, Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1543-1563. https://doi.org/10.1017/etds.2016.98
  12. K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Pacific J. Math. 246 (2010), no. 1, 199-225. https://doi.org/10.2140/pjm.2010.246.199
  13. K. Matsumoto, Asymptotic continuous orbit equivalence of smale spaces and Ruelle algebras, Canad. J. Math. 71 (2019), no. 5, 1243-1296. https://doi.org/10.4153/cjm-2018-012-x
  14. V. V. Nekrashevych, Cuntz-Pimsner algebras of group actions, J. Operator Theory 52 (2004), no. 2, 223-249.
  15. V. 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. 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. J. Renault, A groupoid approach to C*-algebras, Lecture Notes in Mathematics, 793, Springer, Berlin, 1980.
  19. C. Starling, Inverse semigroups associated to subshifts, J. Algebra 463 (2016), 211-233. https://doi.org/10.1016/j.jalgebra.2016.06.014
  20. I. Yi, Orbit equivalence on self-similar groups and their C*-algebras, J. Korean Math. Soc. 57 (2020), no. 2, 383-399. https://doi.org/10.4134/JKMS.j190090