Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Finite-rank Bratteli-Vershik homeomorphisms are expansive

Author: Takashi Shimomura
Journal: Proc. Amer. Math. Soc. 145 (2017), 4353-4362
MSC (2010): Primary 37B05, 37B10
Published electronically: April 7, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Downarowicz and Maass (2008) have shown that every Cantor minimal homeomorphism with finite topological rank $ K > 1$ is expansive. Bezuglyi, Kwiatkowski, and Medynets (2009) extended the result to non-minimal aperiodic cases. In this paper, we show that all finite-rank zero-dimensional systems are expansive or have infinite odometer systems; this is an extension of the two aforementioned results. Nevertheless, the methods follow similar approaches.

References [Enhancements On Off] (What's this?)

  • [AK] Ethan Akin and Sergiĭ Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003), no. 4, 1421-1433. MR 1986303,
  • [BKM] S. Bezuglyi, J. Kwiatkowski, and K. Medynets, Aperiodic substitution systems and their Bratteli diagrams, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 37-72. MR 2470626,
  • [D] Alexandre I. Danilenko, Strong orbit equivalence of locally compact Cantor minimal systems, Internat. J. Math. 12 (2001), no. 1, 113-123. MR 1812067,
  • [DM] Tomasz Downarowicz and Alejandro Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 739-747. MR 2422014,
  • [HPS] Richard H. Herman, Ian F. Putnam, and Christian F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827-864. MR 1194074,
  • [M] Hiroki Matui, Topological orbit equivalence of locally compact Cantor minimal systems, Ergodic Theory Dynam. Systems 22 (2002), no. 6, 1871-1903. MR 1944409,
  • [S] T. Shimomura, A Bratteli-Vershik representation for all zero-dimensional systems, arXiv:1603.03940, submitted, 2016. Available at

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37B05, 37B10

Retrieve articles in all journals with MSC (2010): 37B05, 37B10

Additional Information

Takashi Shimomura
Affiliation: Nagoya University of Economics, Uchikubo 61-1, Inuyama 484-8504, Japan

Keywords: Rank, Bratteli diagram, periodic, expansive
Received by editor(s): June 29, 2016
Received by editor(s) in revised form: October 25, 2016
Published electronically: April 7, 2017
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society