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)

Request Permissions   Purchase Content 
 
 

 

Scrambled sets in shift spaces on a countable alphabet


Authors: Brian E. Raines and Tyler Underwood
Journal: Proc. Amer. Math. Soc. 144 (2016), 217-224
MSC (2010): Primary 37B10, 37B20, 37D40, 54H20
DOI: https://doi.org/10.1090/proc/12690
Published electronically: June 24, 2015
MathSciNet review: 3415590
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we characterize the shift spaces which have Li-Yorke chaos (an uncountable scrambled set). We focus primarily on shifts over a countably infinite alphabet. We represent them as either edge-shifts on an infinite graph (the subshift of finite type case) or as labelled edge-shifts on an infinite graph (the sofic shift case). We show in the setting of a subshift of finite type on a shift over a countable alphabet that the shift space has Li-Yorke chaos if, and only if, it has a single scrambled pair, and in this case the scrambled set is closed and perfect (but not necessarily compact). We give an example of a sofic shift over an infinite alphabet which has a single scrambled pair but does not have Li-Yorke chaos.


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

  • [1] Bruce P. Kitchens, Symbolic dynamics, Universitext, Springer-Verlag, Berlin, 1998. One-sided, two-sided and countable state Markov shifts. MR 1484730 (98k:58079)
  • [2] M. Kuchta and J. Smítal, Two-point scrambled set implies chaos, European Conference on Iteration Theory (Caldes de Malavella, 1987), World Sci. Publ., Teaneck, NJ, 1989, pp. 427-430. MR 1085314 (91j:58112)
  • [3] Milan Kuchta, Characterization of chaos for continuous maps of the circle, Comment. Math. Univ. Carolin. 31 (1990), no. 2, 383-390. MR 1077909 (92f:58112)
  • [4] Tien Yien Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985-992. MR 0385028 (52 #5898)
  • [5] Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092 (97a:58050)
  • [6] A. R. D. Mathias, Delays, recurrence and ordinals, Proc. London Math. Soc. (3) 82 (2001), no. 2, 257-298. MR 1806873 (2001j:03087), https://doi.org/10.1112/S0024611501012734
  • [7] A. R. D. Mathias, Analytic sets under attack, Math. Proc. Cambridge Philos. Soc. 138 (2005), no. 3, 465-485. MR 2138574 (2006a:54026), https://doi.org/10.1017/S0305004104008254
  • [8] Piotr Oprocha and Paweł Wilczyński, Shift spaces and distributional chaos, Chaos Solitons Fractals 31 (2007), no. 2, 347-355. MR 2259760 (2007i:37019), https://doi.org/10.1016/j.chaos.2005.09.069
  • [9] Sylvie Ruette and L'ubomír Snoha, For graph maps, one scrambled pair implies Li-Yorke chaos, Proc. Amer. Math. Soc. 142 (2014), no. 6, 2087-2100. MR 3182027, https://doi.org/10.1090/S0002-9939-2014-11937-X

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37B10, 37B20, 37D40, 54H20

Retrieve articles in all journals with MSC (2010): 37B10, 37B20, 37D40, 54H20


Additional Information

Brian E. Raines
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798–7328
Email: brian_raines@baylor.edu

Tyler Underwood
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798–7328
Address at time of publication: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
Email: tyler_underwood@umail.ucsb.edu

DOI: https://doi.org/10.1090/proc/12690
Keywords: Uncountable chaotic sets, subshifts, transition graph, chaotic pair, Baire space, Li-Yorke chaos, edge-shift, sofic shift, shift of finite type
Received by editor(s): July 7, 2014
Received by editor(s) in revised form: December 2, 2014
Published electronically: June 24, 2015
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society