Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 3 (2019), 395-410.

Toeplitz subshifts with trivial centralizers and positive entropy

Kostya Medynets and James P. Talisse

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Abstract

Given a dynamical system ( X , G ) , the centralizer C ( G ) denotes the group of all homeomorphisms of X which commute with the action of G . This group is sometimes called the automorphism group of the dynamical system ( X , G ) . We generalize the construction of Bułatek and Kwiatkowski (1992) to d -Toeplitz systems by identifying a class of d -Toeplitz systems that have trivial centralizers. We show that this class of d -Toeplitz systems with trivial centralizers contains systems with positive topological entropy.

Article information

Source
Involve, Volume 12, Number 3 (2019), 395-410.

Dates
Received: 3 May 2017
Revised: 13 June 2017
Accepted: 25 June 2018
First available in Project Euclid: 5 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.involve/1549335629

Digital Object Identifier
doi:10.2140/involve.2019.12.395

Mathematical Reviews number (MathSciNet)
MR3905337

Zentralblatt MATH identifier
07033138

Subjects
Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37B40: Topological entropy 37B50: Multi-dimensional shifts of finite type, tiling dynamics

Keywords
topological dynamics symbolic dynamics automorphism group centralizer topological entropy

Citation

Medynets, Kostya; Talisse, James P. Toeplitz subshifts with trivial centralizers and positive entropy. Involve 12 (2019), no. 3, 395--410. doi:10.2140/involve.2019.12.395. https://projecteuclid.org/euclid.involve/1549335629


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References

  • J. Auslander, “Endomorphisms of minimal sets”, Duke Math. J. 30:4 (1963), 605–614.
  • J. Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies 153, North-Holland, Amsterdam, 1988.
  • M. Boyle, D. Lind, and D. Rudolph, “The automorphism group of a shift of finite type”, Trans. Amer. Math. Soc. 306:1 (1988), 71–114.
  • L. E. J. Brouwer, “On the structure of perfect sets of points”, KNAW, Proc. 12 (1910), 785–794.
  • W. Bułatek and J. Kwiatkowski, “The topological centralizers of Toeplitz flows and their $Z_2$-extensions”, Publ. Mat. 34:1 (1990), 45–65.
  • W. Bułatek and J. Kwiatkowski, “Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers”, Studia Math. 103:2 (1992), 133–142.
  • M. I. Cortez, “$\mathbb{Z}^d$ Toeplitz arrays”, Discrete Contin. Dyn. Syst. 15:3 (2006), 859–881.
  • M. I. Cortez and S. Petite, “$G$-odometers and their almost one-to-one extensions”, J. Lond. Math. Soc. $(2)$ 78:1 (2008), 1–20.
  • V. Cyr and B. Kra, “The automorphism group of a shift of linear growth: beyond transitivity”, Forum Math. Sigma 3 (2015), art. id. e5.
  • V. Cyr and B. Kra, “The automorphism group of a minimal shift of stretched exponential growth”, J. Mod. Dyn. 10 (2016), 483–495.
  • V. Cyr and B. Kra, “The automorphism group of a shift of subquadratic growth”, Proc. Amer. Math. Soc. 144:2 (2016), 613–621.
  • S. Donoso, F. Durand, A. Maass, and S. Petite, “On automorphism groups of low complexity subshifts”, Ergodic Theory Dynam. Systems 36:1 (2016), 64–95.
  • S. Donoso, F. Durand, A. Maass, and S. Petite, “On automorphism groups of Toeplitz subshifts”, Discrete Anal. (2017), art. id. 11.
  • T. Downarowicz, “Survey of odometers and Toeplitz flows”, pp. 7–37 in Algebraic and topological dynamics (Bonn, 2004), edited by S. Kolyada et al., Contemp. Math. 385, Amer. Math. Soc., Providence, RI, 2005.
  • G. A. Hedlund, “Endomorphisms and automorphisms of the shift dynamical system”, Math. Systems Theory 3 (1969), 320–375.
  • E. Hewitt and K. A. Ross, Abstract harmonic analysis, I: Structure of topological groups, integration theory, group representations, 2nd ed., Grundlehren der Mathematischen Wissenschaften 115, Springer, 1979.
  • M. Hochman, “On the automorphism groups of multidimensional shifts of finite type”, Ergodic Theory Dynam. Systems 30:3 (2010), 809–840.
  • K. Jacobs and M. Keane, “$0$-$1$-sequences of Toeplitz type”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13 (1969), 123–131.
  • B. P. Kitchens, Symbolic dynamics: one-sided, two-sided and countable state Markov shifts, Springer, 1998.
  • F. Krieger, “Toeplitz subshifts and odometers for residually finite groups”, pp. 147–161 in École de Théorie Ergodique (Marseilles, 2006), edited by Y. Lacroix et al., Sémin. Congr. 20, Soc. Math. France, Paris, 2010.
  • N. G. Markley, “Substitution-like minimal sets”, Israel J. Math. 22:3-4 (1975), 332–353.
  • N. G. Markley and M. E. Paul, “Almost automorphic symbolic minimal sets without unique ergodicity”, Israel J. Math. 34:3 (1979), 259–272.
  • J. Olli, “Endomorphisms of Sturmian systems and the discrete chair substitution tiling system”, Discrete Contin. Dyn. Syst. 33:9 (2013), 4173–4186.
  • W. A. Veech, “Point-distal flows”, Amer. J. Math. 92 (1970), 205–242.