Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T03:04:51.832Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal flows with arbitrary centralizer

Part of: Noncompact transformation groups Topological dynamics

Published online by Cambridge University Press:  29 December 2020

ANDY ZUCKER Open the ORCID record for ANDY ZUCKER [Opens in a new window]
ANDY ZUCKER*
Affiliation:
Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA92093, USA
*
e-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a G-flow X, let $\mathrm{Aut}(G, X)$ , or simply $\mathrm{Aut}(X)$ , denote the group of homeomorphisms of X which commute with the G action. We show that for any pair of countable groups G and H with G infinite, there is a minimal, free, Cantor G-flow X so that H embeds into $\mathrm{Aut}(X)$ . This generalizes results of [2, 7].

Keywords

minimal flowcentralizerblueprintstrongly irreducible shift

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37B10: Symbolic dynamics 22F50: Groups as automorphisms of other structures
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 310 - 320
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Boyle, M., Lind, D. and Rudolph, D.. The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306(1) (1988), 71–114.CrossRefGoogle Scholar
Cortez, M. I. and Petite, S.. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete Contin. Dyn. Syst. A 40(5) (2020), 2891–2901.CrossRefGoogle Scholar
Cyr, V. and Kra, B.. The automorphism group of a shift of subquadratic growth. Proc. Amer. Math. Soc. 144(2) (2016), 613–621.CrossRefGoogle Scholar
Donoso, S., Durand, F., Maass, A. and Petite, S.. On automorphism groups of low complexity subshifts. Ergod. Th. & Dynam. Sys. 36 (2016), 64–95.CrossRefGoogle Scholar
Frisch, J., Schlank, T., and Tamuz, O.. Normal amenable subgroups of the automorphism group of the full shift. Ergod. Th. & Dynam. Sys. 39(5) (2019), 1290–1298.CrossRefGoogle Scholar
Gao, S., Jackson, S. and Seward, B.. Group Colorings and Bernoulli Subflows (Memoirs of the American Mathematical Society, 241), No. 1141 (2 of 4). American Mathematical Society, Providence, RI, 2016.Google Scholar
Glasner, E., Tsankov, T., Weiss, B. and Zucker, A.. Bernoulli disjointness. Duke Math. J., to appear, arXiv:1901.03406.Google Scholar
Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 (1969), 320–375.CrossRefGoogle Scholar
Hjorth, G. and Molberg, M.. Free continuous actions on zero-dimensional spaces. Topology Appl. 153(7) (2006), 1116–1131.CrossRefGoogle Scholar