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May  2020, 40(5): 2891-2901. doi: 10.3934/dcds.2020153

Realization of big centralizers of minimal aperiodic actions on the Cantor set

María Isabel Cortez 1, and  Samuel Petite 2,

1. 

Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Avenida Libertador Bernardo O'Higgins 3363, Santiago, Chile
2. 

Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS-UMR 7352, Université de Picardie Jules Verne, 33, rue Saint Leu 80039 Amiens Cedex 1, France

Received  August 2019 Revised  November 2019 Published  March 2020

Fund Project: The first author was supported by proyecto Fondecyt Regular No. 1190538.This research was supported through the cooperation project MathAmSud DCS 38889 TM

In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal $ \mathbb Z $ action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group $ G $, the centralizer of any minimal aperiodic $ G $-action on the Cantor set is a subgroup of the centralizer of a minimal $ \mathbb Z $-action.

Keywords: Automorphism group, minimality, Cantor dynamical system, realization, symbolic dynamics.
Mathematics Subject Classification: Primary: 37B05; Secondary: 22F50.
Citation: María Isabel Cortez, Samuel Petite. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2891-2901. doi: 10.3934/dcds.2020153
References:
[1]

N. Aubrun, S. Barbieri and S. Thomassé, Realization of aperiodic subshifts and uniform densities in groups, Groups Geom. Dyn., 13 (2019), 107-129.  doi: 10.4171/GGD/487.

[2]

M. Baake, J. A. G. Roberts and R. Yassawi, Reversing and extended symmetries of shift spaces, Discrete and Continuous Dynamical Systems, 38 (2018), 835-866.  doi: 10.3934/dcds.2018036.

[3]

M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114.  doi: 10.1090/S0002-9947-1988-0927684-2.

[4]

W. Bulatek and J. Kwiatkowski, Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers, Studia Math., 103 (1992), 133-142.  doi: 10.4064/sm-103-2-133-142.

[5]

T. Ceccherini-Silberstein and M. Coornaert, Cellular automata and groups, Cellular automata, Encycl. Complex. Syst. Sci., Springer, New York, (2018), 221–238.

[6]

M. I. Cortez and S. Petite, $G$-odometers and their almost one-to-one extensions, J. London Math. Soc. (2), 78 (2008), 1-20.  doi: 10.1112/jlms/jdn002.

[7]

E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes, Discrete Anal., (2016), 28 pp. doi: 10.19086/da.611.

[8]

V. Cyr and B. Kra, The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 144 (2016), 613-621.  doi: 10.1090/proc12719.

[9]

V. Cyr and B. Kra, The automorphism group of a shift of linear growth: Beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27 pp. doi: 10.1017/fms.2015.3.

[10]

V. Cyr and B. Kra, The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495.  doi: 10.3934/jmd.2016.10.483.

[11]

V. Cyr, J. Franks, B. Kra and S. Petite, Distortion and the automorphism group of a shift, J. Mod. Dyn., 13 (2018), 147-161.  doi: 10.3934/jmd.2018015.

[12]

S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of low complexity subshifts, Ergodic Theory and Dynam. Systems, 36 (2016), 64-95.  doi: 10.1017/etds.2015.70.

[13]

S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of Toeplitz subshifts, Discrete Anal., 2017 (2017), 19 pp.

[14]

F. Durand, N. Ormes and S. Petite, Self-induced systems, J. Anal. Math., 135 (2018), 725-756.  doi: 10.1007/s11854-018-0051-x.

[15]

T. Giordano, I. F. Putnam and C. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.  doi: 10.1007/BF02810689.

[16]

E. Glasner, T. Tsankov, B. Weiss and A. Zucker, Bernoulli disjointness, Preprint, arXiv: 1901.03406.

[17]

G. Hjorth and M. Molberg, Free continuous actions on zero-dimensional spaces, Topology Appl., 153 (2006), 1116-1131.  doi: 10.1016/j.topol.2005.03.003.

[18]

K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math., 178 (2013), 775-787.  doi: 10.4007/annals.2013.178.2.7.

[19]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24.  doi: 10.1007/BF02810577.

[20]

H. Matui, Some remarks on topological full groups of Cantor minimal systems, Internat. J. Math., 17 (2006), 231-251.  doi: 10.1142/S0129167X06003448.

[21]

K. Medynets, Reconstruction of orbits of Cantor systems from full groups, Bull. Lond. Math. Soc., 43 (2011), 1104-1110.  doi: 10.1112/blms/bdr045.

[22]

K. Medynets and J. P. Talisse, Toeplitz subshifts with trivial centralizers and positive entropy, Involve, 12 (2019), 395-410.  doi: 10.2140/involve.2019.12.395.

[23]

V. Salo and M. Schraudner, Automorphism groups of subshifts via group extensions, preprint.

[24]

D. Witte, Arithmetic groups of higher Q-rank cannot act on $1$-manifolds, Proc. Amer. Math. Soc., 122 (1994), 333-340.  doi: 10.2307/2161021.

show all references

References:
[1]

N. Aubrun, S. Barbieri and S. Thomassé, Realization of aperiodic subshifts and uniform densities in groups, Groups Geom. Dyn., 13 (2019), 107-129.  doi: 10.4171/GGD/487.

[2]

M. Baake, J. A. G. Roberts and R. Yassawi, Reversing and extended symmetries of shift spaces, Discrete and Continuous Dynamical Systems, 38 (2018), 835-866.  doi: 10.3934/dcds.2018036.

[3]

M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114.  doi: 10.1090/S0002-9947-1988-0927684-2.

[4]

W. Bulatek and J. Kwiatkowski, Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers, Studia Math., 103 (1992), 133-142.  doi: 10.4064/sm-103-2-133-142.

[5]

T. Ceccherini-Silberstein and M. Coornaert, Cellular automata and groups, Cellular automata, Encycl. Complex. Syst. Sci., Springer, New York, (2018), 221–238.

[6]

M. I. Cortez and S. Petite, $G$-odometers and their almost one-to-one extensions, J. London Math. Soc. (2), 78 (2008), 1-20.  doi: 10.1112/jlms/jdn002.

[7]

E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes, Discrete Anal., (2016), 28 pp. doi: 10.19086/da.611.

[8]

V. Cyr and B. Kra, The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 144 (2016), 613-621.  doi: 10.1090/proc12719.

[9]

V. Cyr and B. Kra, The automorphism group of a shift of linear growth: Beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27 pp. doi: 10.1017/fms.2015.3.

[10]

V. Cyr and B. Kra, The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495.  doi: 10.3934/jmd.2016.10.483.

[11]

V. Cyr, J. Franks, B. Kra and S. Petite, Distortion and the automorphism group of a shift, J. Mod. Dyn., 13 (2018), 147-161.  doi: 10.3934/jmd.2018015.

[12]

S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of low complexity subshifts, Ergodic Theory and Dynam. Systems, 36 (2016), 64-95.  doi: 10.1017/etds.2015.70.

[13]

S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of Toeplitz subshifts, Discrete Anal., 2017 (2017), 19 pp.

[14]

F. Durand, N. Ormes and S. Petite, Self-induced systems, J. Anal. Math., 135 (2018), 725-756.  doi: 10.1007/s11854-018-0051-x.

[15]

T. Giordano, I. F. Putnam and C. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.  doi: 10.1007/BF02810689.

[16]

E. Glasner, T. Tsankov, B. Weiss and A. Zucker, Bernoulli disjointness, Preprint, arXiv: 1901.03406.

[17]

G. Hjorth and M. Molberg, Free continuous actions on zero-dimensional spaces, Topology Appl., 153 (2006), 1116-1131.  doi: 10.1016/j.topol.2005.03.003.

[18]

K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math., 178 (2013), 775-787.  doi: 10.4007/annals.2013.178.2.7.

[19]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24.  doi: 10.1007/BF02810577.

[20]

H. Matui, Some remarks on topological full groups of Cantor minimal systems, Internat. J. Math., 17 (2006), 231-251.  doi: 10.1142/S0129167X06003448.

[21]

K. Medynets, Reconstruction of orbits of Cantor systems from full groups, Bull. Lond. Math. Soc., 43 (2011), 1104-1110.  doi: 10.1112/blms/bdr045.

[22]

K. Medynets and J. P. Talisse, Toeplitz subshifts with trivial centralizers and positive entropy, Involve, 12 (2019), 395-410.  doi: 10.2140/involve.2019.12.395.

[23]

V. Salo and M. Schraudner, Automorphism groups of subshifts via group extensions, preprint.

[24]

D. Witte, Arithmetic groups of higher Q-rank cannot act on $1$-manifolds, Proc. Amer. Math. Soc., 122 (1994), 333-340.  doi: 10.2307/2161021.

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