American Institute of Mathematical Sciences

Advanced
April  2005, 13(3): 779-793. doi: 10.3934/dcds.2005.13.779

Mean topological dimension for actions of discrete amenable groups

Michel Coornaert 1, and  Fabrice Krieger 1,

1. 

Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France, France

Received  October 2004 Revised  April 2005 Published  May 2005

Let $G$ be a countable amenable group containing subgroups of arbitrarily large finite index. Given a polyhedron $P$ and a real number $\rho$ such that $0 \leq \rho \leq$dim$(P)$, we construct a closed subshift $X \subset P^G$ having mean topological dimension $\rho$. This shows in particular that mean topological dimension of compact metrisable $G$-spaces take all values in $[0,\infty]$.
Keywords: subshift of block type., Mean topological dimension, amenable group.
Mathematics Subject Classification: Primary: 37B05, 37B10; Secondary: 43A07, 20F19, 20E2.
Citation: Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779
[1]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[2]

Yunping Wang, Ercai Chen, Xiaoyao Zhou. Mean dimension theory in symbolic dynamics for finitely generated amenable groups. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022050
[3]

Dou Dou. Minimal subshifts of arbitrary mean topological dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1411-1424. doi: 10.3934/dcds.2017058
[4]

Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
[5]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[6]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008
[7]

Tao Yu, Guohua Zhang, Ruifeng Zhang. Discrete spectrum for amenable group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5871-5886. doi: 10.3934/dcds.2021099
[8]

Kazuhiro Kawamura. Mean dimension of shifts of finite type and of generalized inverse limits. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4767-4775. doi: 10.3934/dcds.2020200
[9]

Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[10]

Marcelo Sobottka. Topological quasi-group shifts. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77
[11]

Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
[12]

Riccardo Aragona, Alessio Meneghetti. Type-preserving matrices and security of block ciphers. Advances in Mathematics of Communications, 2019, 13 (2) : 235-251. doi: 10.3934/amc.2019016
[13]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[14]

Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068
[15]

Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki. The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions. Electronic Research Announcements, 1995, 1: 43-47.

[16]

Fang Chen, Ning Gao, Yao- Lin Jiang. On product-type generalized block AOR method for augmented linear systems. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 797-809. doi: 10.3934/naco.2012.2.797
[17]

Tao Wang. Variational relations for metric mean dimension and rate distortion dimension. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4593-4608. doi: 10.3934/dcds.2021050
[18]

Joe Gildea, Abidin Kaya, Adam Michael Roberts, Rhian Taylor, Alexander Tylyshchak. New self-dual codes from $ 2 \times 2 $ block circulant matrices, group rings and neighbours of neighbours. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021039
[19]

Maria Bortos, Joe Gildea, Abidin Kaya, Adrian Korban, Alexander Tylyshchak. New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings. Advances in Mathematics of Communications, 2022, 16 (2) : 269-284. doi: 10.3934/amc.2020111
[20]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy