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March  2017, 37(3): 1411-1424. doi: 10.3934/dcds.2017058

Minimal subshifts of arbitrary mean topological dimension

Dou Dou 1,,

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Dou Dou

Received  May 2016 Revised  August 2016 Published  December 2016

Let $G$ be a countable infinite amenable group and $P$ be a polyhedron. We give a construction of minimal subshifts of $P^G$ with arbitrary mean topological dimension less than $\dim P$.

Keywords: Minimality, subshift, amenable group, mean dimension, tiling.
Mathematics Subject Classification: Primary:37B05, 28D20;Secondary:54H20.
Citation: Dou Dou. Minimal subshifts of arbitrary mean topological dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1411-1424. doi: 10.3934/dcds.2017058
References:
[1]

J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies 153, Elsevier Science Publishers, Amsterdam, 1988.  Google Scholar

[2]

M. Coornaert, Topological Dimension and Dynamical Systems, Universitext, Springer, 2015. Translation from the French language edition: Dimension topologique et systémes dynamiques by M. Coornaert, Cours spécialisés 14, Société Mathématique de France, Paris, 2005. doi: 10.1007/978-3-319-19794-4.  Google Scholar

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M. Coornaert and F. Krieger, Mean topological dimension for actions of discrete amenable groups, Discrete Continuous Dynam. Systems -A, 13 (2005), 779-793.  doi: 10.3934/dcds.2005.13.779.  Google Scholar

[4]

T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, preprint, arXiv:1502.02413v1. Google Scholar

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M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps, Part I, Math. Phys. Anal. Geom., 2 (1999), 323-415.  doi: 10.1023/A:1009841100168.  Google Scholar

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Y. Gutman, Embedding $\mathbb{Z}^k$-actions in cubical shifts and $\mathbb{Z}^k$-symbolic extensions, Ergod. Th. Dynam. Sys., 31 (2011), 383-403.  doi: 10.1017/S0143385709001096.  Google Scholar

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Y. Gutman, Mean dimension and Jaworski-type theorems, Proceedings of the London Mathematical Society, 111 (2015), 831-850.  doi: 10.1112/plms/pdv043.  Google Scholar

[8]

Y. Gutman, Embedding topological dynamical systems with periodic points in cubical shifts Ergod. Th. Dynam. Sys. (2015). doi: 10.1017/etds.2015.40.  Google Scholar

[9]

Y. Gutman, E. Lindenstrauss and M. Tsukamoto, Mean dimension of $\mathbb{Z}^k$-actions, preprint, arXiv:1510.01605v1. Google Scholar

[10]

Y. Gutman and M. Tsukamoto, Mean dimension and a sharp embedding theorem: Extensions of aperiodic subshifts, Ergod. Th. Dynam. Sys., 34 (2014), 1888-1896.  doi: 10.1017/etds.2013.30.  Google Scholar

[11]

F. Krieger, Groupes moyennables, dimension topologique moyenne et sous-décalages, Geom. Dedicata, 122 (2006), 15-31.  doi: 10.1007/s10711-006-9071-2.  Google Scholar

[12]

F. Krieger, Minimal systems of arbitrary large mean topological dimension, Israel J. Math., 172 (2009), 425-444.  doi: 10.1007/s11856-009-0081-2.  Google Scholar

[13]

E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227-262.   Google Scholar

[14]

E. Lindenstrauss and M. Tsukamoto, Mean dimension and an embedding problem: An example, Israel J. Math., 199 (2014), 573-584.  doi: 10.1007/s11856-013-0040-9.  Google Scholar

[15]

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

[16]

D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

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B. Weiss, Actions of amenable groups, in Topics in Dynamics and Ergodic Theory, 226-262.London Math. Soc. Lecture Note Ser., 310, Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511546716.012.  Google Scholar

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S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107.  doi: 10.1007/BF00534085.  Google Scholar

show all references

References:
[1]

J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies 153, Elsevier Science Publishers, Amsterdam, 1988.  Google Scholar

[2]

M. Coornaert, Topological Dimension and Dynamical Systems, Universitext, Springer, 2015. Translation from the French language edition: Dimension topologique et systémes dynamiques by M. Coornaert, Cours spécialisés 14, Société Mathématique de France, Paris, 2005. doi: 10.1007/978-3-319-19794-4.  Google Scholar

[3]

M. Coornaert and F. Krieger, Mean topological dimension for actions of discrete amenable groups, Discrete Continuous Dynam. Systems -A, 13 (2005), 779-793.  doi: 10.3934/dcds.2005.13.779.  Google Scholar

[4]

T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, preprint, arXiv:1502.02413v1. Google Scholar

[5]

M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps, Part I, Math. Phys. Anal. Geom., 2 (1999), 323-415.  doi: 10.1023/A:1009841100168.  Google Scholar

[6]

Y. Gutman, Embedding $\mathbb{Z}^k$-actions in cubical shifts and $\mathbb{Z}^k$-symbolic extensions, Ergod. Th. Dynam. Sys., 31 (2011), 383-403.  doi: 10.1017/S0143385709001096.  Google Scholar

[7]

Y. Gutman, Mean dimension and Jaworski-type theorems, Proceedings of the London Mathematical Society, 111 (2015), 831-850.  doi: 10.1112/plms/pdv043.  Google Scholar

[8]

Y. Gutman, Embedding topological dynamical systems with periodic points in cubical shifts Ergod. Th. Dynam. Sys. (2015). doi: 10.1017/etds.2015.40.  Google Scholar

[9]

Y. Gutman, E. Lindenstrauss and M. Tsukamoto, Mean dimension of $\mathbb{Z}^k$-actions, preprint, arXiv:1510.01605v1. Google Scholar

[10]

Y. Gutman and M. Tsukamoto, Mean dimension and a sharp embedding theorem: Extensions of aperiodic subshifts, Ergod. Th. Dynam. Sys., 34 (2014), 1888-1896.  doi: 10.1017/etds.2013.30.  Google Scholar

[11]

F. Krieger, Groupes moyennables, dimension topologique moyenne et sous-décalages, Geom. Dedicata, 122 (2006), 15-31.  doi: 10.1007/s10711-006-9071-2.  Google Scholar

[12]

F. Krieger, Minimal systems of arbitrary large mean topological dimension, Israel J. Math., 172 (2009), 425-444.  doi: 10.1007/s11856-009-0081-2.  Google Scholar

[13]

E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227-262.   Google Scholar

[14]

E. Lindenstrauss and M. Tsukamoto, Mean dimension and an embedding problem: An example, Israel J. Math., 199 (2014), 573-584.  doi: 10.1007/s11856-013-0040-9.  Google Scholar

[15]

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

[16]

D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

[17]

B. Weiss, Actions of amenable groups, in Topics in Dynamics and Ergodic Theory, 226-262.London Math. Soc. Lecture Note Ser., 310, Cambridge Univ. Press, Cambridge, 2003. doi: 10.1017/CBO9780511546716.012.  Google Scholar

[18]

S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107.  doi: 10.1007/BF00534085.  Google Scholar

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