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Minimal subshifts of arbitrary mean topological dimension
Department of Mathematics, Nanjing University, Nanjing 210093, China |
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 $\dim P$.
References:
[1] |
J. Auslander,
Minimal Flows and their Extensions, North-Holland Mathematics Studies 153, Elsevier Science Publishers, Amsterdam, 1988. |
[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. |
[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. |
[4] |
T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, preprint, arXiv:1502.02413v1. |
[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. |
[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. |
[7] |
Y. Gutman,
Mean dimension and Jaworski-type theorems, Proceedings of the London Mathematical Society, 111 (2015), 831-850.
doi: 10.1112/plms/pdv043. |
[8] |
Y. Gutman, Embedding topological dynamical systems with periodic points in cubical shifts Ergod. Th. Dynam. Sys. (2015).
doi: 10.1017/etds.2015.40. |
[9] |
Y. Gutman, E. Lindenstrauss and M. Tsukamoto, Mean dimension of $\mathbb{Z}^k$-actions, preprint, arXiv:1510.01605v1. |
[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. |
[11] |
F. Krieger,
Groupes moyennables, dimension topologique moyenne et sous-décalages, Geom. Dedicata, 122 (2006), 15-31.
doi: 10.1007/s10711-006-9071-2. |
[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. |
[13] |
E. Lindenstrauss,
Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227-262.
|
[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. |
[15] |
E. Lindenstrauss and B. Weiss,
Mean topological dimension, Israel J. Math., 115 (2000), 1-24.
doi: 10.1007/BF02810577. |
[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. |
[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. |
[18] |
S. Williams,
Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107.
doi: 10.1007/BF00534085. |
show all references
References:
[1] |
J. Auslander,
Minimal Flows and their Extensions, North-Holland Mathematics Studies 153, Elsevier Science Publishers, Amsterdam, 1988. |
[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. |
[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. |
[4] |
T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, preprint, arXiv:1502.02413v1. |
[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. |
[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. |
[7] |
Y. Gutman,
Mean dimension and Jaworski-type theorems, Proceedings of the London Mathematical Society, 111 (2015), 831-850.
doi: 10.1112/plms/pdv043. |
[8] |
Y. Gutman, Embedding topological dynamical systems with periodic points in cubical shifts Ergod. Th. Dynam. Sys. (2015).
doi: 10.1017/etds.2015.40. |
[9] |
Y. Gutman, E. Lindenstrauss and M. Tsukamoto, Mean dimension of $\mathbb{Z}^k$-actions, preprint, arXiv:1510.01605v1. |
[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. |
[11] |
F. Krieger,
Groupes moyennables, dimension topologique moyenne et sous-décalages, Geom. Dedicata, 122 (2006), 15-31.
doi: 10.1007/s10711-006-9071-2. |
[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. |
[13] |
E. Lindenstrauss,
Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227-262.
|
[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. |
[15] |
E. Lindenstrauss and B. Weiss,
Mean topological dimension, Israel J. Math., 115 (2000), 1-24.
doi: 10.1007/BF02810577. |
[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. |
[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. |
[18] |
S. Williams,
Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107.
doi: 10.1007/BF00534085. |
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