American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Multiple periodic solutions of Hamiltonian systems confined in a box
  • DCDS Home
  • This Issue
  • Next Article
    A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation
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

[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

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

[1]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779
[2]

Benjamin Hellouin de Menibus, Hugo Maturana Cornejo. Necessary conditions for tiling finitely generated amenable groups. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2335-2346. doi: 10.3934/dcds.2020116
[3]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[4]

Gabriel Núñez, Jana Rodriguez Hertz. Minimality and stable Bernoulliness in dimension 3. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1879-1887. doi: 10.3934/dcds.2020097
[5]

Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[6]

Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
[7]

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
[8]

Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068
[9]

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.

[10]

Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779
[11]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195
[12]

Silvère Gangloff, Benjamin Hellouin de Menibus. Effect of quantified irreducibility on the computability of subshift entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1975-2000. doi: 10.3934/dcds.2019083
[13]

Elon Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements, 1999, 5: 82-90.

[14]

Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5787-5798. doi: 10.3934/dcds.2015.35.5787
[15]

Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162
[16]

Rafael De La Llave, A. Windsor. An application of topological multiple recurrence to tiling. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 315-324. doi: 10.3934/dcdss.2009.2.315
[17]

S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361
[18]

Fernando Alcalde Cuesta, Ana Rechtman. Minimal Følner foliations are amenable. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 685-707. doi: 10.3934/dcds.2011.31.685
[19]

Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
[20]

Rui Pacheco, Helder Vilarinho. Statistical stability for multi-substitution tiling spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4579-4594. doi: 10.3934/dcds.2013.33.4579

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (28)
  • HTML views (12)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences