Minimal subshifts of arbitrary mean topological dimension

Dou Dou

doi:10.3934/dcds.2017058

Article Contents

2017, Volume 37, Issue 3: 1411-1424. Doi: 10.3934/dcds.2017058

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

Minimal subshifts of arbitrary mean topological dimension

Dou Dou^1, ,

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Dou Dou
* Corresponding author: Dou Dou

Received: May 2016

Revised: August 2016

Published online: December 2016

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary:37B05, 28D20;Secondary:54H20.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.