Abstract
For any positive integer D, we construct a minimal dynamical system with mean dimension equal to D/2 that cannot be embedded into (([0, 1]D)ℤ, shift).
References
J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, Vol. 153, North-Holland, Amsterdam, 1988.
M. Coornaert and F. Krieger, Mean topological dimension for actions of discrete amenable groups, Discrete and Continuous Dynamical Systems 13 (2005), 779–793.
M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I, Mathematical Physics, Analysis and Geometry 2 (1999), 323–415.
Y. Gutman, Embedding ℤk-actions in cubical shifts and ℤk-symbolic extensions, Ergodic Theory and Dynamical Systems 31 (2011), 383–403.
W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, Vol. 4, Princeton University Press, Princeton, NJ, 1941.
A. Jaworski, Ph.D. Thesis, University of Maryland, 1974.
E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Institut des Hautes Études Scientifiques. Publications Mathématiques 89 (1999), 227–262.
E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel Journal of Mathematics 115 (2000), 1–24.
J. Matoušek, Using the Borsuk-Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry, Corrected 2nd printing, Springer-Verlag, Berlin-Heidelberg, 2008.
A. B. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in Surveys in Contemporary Mathematics, London Mathematical Society Lecture Note Series, Vol. 347, Cambridge University Press, Cambridge, 2008, pp. 248–342.
M. Skopenkov, Embedding products of graphs into Euclidean spaces, Fundamenta Mathematicae 179 (2003), 191–198.
Author information
Authors and Affiliations
Corresponding author
Additional information
Elon Lindenstrauss acknowledges the support of the ISF and ERC.
Masaki Tsukamoto was supported by Grant-in-Aid for Young Scientists (B) (21740048).
Rights and permissions
About this article
Cite this article
Lindenstrauss, E., Tsukamoto, M. Mean dimension and an embedding problem: An example. Isr. J. Math. 199, 573–584 (2014). https://doi.org/10.1007/s11856-013-0040-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-013-0040-9