Expansive multiparameter actions and mean dimension
HTML articles powered by AMS MathViewer
- by Tom Meyerovitch and Masaki Tsukamoto PDF
- Trans. Amer. Math. Soc. 371 (2019), 7275-7299 Request permission
Abstract:
Mañé proved in 1979 that if a compact metric space admits an expansive homeomorphism, then it is finite dimensional. We generalize this theorem to multiparameter actions. The generalization involves mean dimension theory, which counts the “averaged dimension” of a dynamical system. We prove that if $T:\mathbb {Z}^k\times X\to X$ is expansive and if $R:\mathbb {Z}^{k-1}\times X\to X$ commutes with $T$, then $R$ has finite mean dimension. When $k=1$, this statement reduces to Mañé’s theorem. We also study several related issues, especially the connection with entropy theory.References
- Alfonso Artigue, Minimal expansive systems and spiral points, Topology Appl. 194 (2015), 166–170. MR 3404611, DOI 10.1016/j.topol.2015.08.018
- Rufus Bowen, Markov partitions and minimal sets for Axiom $\textrm {A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 907–918. MR 277002, DOI 10.2307/2373402
- Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
- Mike Boyle and Douglas Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), no. 1, 55–102. MR 1355295, DOI 10.1090/S0002-9947-97-01634-6
- Ryszard Engelking, Dimension theory, North-Holland Mathematical Library, vol. 19, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author. MR 0482697
- Albert Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Comm. Math. Phys. 126 (1989), no. 2, 249–262. MR 1027497
- David Fried, Métriques naturelles sur les espaces de Smale, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 1, 77–79 (French, with English summary). MR 719952
- David Fried, Finitely presented dynamical systems, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 489–507. MR 922362, DOI 10.1017/S014338570000417X
- A. H. Frink, Distance functions and the metrization problem, Bull. Amer. Math. Soc. 43 (1937), no. 2, 133–142. MR 1563501, DOI 10.1090/S0002-9904-1937-06509-8
- Misha Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Math. Phys. Anal. Geom. 2 (1999), no. 4, 323–415. MR 1742309, DOI 10.1023/A:1009841100168
- Yonatan Gutman, Embedding $\Bbb Z^k$-actions in cubical shifts and $\Bbb Z^k$-symbolic extensions, Ergodic Theory Dynam. Systems 31 (2011), no. 2, 383–403. MR 2776381, DOI 10.1017/S0143385709001096
- Yonatan Gutman, Elon Lindenstrauss, and Masaki Tsukamoto, Mean dimension of $\Bbb {Z}^k$-actions, Geom. Funct. Anal. 26 (2016), no. 3, 778–817. MR 3540453, DOI 10.1007/s00039-016-0372-9
- Y. Gutman, Y. Qiao, and M. Tsukamoto, Application of signal analysis to the embedding problem of $\mathbb {Z}^k$-actions, arXiv:1709.00125, 2017.
- Y. Gutman and M. Tsukamoto, Embedding minimal dynamical systems into Hilbert cubes, arXiv:1511.01802, 2015.
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- Elon Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 227–262 (2000). MR 1793417
- Elon Lindenstrauss and Benjamin Weiss, Mean topological dimension, Israel J. Math. 115 (2000), 1–24. MR 1749670, DOI 10.1007/BF02810577
- Ricardo Mañé, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313–319. MR 534124, DOI 10.1090/S0002-9947-1979-0534124-9
- John Milnor, On the entropy geometry of cellular automata, Complex Systems 2 (1988), no. 3, 357–385. MR 955558
- Ronnie Pavlov, Approximating the hard square entropy constant with probabilistic methods, Ann. Probab. 40 (2012), no. 6, 2362–2399. MR 3050506, DOI 10.1214/11-AOP681
- Klaus Schmidt, Automorphisms of compact abelian groups and affine varieties, Proc. London Math. Soc. (3) 61 (1990), no. 3, 480–496. MR 1069512, DOI 10.1112/plms/s3-61.3.480
- Mark A. Shereshevsky, On continuous actions commuting with actions of positive entropy, Colloq. Math. 70 (1996), no. 2, 265–269. MR 1380381, DOI 10.4064/cm-70-2-265-269
- En Hui Shi and Li Zhen Zhou, The nonexistence of expansive $\Bbb Z^d$ actions on graphs, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1509–1514. MR 2190021, DOI 10.1007/s10114-005-0640-3
Additional Information
- Tom Meyerovitch
- Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 8410501, Israel
- MR Author ID: 824249
- Email: mtom@math.bgu.ac.il
- Masaki Tsukamoto
- Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
- MR Author ID: 828585
- Email: tukamoto@math.kyoto-u.ac.jp
- Received by editor(s): October 28, 2017
- Received by editor(s) in revised form: March 6, 2018
- Published electronically: October 2, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7275-7299
- MSC (2010): Primary 37B05, 54F45
- DOI: https://doi.org/10.1090/tran/7588
- MathSciNet review: 3939578