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Mean topological dimension for random bundle transformations

Published online by Cambridge University Press:  20 June 2017

XIANFENG MA ,
JUNQI YANG  and
ERCAI CHEN
XIANFENG MA
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China email xianfengma@ecust.edu.cn, y30140124@mail.ecust.edu.cn
JUNQI YANG
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China email xianfengma@ecust.edu.cn, y30140124@mail.ecust.edu.cn
ERCAI CHEN
Affiliation:
School of Mathematical Science, Nanjing Normal University, Nanjing 210097, China email ecchen@njnu.edu.cn Center of Nonlinear Science, Nanjing University, Nanjing 210093, China
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Abstract

We introduce the mean topological dimension for random bundle transformations, and show that continuous bundle random dynamical systems with finite topological entropy or satisfying the small boundary property have zero mean topological dimensions.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 4 , April 2019 , pp. 1020 - 1041
Copyright
© Cambridge University Press, 2017 

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References

Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114(2) (1965), 309–319.Google Scholar
Arnold, L.. Random Dynamical Systems (Springer Monographs in Mathematics) . Springer, Berlin, 1998.Google Scholar
Aubin, J. P. and Frankowska, H.. Set-Valued Analysis. Birkhäuser, Basel, 1990.Google Scholar
Auslander, J.. Minimal Flows and their Extensions. North-Holland, Amsterdam, 1988.Google Scholar
Bogenschütz, T.. Entropy, pressure, and a variational principle for random dynamical systems. Random Comput. Dyn. 1(1) (1992), 99–116.Google Scholar
Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.Google Scholar
Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2002.Google Scholar
Castaing, C. and Valadier, M.. Convex Analysis and Measurable Multifunctions (Lecture Notes in Mathematics, 580) . Springer, Berlin–New York, 1977.Google Scholar
Cong, N. D.. Topological Dynamics of Random Dynamical Systems. Clarendon Press, Oxford, 1997.Google Scholar
Coornaert, M.. Topological Dimension and Dynamical Systems (Universitext) . Springer, 2015. Translation from the French language edition: M. Coornaert. Dimension topologique et systèmes dynamiques (Cours spécialisés, 14). Société Mathématique de France, Paris, 2005.Google Scholar
Coornaert, M. and Krieger, F.. Mean topological dimension for actions of discrete amenable groups. Discrete Contin. Dyn. Syst. 13(3) (2005), 779–793.Google Scholar
Crauel, H.. Random Probability Measures on Polish Spaces. Taylor & Francis, London, 2002.Google Scholar
Dinaburg, E. I.. Relationship between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR 190(1) (1970), 19–22.Google Scholar
Dooley, A. and Zhang, G.. Local Entropy Theory of a Random Dynamical System (Memoirs of the American Mathematical Society, 233) . American Mathematical Society, Providence, RI, 2015.Google Scholar
Dudley, R. M.. Real Analysis and Probability. Cambridge University Press, Cambridge, 2002.Google Scholar
Elliott, G. A. and Niu, Z.. The c*-algebra of a minimal homeomorphism of zero mean dimension. Preprint, 2014, arXiv:1406.2382.Google Scholar
Furstenberg, H. and Kesten, H.. Products of random matrices. Ann. Math. Statist. 31(2) (1960), 457–469.Google Scholar
Furstenberg, H. and Kifer, Y.. Random matrix products and measures on projective spaces. Israel J. Math. 46(1–2) (1983), 12–32.Google Scholar
Gournay, A.. On a hölder covariant version of mean dimension. C. R. Math. 347(23) (2009), 1389–1392.Google Scholar
Gromov, M.. Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2(4) (1999), 323–415.Google Scholar
Gutman, Y.. Mean dimension and Jaworski-type theorems. Proc. Lond. Math. Soc. 111(4) (2015), 831–850.Google Scholar
Gutman, Y. and Tsukamoto, M.. Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts. Ergod. Th. & Dynam. Sys. 34(06) (2014), 1888–1896.Google Scholar
Kakutani, S.. Random ergodic theorems and Markoff processes with a stable distribution. Proc. Second Berkeley Symp. on Mathematical Statistics and Probability. University of California Press, Berkeley and Los Angeles, 1950, pp. 247–261.Google Scholar
Kallenberg, O.. Foundations of Modern Probability, 2nd edn. Springer, New York, 1997.Google Scholar
Kechris, A.. Classical Descriptive Set Theory. Springer, New York, 1995.Google Scholar
Khanin, K. and Kifer, Y.. Thermodynamic formalism for random transformations and statistical mechanics. Amer. Math. Soc. Transl. Ser. 2, 171 (1996), 107–140.Google Scholar
Kifer, Y.. On the topological pressure for random bundle transformations. Trans. Amer. Math. Soc. Ser. 2 202 (2001), 197–214.Google Scholar
Kifer, Y.. Ergodic Theory of Random Transformations (Progress in Probability and Statistics, 10) . Birkhäuser, Boston, 1986.Google Scholar
Kuratowski, C.. Topologie I. Pánstwowe Wydawnictvo Naukowe, Warszawa, 1948, pp. 160–172.Google Scholar
Kuratowski, C.. Topologie II. Pánstwowe Wydawnictvo Naukowe, Warszawa, 1961, pp. 45–56.Google Scholar
Li, H.. Sofic mean dimension. Adv. Math. 244 (2013), 570–604.Google Scholar
Li, H. and Liang, B.. Mean dimension, mean rank, and von Neumann–Lück rank. J. Reine Angew. Math. (2015), doi:10.1515/crelle-2015-0046.Google Scholar
Lindenstrauss, E.. Mean dimension, small entropy factors and an embedding theorem. Publ. Math. Inst. Hautes Études Sci. 89(1) (1999), 227–262.Google Scholar
Lindenstrauss, E. and Tsukamoto, M.. Mean dimension and an embedding problem: an example. Israel J. Math. 199 (2014), 573–584.Google Scholar
Lindenstrauss, E. and Weiss, B.. Mean topological dimension. Israel J. Math. 115(1) (2000), 1–24.Google Scholar
Liu, P.-D.. Dynamics of random transformations: smooth ergodic theory. Ergod. Th. & Dynam. Sys. 21(05) (2001), 1279–1319.Google Scholar
Liu, P.-D.. A note on the entropy of factors of random dynamical systems. Ergod. Th. & Dynam. Sys. 25(02) (2005), 593–603.Google Scholar
Liu, P.-D. and Qian, M.. Smooth Ergodic Theory of Random Dynamical Systems (Lecture Notes in Mathematics, 1606) . Springer, New York, 1995.Google Scholar
Matsuo, S. and Tsukamoto, M.. Instanton approximation, periodic ASD connections, and mean dimension. J. Funct. Anal. 260(5) (2011), 1369–1427.Google Scholar
Michael, E.. Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71(1) (1951), 152–182.Google Scholar
Niu, Z.. Mean dimension and ah-algebras with diagonal maps. J. Funct. Anal. 266(8) (2014), 4938–4994.Google Scholar
Phillips, N. C.. The c*-algebra of a minimal homeomorphism with finite mean dimension has finite radius of comparison. Preprint, 2016, arXiv:1605.07976.Google Scholar
Shub, M. and Weiss, B.. Can one always lower topological entropy? Ergod. Th. & Dynam. Sys. 11(03) (1991), 535–546.Google Scholar
Klein, E. and Thompson, A. C.. Theory of Correspondences. John Wiley & Sons, New York, 1984.Google Scholar
Tsukamoto, M.. Mean Dimension of the Unit Ball in  $l^{p}$ . Preprint,  2007,  https://www.math.kyoto-u.ac.jp/preprint/2007/10tsukamoto.pdf.Google Scholar
Tsukamoto, M.. Deformation of Brody curves and mean dimension. Ergod. Th. & Dynam. Sys. 29(05) (2009), 1641–1657.Google Scholar
Tsukamoto, M.. Gauge theory on infinite connected sum and mean dimension. Math. Phys. Anal. Geom. 12(4) (2009), 325–380.Google Scholar
Ulam, S. M. and von Neumann, J.. Random ergodic theorems. Bull. Amer. Math. Soc. 51 (1945), p. 660.Google Scholar
Gutman, Y.. Embedding topological dynamical systems with periodic points in cubical shifts. Ergod. Th. & Dynam. Sys. 37(2) (2017), 512–538.Google Scholar
Gutman, Y. and Tsukamoto, M.. Embedding minimal dynamical systems into Hilbert cubes. Preprint, 2015,arXiv:1511.01802.Google Scholar