Skip to main content Accessibility help

Mean topological dimension for random bundle transformations

  • XIANFENG MA (a1), JUNQI YANG (a1) and ERCAI CHEN (a2) (a3)


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.



Hide All
[1] Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114(2) (1965), 309319.
[2] Arnold, L.. Random Dynamical Systems (Springer Monographs in Mathematics) . Springer, Berlin, 1998.
[3] Aubin, J. P. and Frankowska, H.. Set-Valued Analysis. Birkhäuser, Basel, 1990.
[4] Auslander, J.. Minimal Flows and their Extensions. North-Holland, Amsterdam, 1988.
[5] Bogenschütz, T.. Entropy, pressure, and a variational principle for random dynamical systems. Random Comput. Dyn. 1(1) (1992), 99116.
[6] Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401414.
[7] Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2002.
[8] Castaing, C. and Valadier, M.. Convex Analysis and Measurable Multifunctions (Lecture Notes in Mathematics, 580) . Springer, Berlin–New York, 1977.
[9] Cong, N. D.. Topological Dynamics of Random Dynamical Systems. Clarendon Press, Oxford, 1997.
[10] 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.
[11] Coornaert, M. and Krieger, F.. Mean topological dimension for actions of discrete amenable groups. Discrete Contin. Dyn. Syst. 13(3) (2005), 779793.
[12] Crauel, H.. Random Probability Measures on Polish Spaces. Taylor & Francis, London, 2002.
[13] Dinaburg, E. I.. Relationship between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR 190(1) (1970), 1922.
[14] 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.
[15] Dudley, R. M.. Real Analysis and Probability. Cambridge University Press, Cambridge, 2002.
[16] Elliott, G. A. and Niu, Z.. The c*-algebra of a minimal homeomorphism of zero mean dimension. Preprint, 2014, arXiv:1406.2382.
[17] Furstenberg, H. and Kesten, H.. Products of random matrices. Ann. Math. Statist. 31(2) (1960), 457469.
[18] Furstenberg, H. and Kifer, Y.. Random matrix products and measures on projective spaces. Israel J. Math. 46(1–2) (1983), 1232.
[19] Gournay, A.. On a hölder covariant version of mean dimension. C. R. Math. 347(23) (2009), 13891392.
[20] Gromov, M.. Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2(4) (1999), 323415.
[21] Gutman, Y.. Mean dimension and Jaworski-type theorems. Proc. Lond. Math. Soc. 111(4) (2015), 831850.
[22] Gutman, Y. and Tsukamoto, M.. Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts. Ergod. Th. & Dynam. Sys. 34(06) (2014), 18881896.
[23] 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. 247261.
[24] Kallenberg, O.. Foundations of Modern Probability, 2nd edn. Springer, New York, 1997.
[25] Kechris, A.. Classical Descriptive Set Theory. Springer, New York, 1995.
[26] Khanin, K. and Kifer, Y.. Thermodynamic formalism for random transformations and statistical mechanics. Amer. Math. Soc. Transl. Ser. 2, 171 (1996), 107140.
[27] Kifer, Y.. On the topological pressure for random bundle transformations. Trans. Amer. Math. Soc. Ser. 2 202 (2001), 197214.
[28] Kifer, Y.. Ergodic Theory of Random Transformations (Progress in Probability and Statistics, 10) . Birkhäuser, Boston, 1986.
[29] Kuratowski, C.. Topologie I. Pánstwowe Wydawnictvo Naukowe, Warszawa, 1948, pp. 160172.
[30] Kuratowski, C.. Topologie II. Pánstwowe Wydawnictvo Naukowe, Warszawa, 1961, pp. 4556.
[31] Li, H.. Sofic mean dimension. Adv. Math. 244 (2013), 570604.
[32] 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.
[33] Lindenstrauss, E.. Mean dimension, small entropy factors and an embedding theorem. Publ. Math. Inst. Hautes Études Sci. 89(1) (1999), 227262.
[34] Lindenstrauss, E. and Tsukamoto, M.. Mean dimension and an embedding problem: an example. Israel J. Math. 199 (2014), 573584.
[35] Lindenstrauss, E. and Weiss, B.. Mean topological dimension. Israel J. Math. 115(1) (2000), 124.
[36] Liu, P.-D.. Dynamics of random transformations: smooth ergodic theory. Ergod. Th. & Dynam. Sys. 21(05) (2001), 12791319.
[37] Liu, P.-D.. A note on the entropy of factors of random dynamical systems. Ergod. Th. & Dynam. Sys. 25(02) (2005), 593603.
[38] Liu, P.-D. and Qian, M.. Smooth Ergodic Theory of Random Dynamical Systems (Lecture Notes in Mathematics, 1606) . Springer, New York, 1995.
[39] Matsuo, S. and Tsukamoto, M.. Instanton approximation, periodic ASD connections, and mean dimension. J. Funct. Anal. 260(5) (2011), 13691427.
[40] Michael, E.. Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71(1) (1951), 152182.
[41] Niu, Z.. Mean dimension and ah-algebras with diagonal maps. J. Funct. Anal. 266(8) (2014), 49384994.
[42] Phillips, N. C.. The c*-algebra of a minimal homeomorphism with finite mean dimension has finite radius of comparison. Preprint, 2016, arXiv:1605.07976.
[43] Shub, M. and Weiss, B.. Can one always lower topological entropy? Ergod. Th. & Dynam. Sys. 11(03) (1991), 535546.
[44] Klein, E. and Thompson, A. C.. Theory of Correspondences. John Wiley & Sons, New York, 1984.
[45] Tsukamoto, M.. Mean Dimension of the Unit Ball in  $l^{p}$ . Preprint,  2007,
[46] Tsukamoto, M.. Deformation of Brody curves and mean dimension. Ergod. Th. & Dynam. Sys. 29(05) (2009), 16411657.
[47] Tsukamoto, M.. Gauge theory on infinite connected sum and mean dimension. Math. Phys. Anal. Geom. 12(4) (2009), 325380.
[48] Ulam, S. M. and von Neumann, J.. Random ergodic theorems. Bull. Amer. Math. Soc. 51 (1945), p. 660.
[49] Gutman, Y.. Embedding topological dynamical systems with periodic points in cubical shifts. Ergod. Th. & Dynam. Sys. 37(2) (2017), 512538.
[50] Gutman, Y. and Tsukamoto, M.. Embedding minimal dynamical systems into Hilbert cubes. Preprint, 2015,arXiv:1511.01802.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed