Hostname: page-component-76c49bb84f-6sxsx Total loading time: 0 Render date: 2025-07-10T07:47:48.580Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean topological dimension of a random dynamical system for amenable groups

Part of: Random dynamical systems Ergodic theory

Published online by Cambridge University Press:  07 July 2025

Yu Liu  and
Xiaojun Huang
Yu Liu
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing, China
Xiaojun Huang*
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing, China
*
Corresponding author: Xiaojun Huang, email: mathhxj@163.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the mean topological dimension of random bundle transformations associated with an infinite countable discrete amenable group action and show that continuous bundle random dynamical systems for amenable groups with finite fibre topological entropy have zero mean topological dimensions.

Keywords

bundle random dynamical systemmean topological dimensionmetric mean dimensionamenable group

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37H05: Foundations, general theory of cocycles, algebraic ergodic theory

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 32
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramov, L. M. and Rohlin, V. A., Entropy of a skew product of mappings with invariant measure, Vestnik Leningrad. Univ. 17(7): (1962), 5–13.Google Scholar
Adler, R. L., Konheim, A. G. and McAndrew, M. H., Topological entropy, Trans. Am. Math. Soc. 114 (1965), 309–319.10.1090/S0002-9947-1965-0175106-9CrossRefGoogle Scholar
Arnold, L., Random dynamical systems. Springer Monographs in Mathematics (Springer-Verlag, Berlin, 1998).Google Scholar
Aubin, J. P. and Frankowska, H., Set-valued analysis, systems & control. Foundations & Applications, Volume 2 (Birkhäuser Boston, Inc., Boston, MA, 1990)Google Scholar
Auslander, J., Minimal Flows and Their Extensions, Volume 153 (Elsevier).Google Scholar
Bogenschütz, T., Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dyn. 1(1): (1992), 99–116.Google Scholar
Bogenschütz, T., Equilibrium states for random dynamical systems. PhD Thesis 1993.Google Scholar
Bogenschütz, T. and Gundlach, V. M., Ruelle’s transfer operator for random subshifts of finite type, Ergodic Theory Dynam. Syst. 15(3): (1995), 413–447.10.1017/S0143385700008464CrossRefGoogle Scholar
Bowen, R., Entropy for group endomorphisms and homogeneous spaces, Trans. Am. Math. Soc. 153 (1971), 401–414.10.1090/S0002-9947-1971-0274707-XCrossRefGoogle Scholar
Brin, M. and Stuck, G., Introduction to Dynamical Systems (Cambridge University Press, Cambridge, 2002).10.1017/CBO9780511755316CrossRefGoogle Scholar
Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions. Lecture Notes in Mathematics, Volume 580 (Springer-Verlag, Berlin-New York, 1977)Google Scholar
Chung, N. P. and Li, H., Homoclinic groups, IE groups, and expansive algebraic actions, Invent. Math. 199(3): (2015), 805–858.10.1007/s00222-014-0524-1CrossRefGoogle Scholar
Coornaert, M., Dimension topologique et systèmes dynamiques. Cours Spécialisés [Specialized Courses], Volume 14 (Société Mathématique de France, Paris, 2005)Google Scholar
Crauel, H., Random probability measures on Polish spaces. Stochastics Monographs, Volume 11 (Taylor & Francis, 2002)Google Scholar
Danilenko, A.I., Entropy theory from the orbital point of view, Monatsh. Math. 134(2): (2001), 121–141.10.1007/s006050170003CrossRefGoogle Scholar
Danilenko, A.I. and Park, K. K., Generators and Bernoullian factors for amenable actions and cocycles on their orbits, Ergodic Theory Dynam. Syst. 22(2): (2002), 1715–1745.10.1017/S014338570200072XCrossRefGoogle Scholar
Dinaburg, E.I., A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR 190 (1970), 19–22.Google Scholar
Dooley, A. H., Golodets, V. Y. and Zhang, G., Sub-additive ergodic theorems for countable amenable groups, J. Funct. Anal. 267 (2014), 1291–1320.10.1016/j.jfa.2014.06.008CrossRefGoogle Scholar
Dooley, A. H. and Zhang, G., Local entropy theory of a random dynamical system, Mem. Am. Math. Soc. 233(1099): (2015), vi+106.Google Scholar
Dou, D., Ye, X. and Zhang, G., Entropy sequences and maximal entropy sets, Nonlinearity 19 (2006), 53–74.10.1088/0951-7715/19/1/004CrossRefGoogle Scholar
Glasner, E., Thouvenot, J. P. and Weiss, B., Entropy theory without a past, Ergodic Theory Dyn. Syst. 20 (2000), 1355–1370.10.1017/S0143385700000730CrossRefGoogle Scholar
Gromov, M., Topological invariants of dynamical systems and spaces of holomorphic maps: I, Math. Phys. Anal. Geom. 2 (1999), 323–415.10.1023/A:1009841100168CrossRefGoogle Scholar
Huang, W., Ye, X. and Zhang, G., Local entropy theory for a countable discrete amenable group action, J. Funct. Anal. 261 (2011), 1028–1082.10.1016/j.jfa.2011.04.014CrossRefGoogle Scholar
Kakutani, S.. Random ergodic theorems and Markoff processes with a stable distribution, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1951, pp. 247–261, Berkeley-Los Angeles, Calif.10.1525/9780520411586-019CrossRefGoogle Scholar
Kallenberg, O., Foundations of Modern Probability, Probability and its Applications (New York), (Springer-Verlag, New York, 1997).Google Scholar
Kechris, A. S., Classical descriptive set theory. Graduate Texts in Mathematics, Volume 156 (Springer-Verlag, New York, 1995)Google Scholar
Kerr, D. and Li, H., Independence in topological and $C^*$-dynamics, Math. Ann. 338 (2007), 869–926.10.1007/s00208-007-0097-zCrossRefGoogle Scholar
Kerr, D. and Li, H., Combinatorial independence in measurable dynamics, J. Funct. Anal. 256 (2009), 1341–1386.10.1016/j.jfa.2008.12.014CrossRefGoogle Scholar
Kerr, D. and Li, H., Combinatorial independence and sofic entropy, Commun. Math. Stat. 1 (2013), 213–257.10.1007/s40304-013-0001-yCrossRefGoogle Scholar
Khanin, K. and Kifer, Y., Thermodynamic formalism for random transformations and statistical mechanics. Sinai’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser., Volume 2, pp. 107–140 (Amer. Math. Soc., Providence, RI, 1996)Google Scholar
Kifer, Y., Ergodic Theory of Random Transformations (Springer Science & Business Media, 1986).10.1007/978-1-4684-9175-3CrossRefGoogle Scholar
Kifer, Y., On the topological pressure for random bundle transformations. Topology, Ergodic Theory, Real Algebraic Geometry (Amer. Math. Soc. Transl. Ser.), Volume 2, pp. 197–214 (Amer. Math. Soc., Providence, RI, 2001)Google Scholar
Kifer, Y. and Liu, P. D., Random dynamics. Handbook of Dynamical Systems, Volume 1B, pp. 379–499 (Elsevier B.V., Amsterdam, 2006)Google Scholar
Kifer, Y. and Weiss, B., Generating partitions for random transformations, Ergodic Theory Dynam. Syst. 22 (2002), 1813–1830.10.1017/S0143385702000755CrossRefGoogle Scholar
Klein, E. and Thompson, A. C., Theory of correspondences. Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley & Sons, Inc., 1984), Including applications to mathematical economics, A Wiley-Interscience Publication.Google Scholar
Kuratowski, C., Topologie. Vol. II, Monografie Matematyczne, Volume 21 (Państwowe Wydawnictwo Naukowe, Warsaw, 1961) Troisième édition, corrigèe et complétée de deux appendicesGoogle Scholar
Kuratowski, C., Topologie. I. Espaces Métrisables, Espaces Complets. Monografie Matematyczne [Mathematical Monographs], 2nd edn Warsaw (1948).Google Scholar
Lian, Z. and Lu, K., Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Am. Math. Soc. 206 (2010), vi+106.Google Scholar
Lindenstrauss, E., Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math. (89): (1999), 227–262.10.1007/BF02698858CrossRefGoogle Scholar
Lindenstrauss, E. and Weiss, B., Mean topological dimension, Israel J. Math. 115 (2000), 1–24.10.1007/BF02810577CrossRefGoogle Scholar
Liu, P. D., Dynamics of random transformations: Smooth ergodic theory, Ergodic Theory Dynam. Syst. 21 (2001), 1279–1319.10.1017/S0143385701001614CrossRefGoogle Scholar
Liu, P. D., A note on the entropy of factors of random dynamical systems, Ergodic Theory Dynam. Syst. 25 (2005), 593–603.10.1017/S0143385704000586CrossRefGoogle Scholar
Liu, P. D. and Qian, M., Smooth ergodic theory of random dynamical systems. Lecture Notes in Mathematics, Volume 1606 (Springer-Verlag, Berlin, 1995)Google Scholar
Ma, X., Yang, J. and Chen, E., Mean topological dimension for random bundle transformations, Ergodic Theory Dynam. Syst. 39 (2019), 1020–1041.10.1017/etds.2017.51CrossRefGoogle Scholar
Michael, E., Topologies on spaces of subsets, Trans. Am. Math. Soc. 71 (1951), 152–182.10.1090/S0002-9947-1951-0042109-4CrossRefGoogle Scholar
Ollagnier, J. M., Ergodic theory and statistical mechanics Lecture Notes in Mathematics, Volume 1115, (Springer-Verlag, Berlin, 1985)Google Scholar
Ollagnier, J. M. and Pinchon, D., Groupes pavables et principe variationnel, Z. Wahrsch. Verw. Gebiete 48(1): (1979), 71–79.10.1007/BF00534883CrossRefGoogle Scholar
Ollagnier, J. M. and Pinchon, D., The variational principle, Studia Math. 72(2): (1982), 151–159.10.4064/sm-72-2-151-159CrossRefGoogle Scholar
Ornstein, D. S. and Weiss, B., Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1–141.10.1007/BF02790325CrossRefGoogle Scholar
Rudolph, D. J. and Weiss, B., Entropy and mixing for amenable group actions, Ann. Math. 151(3): (2000), 1119–1150.10.2307/121130CrossRefGoogle Scholar
Shub, M. and Weiss, B., Can one always lower topological entropy? Ergodic Theory Dynam. Syst. 11(3): (1991), 535–546.10.1017/S0143385700006325CrossRefGoogle Scholar
Stepin, A. M. and Tagi-Zade, A. T., Variational characterization of topological pressure of the amenable groups of transformations, Dokl. Akad. Nauk SSSR 254(3): (1980), 545–549.Google Scholar
Ulam, S. M. and von Neumann, J., Random ergodic theorems, Bull. Am. Math. Soc. 51 (1945).Google Scholar
Walters, P., An introduction to ergodic theory. Graduate Texts in Mathematics, Volume 79 (Springer-Verlag, New York-Berlin, 1982)Google Scholar
Ward, T. and Zhang, Q., The Abramov-Rokhlin entropy addition formula for amenable group actions, Monatsh. Math. 114(3–4): (1992), 317–329.10.1007/BF01299386CrossRefGoogle Scholar
Weiss, B., Actions of amenable groups, Topics in dynamics and ergodic theory. London Math. Soc. Lecture Note Ser., Volume 310, pp. 226–262 (Cambridge University Press, Cambridge, 2003)Google Scholar
Zhao, Y. and Cao, Y., On the topological pressure of random bundle transformations in sub-additive case, J. Math. Anal. Appl. 342(1): (2008), 715–725.10.1016/j.jmaa.2007.11.044CrossRefGoogle Scholar