Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-10T20:15:54.901Z Has data issue: false hasContentIssue false
  • English
  • Français

A variational principle of amenable random metric mean dimensions

Part of: Measure-theoretic ergodic theory Ergodic theory

Published online by Cambridge University Press:  16 May 2024

Dingxuan Tang  and
Zhiming Li
Dingxuan Tang
Affiliation:
School of Mathematics, Northwest University, Xi’an, P.R. China
Zhiming Li*
Affiliation:
School of Mathematics, Northwest University, Xi’an, P.R. China
*
Corresponding author: Zhiming Li, email: china-lizhiming@163.com
Get access Rights & Permissions [Opens in a new window]

Abstract

In the context of random amenable group actions, we introduce the notions of random upper metric mean dimension with potentials and the random upper measure-theoretical metric mean dimension. Besides, we establish a variational principle for the random upper metric mean dimensions. At the end, we study the equilibrium state for random upper metric mean dimensions.

Keywords

random dynamical systemsamenable group actionsmetric mean dimensionmeasure-theoretic metric mean dimensionvariational principleequilibrium statespotentials

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification 28D20: Entropy and other invariants
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 24
Check for updates
Copyright
© The Author(s), 2024. 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.)

References

Danilenko, A. I., Entropy theory from the orbital point of view, Monatsh. Math., 134 (no. 2) (2001), 121–141.CrossRefGoogle Scholar
Arnold, L., Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, (1998).CrossRefGoogle Scholar
Biś, A., Carvalho, M., Mendes, M. and Varandas, P., A convex analysis approach to entropy functions, variational principles and equilibrium states, Comm. Math. Phys., 394 (2022), 215–256.CrossRefGoogle Scholar
Bogenschutz, T., Entropy, pressure and a variational principle for random dynamical systems, Random Comput. Dynam., 1 (1992), 99–116.Google Scholar
Bogenschutz, T., Equilibrium states for random dynamical systems, PhD. Thesis, Bremen University, (1993).Google Scholar
Bogenschutz, T. and Gundlach, V., Ruelle’s transfer operator for random subshifts of finite type, Ergodic Theory Dynam. Systems, 15 (no. 3) (1995), 413–447.CrossRefGoogle Scholar
Carvalho, M., Pessil, G. and Varandas, P., A convex analysis approach to the metric mean dimension: Limits of scaled pressures and variational principles, Adv. Math., 436 (Paper No. 109407) (2024), .CrossRefGoogle Scholar
Cioletti, L., Silva, E. and Stadlbauer, M., Thermodynamic formalism for topological Markov chains on standard Borel spaces, Discrete Cont. Dyn. Syst., 39 (11) (2019), 6277–6298.CrossRefGoogle Scholar
Crauel, H., Random dynamical systems: positivity of Lyapunov exponents, and Markov systems, Doctoral thesis, (1987).Google Scholar
Chen, E., Dou, D. and Zheng, D., Variational principles for amenable metric mean dimensions, J. Differ. Equat. 319 (3) 2017.Google Scholar
Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions, Lecture Notes Math., 580, Springer-Verlag, New York-Berlin, (1977).CrossRefGoogle Scholar
Coornaert, M. and Krieger, F., Mean topological dimension for actions of discrete amenable groups, Discrete Contin. Dyn. Syst. 13 (3) (2005), 779–793.CrossRefGoogle Scholar
Dou, D., Minimal subshifts of arbitrary mean topological dimension, Discrete Contin. Dyn. Syst. 37 (3) (2017), 1411–1424.CrossRefGoogle Scholar
Downarowicz, T., Entropy in dynamical systems, New mathematical monographs, 18. Cambridge University Press, Cambridge, (2011).CrossRefGoogle Scholar
Dunford, N., Schwartz, T., Linear operators. Part I. General theory. John Wiley and Sons, Inc., New York, (1988).Google Scholar
Dooley, A. H. and Zhang, G., Local entropy theory of a random dynamical system, Memoirs of the American Mathematical Society, (2015).Google Scholar
Downarowicz, T., Huczek, D. and Zhang, G., Tilings of amenable groups, Journal für die reine und angewandte Mathematik 747 (2019), 277–298.CrossRefGoogle Scholar
Dudley, R., Real analysis and probability (Cambridge Studies in Advanced Mathematics 74), Cambridge University Press, (2004).Google Scholar
Gutman, Y. and Śpiewak, A., Around the variational principle for metric mean dimension, Studia Mathematica 261 (3), (2021).CrossRefGoogle Scholar
Gromov, M., Topological invariants of dynamical systems and spaces of holomorphic maps, I. Math. Phys. Anal. Geom., 2 (4) (1999), 323–415.CrossRefGoogle Scholar
Huang, W. and Lu, K., Entropy, chaos, and weak horseshoe for infinite-dimensional random dynamical systems, Comm. Pure Appl. Math. 70 (no. 10) (2017), 1987–2036.CrossRefGoogle Scholar
Kerr, D. and Li, H., Ergodic theory independence and dichotomies, Springer, (2016).CrossRefGoogle 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. 2, 171, Amer. Math. Soc., Providence, RI (1996), 107–140.Google Scholar
Kifer, Y., Ergodic theory of random transformations, Progress in Probability and Statistics, 10, Birkhäuser Boston Inc., Boston, MA, (1986).CrossRefGoogle Scholar
Kifer, Y., On the topological pressure for random bundle transformations, Translations of the American Mathematical Society-Series, 2 (2001), 197–214.CrossRefGoogle Scholar
Kifer, Y. and Liu, P., Handbook of dynamical systems: chapter 5 – random dynamics, Editor(s): B. Hasselblatt, A. Katok, Elsevier Science, 1 Part B (2006), 379–499. ISBN 9780444520555.Google Scholar
Krieger, F., Minimal systems of arbitrary large mean topological dimension, Isr. J. Math. 172 (2009) 425–444.CrossRefGoogle Scholar
Liu, P., Dynamics of random transformations: smooth ergodic theory, Ergodic Theory Dynam. Systems 21 no. 5 (2001), 1279–1319.CrossRefGoogle Scholar
Liu, P., A note on the entropy of factors of random dynamical systems, Ergodic Theory Dynam. Systems 25 no. 2 (2005), 593–603.CrossRefGoogle Scholar
Li, H. and Liang, B., Mean dimension, mean rank, and von Neumann-Lück rank, J. Reine Angew. Math. 739 (2018) 207–240.CrossRefGoogle Scholar
Lian, Z. and Lu, K., Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc. 206 (no. 967) (2010), .Google Scholar
Lian, Z., Liu, P. and Lu, K., Existence of SRB measures for a class of partially hyperbolic attractors in Banach spaces, Discrete Contin. Dyn. Syst. 37 (no. 7) (2017), 3905–3920.Google Scholar
Li, Z., Equilibrium states for random lattice models of hyperbolic type, J. Diff. Equat. 265 (no. 10) (2018), 4798–4819.CrossRefGoogle Scholar
Li, Z. and Tang, D., Entropies of random transformations on a non-compact space, Results Math. 74 (no. 3) (2019), Paper No. 120, .CrossRefGoogle Scholar
Li, Z. and Zhu, Y., Entropies of commuting transformations on Hilbert spaces, Discrete Contin. Dyn. Syst. 40 (no. 10) (2020), 5795–5814.CrossRefGoogle Scholar
Lindenstrauss, E., Pointwise theorems for amenable groups, Invent. Math. 146 (no. 2), 259–295 (2001).CrossRefGoogle Scholar
Lindenstrauss, E. and Tsukamoto, M., From rate distortion theory to metric mean dimension: variational principle, IEEE Trans. Inf. Theoryy, 64 (5) (2018), 3590–3609.CrossRefGoogle Scholar
Lindenstrauss, E. and Tsukamoto, M., Double variational principle for mean dimension, Geom. Funct. Anal. 29 (2019), 1048–1109.CrossRefGoogle Scholar
Lindenstrauss, E. and Weiss, B., Mean topological dimension. Israel J. Math. 115 (2000), 1–24.CrossRefGoogle Scholar
Ledrappier, F. and Young, L.-S., Dimension formula for random transformations, Comm. Math. Phys. 117 (1988), 529–548.CrossRefGoogle Scholar
Ma, X., Yang, J. and Chen, E., Relative tail entropy for random bundle transformations, Stoch. Dyn. 18 no. 1 (2018), 1850005, .CrossRefGoogle Scholar
Ornstein, D. S. and Weiss, B., Ergodic theory of amenable group actions, I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 no. 1 161–164 (1980).CrossRefGoogle Scholar
Ornstein, D. S. and Weiss, B., Entropy and isomorphism theorems for actions of amenable groups, Journal d Analyse Mathématique 48 (1) (1987), 1–141.CrossRefGoogle Scholar
Rodrigues, F. B. and Acevedo, J. M., Mean dimension and metric mean dimension for non-autonomous dynamical systems, J. Dyn. Control Syst. 28 no. 4 (2022), 697–723.CrossRefGoogle Scholar
Rudolph, D. J. and Weiss, B., Entropy and mixing for amenable group actions, Ann. of Math. (2) 151no. 3, 1119–1150 (2000).CrossRefGoogle Scholar
Shi, R., On variational principles for metric mean dimension, IEEE Trans. Inf. Theory 68 (7), (2022), 4282–4288.CrossRefGoogle Scholar
Tang, D., Wu, H. and Li, Z., Weighted upper metric mean dimension for amenable group actions, Dyn. Syst., 35 (3) (2019), 1–19.Google Scholar
Weiss, B., Monotileable amenable groups, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, (2001), 257–262.Google Scholar
Weiss, B., Actions of amenable groups, Topics in dynamics and ergodic theory. (2003) 226–262. London Math. Soc. Lecture Note Ser., 310, Cambridge Univ. Press, Cambridge, (2003).CrossRefGoogle Scholar
Walters, P., An introduction to ergodic theory, Springer-Verlag, New York, (1982).CrossRefGoogle Scholar
Wang, X., Wu, W. and Zhu, Y., Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems, Discrete Contin. Dyn. Syst. 40 (no. 1) (2020), 81–105.CrossRefGoogle Scholar
Wang, X., Wu, W. and Zhu, Y., Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems, Stoch. Dyn. 21 (no. 5) (2021), Paper No. 2150021, .CrossRefGoogle Scholar
Wang, X., Wu, W. and Zhu, Y., On entropy and pressure via preimage structure for random dynamical systems, J. Diff. Equat., 346 (2023), 1–63.CrossRefGoogle Scholar
Yang, R., Chen, E. and Zhou, X., On variational principle for upper metric mean dimension with potential, Preprint (2022), arXiv: 2207.01901.Google Scholar
Yang, R., Chen, E. and Zhou, X., On integral formulas of metric mean dimension for random dynamical systems, (2022), DOI: 10.48550/arXiv.2210.13126.CrossRefGoogle Scholar