Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

Mathematical Citation Quotient (MCQ) 2017: 0.71

Open Access
See all formats and pricing
More options …

Metric Entropy of Nonautonomous Dynamical Systems

Christoph Kawan
Published Online: 2013-07-10 | DOI: https://doi.org/10.2478/msds-2013-0003


We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (Xn; μn) of probability spaces and a sequence of measurable maps fn : Xn → Xn+1 with fnμn = μn+1. This notion generalizes the classical concept of metric entropy established by Kolmogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz, and Snoha. Moreover, it shares several properties with the classical notion of metric entropy. In particular, invariance with respect to appropriately defined isomorphisms, a power rule, and a Rokhlin-type inequality are proved

Keywords: Nonautonomous dynamical systems; topological entropy; metric entropy; variational principle


  • [1] R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy. Trans. Am. Math. Soc. 114 (1965), 309-319.Web of ScienceCrossrefGoogle Scholar

  • [2] F. Balibrea, V. Jiménez López, J. S. Cánovas, Some results on entropy and sequence entropy. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1731-1742.CrossrefGoogle Scholar

  • [3] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems. Random Comput. Dynam. 1 (1992/93), no. 1, 99-116.Google Scholar

  • [4] R. Bowen, Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153 (1971), 401-414.Google Scholar

  • [5] J. S. Cánovas, Some results on (X; f; A) nonautonomous systems. Iteration theory (ECIT ’02), 53-60, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, 2004.Google Scholar

  • [6] R.-A. Dana, L. Montrucchio, Dynamic complexity in duopoly games. J. Economic Theory 44 (1986), 44-56.Google Scholar

  • [7] G. Froyland, O. Stancevic, Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems. arXiv:1106.1954v4 [math.DS], 2011/12.Google Scholar

  • [8] T. N. T. Goodman, Topological sequence entropy. Proc. London Math. Soc. (3) 29 (1974), 331-350.CrossrefGoogle Scholar

  • [9] X. Huang, X. Wen, F. Zeng, Topological pressure of nonautonomous dynamical systems. Nonlinear Dyn. Syst. Theory 8 (2008), no. 1, 43-48.Google Scholar

  • [10] X. Huang, X. Wen, F. Zeng, Pre-image entropy of nonautonomous dynamical systems. J. Syst. Sci. Complex. 21 (2008), no. 3, 441-445.CrossrefWeb of ScienceGoogle Scholar

  • [11] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.Google Scholar

  • [12] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 861-864.Google Scholar

  • [13] S. Kolyada, L. Snoha, Topological entropy of nonautonomous dynamical systems. Random Comput. Dynamics 4 (1996), no. 2-3, 205-233.Google Scholar

  • [14] S. Kolyada, M. Misiurewicz, L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval. Fund. Math. 160 (1999), no. 2, 161-181.Google Scholar

  • [15] K. Krzyzewski, W. Szlenk, On invariant measures for expanding differentiable mappings. Studia Math. 33 (1969), 83-92.Google Scholar

  • [16] A. G. Kushnirenko, On metric invariants of entropy type. Russ. Math. Surv. 22 (1967), no. 5, 53-61; translation from Usp. Mat. Nauk 22, no. 5 (137) (1967), 57-65.CrossrefGoogle Scholar

  • [17] P.-D. Liu, Entropy formula of Pesin type for noninvertible random dynamical systems. Math. Z. 230 (1999), no. 2, 201-239.Google Scholar

  • [18] P.-D. Liu, M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995.Google Scholar

  • [19] M. Misiurewicz, Topological entropy and metric entropy. Ergodic theory (Sem., Les Plans-sur-Bex, 1980) (French), 61-66, Monograph. Enseign. Math. 29, Univ. Genéve, Geneva (1981).Google Scholar

  • [20] C. Mouron, Positive entropy on nonautonomous interval maps and the topology of the inverse limit space. Topology Appl. 154 (2007), no. 4, 894-907.Web of ScienceCrossrefGoogle Scholar

  • [21] P. Oprocha, P. Wilczynski, Chaos in nonautonomous dynamical systems. An. Stiint. Univ. “Ovidius” Constanta Ser. Mat. 17 (2009), no. 3, 209-221.Google Scholar

  • [22] P. Oprocha, P. Wilczynski, Topological entropy for local processes. J. Differential Equations 249 (2010), no. 8, 1929-1967.Web of ScienceGoogle Scholar

  • [23] W. Ott, M. Stendlund, L.-S. Young, Memory loss for time-dependent dynamical systems. Math. Res. Lett. 16 (2009), no. 3, 463-475.CrossrefGoogle Scholar

  • [24] A. Y. Pogromsky, A. S. Matveev, Estimation of topological entropy via the direct Lyapunov method. Nonlinearity 24 (2011), no. 7, 1937-1959.CrossrefWeb of ScienceGoogle Scholar

  • [25] Ja. Sinai, On the concept of entropy for a dynamic system. Dokl. Akad. Nauk SSSR 124 (1959), 768-771.Google Scholar

  • [26] J. Zhang, L. Chen, Lower bounds of the topological entropy for nonautonomous dynamical systems. Appl. Math. J. Chinese Univ. Ser. B 24 (2009), no. 1, 76-82.CrossrefWeb of ScienceGoogle Scholar

  • [27] Y. Zhao, The relation of dimension, entropy and Lyapunov exponent in random case. Anal. Theory Appl. 24 (2008), no. 2, 129-138.CrossrefGoogle Scholar

  • [28] Y. Zhu, Z. Liu, X. Xu and W. Zhang, Entropy of nonautonomous dynamical systems. J. Korean Math. Soc. 49 (2012), no. 1, 165-185.CrossrefGoogle Scholar

  • [29] Y. Zhu, J. Zhang, L. He, Topological entropy of a sequence of monotone maps on circles. Korean Math. Soc. 43 (2006), no. 2, 373-382. CrossrefGoogle Scholar

About the article

Received: 2013-05-24

Accepted: 2013-06-24

Published Online: 2013-07-10

Published in Print: 2014-01-01

Citation Information: Nonautonomous Dynamical Systems, Volume 1, Issue 1, ISSN (Online) 2353-0626, DOI: https://doi.org/10.2478/msds-2013-0003.

Export Citation

© 2013 Christoph Kawan. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Juho Leppänen
Nonautonomous Dynamical Systems, 2018, Volume 5, Number 1, Page 8
Ewa Korczak-Kubiak, Anna Loranty, and Ryszard Pawlak
Entropy, 2018, Volume 20, Number 2, Page 128
Juho Leppänen and Mikko Stenlund
Mathematical Physics, Analysis and Geometry, 2016, Volume 19, Number 2

Comments (0)

Please log in or register to comment.
Log in