# Metric Entropy of Nonautonomous Dynamical Systems

Nonautonomous Dynamical Systems (2014)

- Volume: 1, page 26-52, electronic only
- ISSN: 2353-0626

## Access Full Article

top## Abstract

top## How to cite

topChristoph Kawan. "Metric Entropy of Nonautonomous Dynamical Systems." Nonautonomous Dynamical Systems 1 (2014): 26-52, electronic only. <http://eudml.org/doc/266895>.

@article{ChristophKawan2014,

abstract = {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.},

author = {Christoph Kawan},

journal = {Nonautonomous Dynamical Systems},

keywords = {Nonautonomous dynamical systems; topological entropy; metric entropy; variational principle; nonautonomous dynamical system},

language = {eng},

pages = {26-52, electronic only},

title = {Metric Entropy of Nonautonomous Dynamical Systems},

url = {http://eudml.org/doc/266895},

volume = {1},

year = {2014},

}

TY - JOUR

AU - Christoph Kawan

TI - Metric Entropy of Nonautonomous Dynamical Systems

JO - Nonautonomous Dynamical Systems

PY - 2014

VL - 1

SP - 26

EP - 52, electronic only

AB - 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.

LA - eng

KW - Nonautonomous dynamical systems; topological entropy; metric entropy; variational principle; nonautonomous dynamical system

UR - http://eudml.org/doc/266895

ER -

## References

top- [1] R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy. Trans. Am. Math. Soc. 114 (1965), 309–319. [WoS][Crossref] Zbl0127.13102
- [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. [Crossref] Zbl1089.37501
- [3] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems. Random Comput. Dynam. 1 (1992/93), no. 1, 99–116.
- [4] R. Bowen, Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153 (1971), 401–414. [Crossref] Zbl0212.29201
- [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. Zbl1065.37016
- [6] R.-A. Dana, L. Montrucchio, Dynamic complexity in duopoly games. J. Economic Theory 44 (1986), 44–56. Zbl0617.90104
- [7] G. Froyland, O. Stancevic, Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems. arXiv:1106.1954v4 [math.DS], 2011/12.
- [8] T. N. T. Goodman, Topological sequence entropy. Proc. London Math. Soc. (3) 29 (1974), 331–350. Zbl0293.54043
- [9] X. Huang, X. Wen, F. Zeng, Topological pressure of nonautonomous dynamical systems. Nonlinear Dyn. Syst. Theory 8 (2008), no. 1, 43–48. Zbl1300.37007
- [10] X. Huang, X. Wen, F. Zeng, Pre-image entropy of nonautonomous dynamical systems. J. Syst. Sci. Complex. 21 (2008), no. 3, 441–445. [Crossref][WoS] Zbl1175.37025
- [11] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. Zbl0878.58020
- [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. Zbl0083.10602
- [13] S. Kolyada, L. Snoha, Topological entropy of nonautonomous dynamical systems. Random Comput. Dynamics 4 (1996), no. 2–3, 205–233. Zbl0909.54012
- [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. Zbl0936.37004
- [15] K. Krzyzewski, W. Szlenk, On invariant measures for expanding differentiable mappings. Studia Math. 33 (1969), 83–92. Zbl0176.00901
- [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. [Crossref] Zbl0169.46101
- [17] P.–D. Liu, Entropy formula of Pesin type for noninvertible random dynamical systems. Math. Z. 230 (1999), no. 2, 201–239. Zbl0955.37028
- [18] P.–D. Liu, M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995. Zbl0841.58041
- [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).
- [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. [Crossref][WoS] Zbl1117.37011
- [21] P. Oprocha, P. Wilczynski, Chaos in nonautonomous dynamical systems. An. Stiint. Univ. “Ovidius” Constanta Ser. Mat. 17 (2009), no. 3, 209–221. Zbl1199.37021
- [22] P. Oprocha, P. Wilczynski, Topological entropy for local processes. J. Differential Equations 249 (2010), no. 8, 1929–1967. [WoS] Zbl1209.37015
- [23] W. Ott, M. Stendlund, L.–S. Young, Memory loss for time-dependent dynamical systems. Math. Res. Lett. 16 (2009), no. 3, 463–475. [Crossref] Zbl1177.37055
- [24] A. Y. Pogromsky, A. S. Matveev, Estimation of topological entropy via the direct Lyapunov method. Nonlinearity 24 (2011), no. 7, 1937–1959. [Crossref][WoS]
- [25] Ja. Sinai, On the concept of entropy for a dynamic system. Dokl. Akad. Nauk SSSR 124 (1959), 768–771. Zbl0086.10102
- [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. [WoS][Crossref] Zbl1199.37009
- [27] Y. Zhao, The relation of dimension, entropy and Lyapunov exponent in random case. Anal. Theory Appl. 24 (2008), no. 2, 129–138. [Crossref] Zbl1174.37004
- [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. [Crossref] Zbl1252.37009
- [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. Zbl1098.37038

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.