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ENTROPY OF NONAUTONOMOUS DYNAMICAL SYSTEMS

  • Zhu, Yujun ;
  • Liu, Zhaofeng ;
  • Xu, Xueli ;
  • Zhang, Wenda
  • Received : 2010.09.27
  • Published : 2012.01.01

Abstract

In this paper, the topological entropy and measure-theoretic entropy for nonautonomous dynamical systems are studied. Some properties of these entropies are given and the relation between them is discussed. Moreover, the bounds of them for several particular nonautonomous systems, such as affine transformations on metrizable groups (especially on the torus) and smooth maps on Riemannian manifolds, are obtained.

Keywords

nonautonomous dynamical system;random dynamical system;topological entropy;measure-theoretic entropy

References

  1. T. Bogenschutz and H. Crauel, The Abramov-Rokhlin formula, Ergodic theory and related topics, III (Gustrow, 1990), 32-35, Lecture Notes in Math., 1514, Springer, Berlin, 1992.
  2. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.
  3. Y. Kifer and P.-D. Liu, Random dynamical systems, Handbook of Dynamical Systems, vol. 1B, eds, B. Hasselblatt and A. Katok, Elsevier, (2006), 379-499.
  4. S. Kolyada, M. Misiurewicz, and L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval, Fund. Math. 160 (1999), no. 2, 161-181.
  5. S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), no. 2-3, 205-223.
  6. P. D. Liu, Dynamics of random transformations: smooth ergodic theory, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1279-1319.
  7. P. D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lect. Notes in Math. 1606, Springer, New York, 1995.
  8. M. Misiurewicz, A short proof of the variational principle for a $Z^N_+$ action on a compact space, International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), pp. 147-157. Asterisque, No. 40, Soc. Math. France, Paris, 1976.
  9. W. Ott, M. Stenlund, and Lai-sang Young, Memory loss for time-dependent dynamical systems, Math. Res. Lett. 16 (2009), no. 3, 463-475. https://doi.org/10.4310/MRL.2009.v16.n3.a7
  10. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982.
  11. J. L. Zhang and L. X. Chen, Lower bounds of the topological entropy for nonautonomou dynamical systems, Appl. Math. J. Chinese Univ. Ser. B 24 (2009), no. 1, 76-82. https://doi.org/10.1007/s11766-009-2013-7
  12. J. L. Zhang, Y. J. Zhu, and L. F. He, Preimage entropy for nonautonomous dynamical systems, Acta Math. Sinica 48 (2005), no. 4, 693-702.
  13. Y. J. Zhu, Growth in topological complexity and volume growth for random dynamical systems, Stoch. Dyn. 6 (2006), no. 4, 459-471. https://doi.org/10.1142/S0219493706001827
  14. Y. J. Zhu, Preimage entropy for random dynamical systems, Discrete Contin. Dyn. Syst. 18 (2007), no. 4, 529-551.
  15. Y. J. Zhu, On local entropy of random transformations, Stoch. Dyn. 8 (2008), no. 2, 197-207. https://doi.org/10.1142/S0219493708002275
  16. Y. J. Zhu, Z. Li, and X. H. Li, Preimage pressure for random transformations, Ergodic Theory Dynam. Systems 29 (2009), no. 5, 1669-1687. https://doi.org/10.1017/S0143385708000758
  17. Y. J. Zhu, J. L. Zhang, and L. F. He, Topological entropy of a sequence of monotone maps on circles, J. Korean Math. Soc. 43 (2006), no. 2, 373-382. https://doi.org/10.4134/JKMS.2006.43.2.373

Cited by

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  2. On an entropy of ℤ + k -actions vol.30, pp.3, 2014, https://doi.org/10.1007/s10114-014-2357-7
  3. Quasistatic dynamical systems 2016, https://doi.org/10.1017/etds.2016.9
  4. Directional entropy of ℤ+k-actions vol.16, pp.01, 2016, https://doi.org/10.1142/S0219493716500040
  5. Topological pressure for nonautonomous systems vol.76, 2015, https://doi.org/10.1016/j.chaos.2015.03.010
  6. Estimations of topological entropy for non-autonomous discrete systems vol.22, pp.3, 2016, https://doi.org/10.1080/10236198.2015.1107055
  7. Variational Principles for Entropies of Nonautonomous Dynamical Systems 2017, https://doi.org/10.1007/s10884-017-9586-2
  8. On the topological entropy of free semigroup actions vol.435, pp.2, 2016, https://doi.org/10.1016/j.jmaa.2015.11.038
  9. Metric Entropy of Nonautonomous Dynamical Systems vol.1, pp.1, 2014, https://doi.org/10.2478/msds-2013-0003
  10. Topological and Measure-Theoretical Entropies of Nonautonomous Dynamical Systems 2016, https://doi.org/10.1007/s10884-016-9554-2
  11. Relationships among some chaotic properties of non-autonomous discrete dynamical systems vol.24, pp.7, 2018, https://doi.org/10.1080/10236198.2018.1458101

Acknowledgement

Supported by : National Natural Science Foundation of China