On Entropy of Non–autonomous Discrete Systems

  • Jose S. CánovasEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)


In this paper we explore the notion of entropy for non–autonomous discrete systems and solve an open question stated in Zhu et al. (J Korean Math Soc 49:165–185, 2012). Some other open questions are also proposed.


Invariant Measure Autonomous System Discrete Spectrum Open Cover Haar Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This paper has been partially supported by the grants MTM2011-23221 from Ministerio de Economía y Competitividad (Spain) and 08667/PI/08 from Fundación Séneca, Agencia de Ciencia y Tecnología de la Comunidad Autónoma de la Región de Murcia (II PCTRM 2007–10).


  1. 1.
    Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Below, A., Losert, V.: On sequences of density zero in ergodic theory. Contemp. Math. 26, 49–60 (1984)CrossRefGoogle Scholar
  3. 3.
    Blum, J.R., Hanson, D.L.: On the mean ergodic theorem for subsequences. Bull. Am. Math. Soc. 66, 308–311 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgain, J.: On the maximal ergodic theorem for certain subsets of the integers. Irsael J. Math. 61, 39–72 (1988)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bourgain, J.: On the maximal ergodic theorem on L p for arithmetic sets. Irsael J. Math. 61, 73–84 (1988)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bourgain, J.: Almost sure convergence and bounded entropy. Irsael J. Math. 63, 79–97 (1988)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bowen, R.: Entropy for group endomorphism and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cánovas, J.S.: Topological sequence entropy of interval maps. Nonlinearity 17, 49–56 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cánovas, J.S.: Topological sequence entropy of circle maps. Appl. Gen. Topol. 2, 1–7 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics. Springer, Berlin (1976)zbMATHGoogle Scholar
  11. 11.
    Goodman, T.N.T.: Topological sequence entropy. Proc. Lond. Math. Soc. 29, 331–350 (1974)CrossRefzbMATHGoogle Scholar
  12. 12.
    Halmos, P.R.: Lectures on Ergodic Theory. The Mathematical Society of Japan, Tokyo (1956)zbMATHGoogle Scholar
  13. 13.
    Hu, H.: Some ergodic properties of commuting diffeomorphisms. Ergod. Theory Dyn. Syst. 13, 73–100 (1993)CrossRefzbMATHGoogle Scholar
  14. 14.
    Hulse, P.: Counterexamples to the product rule for entropy. Dyn. Syst. 24, 81–95 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kolyada, S., Snoha, L.: Topological entropy of nonautonomous dynamical systems. Random Comput. Dyn. 4, 205–233 (1996)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kushnirenko, A.G.: On metric invariants of entropy type. Russ. Math. Surv. 22, 53–61 (1967)CrossRefzbMATHGoogle Scholar
  17. 17.
    Nagata, J.: Modern Dimension Theory. Wiley, New York (1965)zbMATHGoogle Scholar
  18. 18.
    Pickel, B.S.: Some properties of A–entropy. Mat. Zametki 5, 327–334 (1969). (Russian)Google Scholar
  19. 19.
    Walters, P,: An Introduction to Ergodic Theory. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  20. 20.
    Zhu, Y., Liu, Z., Xu, X., Zhang, W.: Entropy of nonautonomous dynamical systems. J. Korean Math. Soc. 49, 165–185 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de CartagenaCartagena, MurciaSpain

Personalised recommendations